Category Theory: Proceedings of the International Conference Held in Como, Italy, July 22-28, 1990

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich F. Takens, Groningen

1488

A. Carboni

M.C. Pedicchio

G. Rosolini (F~ds.)

CategoryTheory Proceedings of the International Conference held in Como, Italy, July 22-28, 1990

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Editors Aurelio Carboni Dipartimento di Matematica Universit~ di Milano Via C. Saldini 50 20133 Milano, Italy Maria Cristina Pedicchio Dipartimento di Scienze Matematiche Universith di Trieste Piazzale Europa 1 34100 Trieste, Italy Guiseppe Rosolini Dipartimento di Matematica Universith di Parma 43100 Parma, Italy

Mathematics Subject Classification ( 1991 ): 18-06

ISBN 3-540-54706-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54706-1 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper

PREFACE

The Conference "Category Theory '90" was held in Como (Italy) from July 22 to July 28, 1990. The organizing committee was composed of:

A. Carboni, University of Milan, Italy M.C. Pedicchio, University of Trieste, Italy G. Rosolini, University of Parma, Italy. The scientific committee was composed of:

J. Adamek (Prague), Y. Bgnabou (Paris), A. Heller (New York), P.T. Johnstone (Cambridge), G.M. Kelly (Sydney), F.W. Lawvere (Buffalo). 125 persons took part to the Conference and 46 of them gave a talk. The editors would like to express their gratitude to the organisms that supported the meeting:

'~Consiglio Nazionale delle Ricerche " aGruppi Nazionali 40~ di Topologia e Logica" UniversiLies of Milan, Parma and Trieste, as well as to all referees for their competent and prompt collaboration and to Springer for publishing the Proceedings. The volume is divided in two parts: the first one contains research papers and is preceded by an introductory article by F.W. Lawvere; the second part consists of a monograph by A. Joyal and R. Street on Tarmaka duality and quantum groups. It is intended to be an introduction to this rapidly growing subject and is addressed to a wider audience than category theorists; the paper is not only expository, but also contains new results and new proofs of classical results.

TABLE

OF

CONTENTS

PART F.W.

Some remarks on free monoids in a topos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - G. VAN

BORCEUX

DEN

BOSSCHE,

A generic sheaf representation for rings . . . . . . . . . . .

Normalization equivalence, kernel equivalence and affine categories . . . . . . . . . . . . . . . . . . .

D. BOURN, S. C A R M O D Y A.M.

What are locally generated categories? . . . . . . . . . . . . . . . . . . . . . . . . . . .

- J. I~OSICKY,

J. BENABOU, F.

Some thoughts on the future of category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LAWVERE,

J. ADAMEK

I

- R.F.C.

CEGARRA

Computing quotients of actions of a free category . . . . . . . . .

p. 63

A long exact sequence in non-abelian cohomology . . . . . . . . . .

p. 79

J.W.

Order-enriched sketches for typed lambda calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Precategories and Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G. JANELIDZE, G. JANELIDZE C.B. JAY,

JOYAL

Fixpoint and loop constructions as colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TIERNEY,

D. PAVLOVIC~

J. POWER, G.E.

REYES

Concretely functorial programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Weak products over a locally Hausdorff locale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Categorical interpolation: descent and the Beck-Chevalley condition without direct images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

An n-categorical pasting theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - H. ZOLFAGHARI,

S.H. SCHANUEL~ M. THIEBAUD, J.J,C.

Topos-theoretic approaches to modality . . . . . . . . . . . . . . . . . . . .

Negative sets have Euler characteristic and dimension . . . . . . . . . . . . . . . . . . . . . . . . Modular categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VERMEULEN~

Some constructive results related to compactness and the (strong) Hausdorff property for locales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PART

A.

JOYAL

- R.

STREET,

p. 157 p. 174 p. 187 p. 193

p. 237

Intrinsic eo-Heyting boundaries and the Leibniz rule in certain toposes . . . . . . . .

S.B. NIEFIELD~

p. 131

Trees in distributive categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algebras for the partial map classifier monad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

J.L. MACDONALD~

p. 105

p. 213

- S. V I G N A ,

LAWVERE,

p. 95

Strong stacks and classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A note on relations relative to a factorization system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

KELLY,

A. KOCK, F.W.

Preframe presentations present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- S. V I C K E R S ,

- M.

S. K A S A N G I A N G.M.

How algebraic is the change-of-base functor? . . . . . . . . . . . . . . . . .

- W. THOLEN,

P. JOHNSTONE A.

First steps in synthetic domain theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HYLAND~

p. 30 p. 43

GARZON,

Algebraically complete categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

GRAY,

p. 20

WALTERS,

- A.R.

P.J. FREYD,

J.M.E.

p. 1 p. 14

p. 249 p. 262 p. 279 p. 282 p. 298

p. 306 p. 326 p. 359 p. 379 p. 386

p. 401

II

An introduction to Tannaka duality and quantum groups . . . . . . . . . . . . .

p. 413

SOME THOUGHTS ON THE FUTURE CATEGORY THEORY

OF

F. William Lawvere Department of Mathematics, S.U.N.Y. at Buffalo Buffalo, NY 142140

The Como meeting was something of a milestone, coming as it did just twenty-five years after the first international meeting on category theory held at La Jolla, California in 1965. The work of Kan, Grothendieck, and others had greatly intensified the elaboration and application of the subject in the ten years prior to La Jolla, and enormous development has continued uninterruptedly since. I have been asked, as a participant at both meetings, to speculate on how at least some of the threads of the subject might develop in the immediate future. The threads I have selected now were only dimly visible then, for when J. L. Verdier described topos theory on the beach at La Jolla, most of us were slow to grasp its significance. The crystallized philosophical discoveries which still propel our subject include the idea that a category of objects of thought is not specified until one has specified the category of maps which transform these objects into one another and by means of which they can be compared and distinguished. Thus, for applications of mathematics, to objectify is to mapify. Quite non-trivial in fact is also the idea that there must be definite domains and definite codomains and that there must be identity maps; even today there are many who think one could usefully "generalize" by omitting those requirements, sometimes on grounds of dislike for the "stasis" they think they imply. However, in modern Greek "stasis" means "bus-stop"; how useless an intricate network of speeding buses would be without them, and how disembodied would be processes without states. In fact category theory is the first to capture in reproducible form an incessant contradiction in mathematical practice: we must, more than in any other science, hold a given object quite precisely in order to construct, calculate, and deduce; yet we must also constantly transform it into other objects. These precepts, together with the powerful guide to look for and use adjoints in all categories large and small because they are the form of most constructions and deductions and many calculations and estimates, have guided us in our work in all

the varied fields of mathematics. Most of us have struggled to explicitly introduce these principles also into our teaching, and those who have persisted find that this explicit use of the unity and cohesiveness of m a t h e m a t i c s sparks the many particular processes whereby ignorance becomes knowledge, in learning just as it does in investigating. The need to teach, to explain and to respond to students' probing is often the genesis of problems taken up in "pure" research. Though much remains to be done, it seems to some that we (that is, the community of category theorists with our ties to all the fields of pure and applied mathematics) have reached a unique position with regard to 'philosophy. I concentrate here on an outline of what is intended as a positive mathematical program. The history of possible philosophical objections to it will be treated elsewhere. Suffice it to suggest that M6bius, Hamilton, Grassman, Maxwell, etc. would not be among the naysayers. At least we can hope that sober application, of category theory to the ancient philosophical categories, will not only clarify both but also renew respect for serious thought, through solid examples approaching adequacy to their concept. This attempt, by an admirer of rational mechanics, to include objective logic among the tools for arriving at a more accurate conception of space, will, I hope, not be dismissed by confusing it with objective idealism. The general science of the development of scientific ideas has a big overlap with category theory. That general science does not claim that scientific ideas are self-generating nor does it depend on faith for the acceptance of its own conclusions, as idealism would. In the first section I start from the opposition between connected and separable objects to propose the tentative clarification, by a certain disjoint pair of classes of categories, of the conceptions of Being and Becoming respectively; how the one class arises from the other is the content of some resulting mathematical conjectures. In the second section, a specific mathematical formulation of the principle "unity-and-identity-of-opposites" is described in hopes of clarifying d i m e n s i o n a l i t y in general and infinitesimals in particular, with again some mathematical conjectures aiming at further clarification. In the third section it is urged that certain pathologies "commonplace" since 1861/1890 need not be included in a more accurate conception of space and that both more physically-realistic models of computers as well as a more "objective" approach to Diophantine problems are already emerging from certain fascinating calculations.

I. In the remarkable paper [QDC] (Quotients of Decidable Objects, Cambridge) a certain epsilon difference between classes of toposes is mentioned. This epsilon is in a way the victory of geometry over narrow logicism and is what [QDB] (Qualitative Distinctions, Boulder) is groping to clarify. There were actually two kinds of mathematical examples which around 1960 forced the qualitative generalization of the previous notion of sheaf: for a particular algebraic space, the replacement o f mere open subspaces by objects unramified over it, and for the category of all analytic spaces, the enlargement to a much nicer c a t e g o r y within which those p a r t i c u l a r spaces which happen to be "nicer" according to some previously-achieved definition could be styled "representable". This raised the question: given a space in a topos of the second kind, what is the reasonable topos of pseudo-classical sheaves on it? Taking only a few of the simplest ideas from those circulated by the dozen or so who have strenuously worked on this problem, one arrives at the suggestion to study QD subtoposes of the topos of all spaces over a given space. Let's recall what QD means. While the term "decidable" has subjective connotations which are a powerful guide to certain investigations, and "separable" is well-established in commutative algebra, in geometry " u n r a m i f i e d " has an objective history; here I may use "SUD-object" to remind myself of the essential identity of all three. For brevity I'I1 use the term "neat" for objects which are both SUD and connected. There is a reflection from any locally connected topos to a topos in which every object is a colimit of neat objects; on the other hand, there is no coreflection (like Booleanization). In the cases where the reflection map is local, we have a start on the investigation of QD subtoposes. Note that "locally connected" and "sum" are relative to a base topos, which itself is quite special but (as in the Galois base of algebraic geometry) is sometimes much better not the topos of abstract sets; it would be fortunate if the hypothesis that the base itself be QD turned out to be sufficient As usual, if I say "set", we should imagine an object of the base topos. To clarify the above considerations, generalize to distributive | . categories and seek philosophical guidance. Even though the determination o f which maps are epimorphisms is the more profound question studied with Grothendieck topologies, it takes place within a topos of the following kind. Call a small category C "extensive" if it has finite coproducts which yield an equivalence C/A+B = C/A x C / B and C/0 = 1 (this seems a minimum requirement on an op-fibration to c o n f o r m with the notion of " f a m i l y " and with G r a s s m a n n ' s "combinatorics of continuous magnitudes"); for example, "the" homotopy category or the category of spaces of dimension at most 4.

Then the topos G(C) of all those presheaves X on C for which merely X(A+B) = X ( A ) x X(B) & X(0) = 1 contains any conceivable theory of intensive quantities, either c o h o m o l o g i c a l or function-theoretic, as an (algebra) object. Further subtoposes of G(C) and G(C)/X based on "flatness" or the need to classify structures satisfying existential axioms are obviously of great importance, but passed over here to get to the main point. Those extensive categories which also have finite limits are called distributive, as discussed in the paper on their Burnside rigs in this volume [ECD]. A general category of B e i n g , p a r t i c u l a r categories of Becoming: this is a suggested philosophical guide for sorting the two original kinds of toposes and what they have become. The unity and cohesiveness of Being provides the basis for Becoming, and the historicity and controlled variability of B e c o m i n g produces new Being from old. The unity and cohesiveness of space suggests the following condition on a category: "Every object can be included in a connected object". This axiom is not true in many toposes, for example, in the Weil (or infinitesimal) topos. If the category is not a topos, the axiom should perhaps be strengthened to say that every object is an equalizer of maps between connected objects; if the category is a topos, the axiom can be sufficiently checked on the one object 1 + I, but implies that any object is functorially included in a contractible object (i.e. C for which all C X are connected). This axiom for a category of Being will be paired with another one. Here is a dialogue which suggests how the unity-and-cohesiveness axiom may be used: Suppose you claim that the surface of the earth and the point called the sky have "nothing to do with each other", whereas I claim they must. As a first step I consider the contractible container C of E+S which (though simple) may then become the basis for a more concrete connection, such as a scheme C' for a system of airlines and airports. .[By the way, the unity-andcohesiveness axiom can sometimes be demonstrated without invocation of power sets by using properties of the four adjoint functors: components, discrete, points, codiscrete; namely, if points map surjectively to c o m p o n e n t s and if each discrete space maps injectively to the codiscrete space it negates, and finally, if nonempty codiscrete spaces are connected, then we need the injectivity of pushouts of injections.] The connected objects and the u n r a m i f i e d o b j e c t s are "orthogonal" in the sense that any two maps C ~ U are either totally equal or nowhere equal. Hence the subcategory of neat objects in a given distributive category is always a category in which every map

is an epimorphism, i.e. a [QDC] site. The orthogonal axiom, that "every object X can be covered by a SUD object U", is proposed as a characterization of a category of Becoming. Why? Considering the maps between objects in the site as control processes or deformations, a figure U > X in a sheaf X may be considered to be a state of X, and a composite U' t ~ U____x_~X to be the state x' which x "becomes" under the process t. Dialogue: I think that if x~t = x2t now, surely xl = x2 originally. You say no, there are many dissipative systems X. But no, I reply, you forgot to maintain enough information about history in your definition of present state; if you correct this neglect, you will obtain an epic hX ~ X where hX satisfies my original injectivity-ofbecoming claim. I resolve this dialogue iff my category satisfies the above "QD" axiom (a really ineradicable dissipation would require another sort of site, possibly with "relaxation" idempotents in it.) Note that a fundamental process of analysis, where a neighborhood becomes a smaller neighborhood U ' c U, inducing a section of any sheaf to become its restriction, is of this kind. [In a distributive category, an object U is SUD iff for any two maps A -""~.U the equalizer E is a coproduct summand: A = E+E'. E ' ~ A has the property that for any map T > E', if the composites T >E' > A--'-~ U are equal, then T = 0. Thus the requirement that all objects in a distributive category be SUD could be extended to merely extensive categories by demanding that every pair of maps have such a pair E, E'.] For any distributive category C and any space X in G(C), the locally distributive site of SUD objects in C/X determines a QD subtopos P(X) of G(C)/X which is an approximation to "the particular category of Becoming which X is". Of course we have plucked X from its environment, so P(X) by itself is a too-clean abstraction from which to recover X; however, the composite P(X) :- ; G(C)/X ~ G(C), which we may call 0 X, retains the ties: for any R in G(C), O x * ( R ) is the pseudoclassical sheaf of intensive quantities of type R. 0 X as a classifying map shows that P(X) is canonically given the additional structure of a sheaf of C-algebras ("without idempotents"). Note that I still have not succeeded to describe this in a site-invariant manner starting from a given pair of toposes X , S satisfying suitable axioms, with the nature of X itself determining the corresponding refined version of the fiber P(X). I hope that the above clarifies the problem and that the several efforts in this direction will combine to solve it. The normalization P(1) = S and the QD reflection suggest that a suitable axiom on X

might just be that its QD reflection map is local

(which is similar to the possible dual axiom that ---1--1 is essential). This strong localness tie persists when 1 is generalized to a discrete space. However, for X of higher dimension, the extra essentialness adjoint of the refined P ( X ) ' ' ~ X/X, while (remarkably) product-preserving, is not exact, and X / X ) ( X / X ) Q D may not be local; the image of the composite may be a significant topos. We'll return in the next section to the meaning of "higher dimension." There are m a n y distributive categories which satisfy both axioms: in that case every object X is the image U ~ X ¢,.~C of a map from a SUD object to a connected object. For example, consider the topos of quivers (i.e. irreflexive graphs). However, they don't satisfy the further requirement on a general category of Being that "the product of connected spaces is connected". For example, if A is the connected quiver with a single arrow, then A 2 = A+2D where D is the n a k e d - d o t quiver. The condition, that a category of Being should not only be cohesively unifying but also have its connected objects closed under finite product, justifies Hurewicz's definition IX,Y] = ~ o ( Y X) of the homotopy category, expressing a definite kind of qualitative aspect of spaces. Such a category of Being cannot be simultaneously also a pure category of Becoming. For, in that case, the neat objects would be subobjects of 1, the topos thus localic; but a product-preserving cocontinuous functor on a localic topos is always left exact, hence preserves any equalizer 2 : " I r ~ I' of connected objects representing 2, so an inconsistency would be reached by taking 7go as the functor. I I. The intuitive idea that any one-dimensional c o n n e c t e d group must be abelian could probably be proved in any suitable topos. We know what "connected" means, but what is "one-dimensional" for an object? It seems that a significant portion of algebraic geometry and differential geometry does not depend so m u c h on the particular algebraic theory used to construct models for it but is of a more fundamental conceptual nature. "One-dimensional", like " c o n n e c t e d " , is actually a p h i l o s o p h i c a l concept, related to the minimal Hegelian level of figures which must be considered within an arbitrary space in order to determine that space's connectedness. By a level in a category of Being, I mean a ("downward") functor from it to a smaller category which has both left and right

adjoints which are full inclusions. Such a pair of categories and triple of functors is a unity-and-identity-of-opposites (UIO) in the sense that the big category unites the two opposite subcategories which in themselves are identical with the smaller category. One can picture the big category as a (horizontal) cylinder, some objects of which lie on the identical right or left ends. The two ends are opposite not only because we picture them so, but for the intrinsic reason of adjointness; every object in the category lies on a unique horizontal thread, two objects lying on the same thread iff the downward functor assigns to them isomorphic objects in the smaller (or lower) category. All is determined by the one functor. If the big category is a topos, the right-hand end will automatically be a subtopos, but the "identical" left hand end will usually not be. To say that a particular object belongs to the level has two sharply opposed meanings: we may say that it is a sheaf for the level if it belongs to the right-hand end, but that it negates a sheaf for the level if it belongs to the left hand end. The two idempotent adjoint endofunctors of the big category obtained by c o m p o s i n g the three are called the coskeleton (right) and skeleton (left) functors for the level; the skeleton and coskeleton of any given space (object) in Being provide a kind of interval, graspable at this level, within which the possibly more complex space being studied must lie. The basic starting example of all this is that where the downward functor is the unique one to the terminal category; then the whole big category of Being constitutes just one thread, the unique sheaf being the terminal object ("pure Being") and its negative being the initial object ("non Being"); the two opposed subcategories are singletons in this case. Within a given category of Being, c o n s i d e r the partiallyordered class of all levels within it whose adjointness is enriched over a given base topos which is a category of Becoming and which itself has the structure of a level, to be thought of as the lowest nontrivial level; assume that both the initial object as well as the terminal object are sheaves for this base level; the general sheaves for this base level are commonly called "codiscrete" or "chaotic" objects within the big category of Being, and the subtopos of them may be called "pure Becoming". The negative objects for this level are commonly called "discrete" and the subcategory of them deserves to be styled "non Becoming". The objects of the base category (which is identical with the two opposite subcategories of pure Becoming and non B e c o m i n g when the inclusion functors are neglected) can just be called "sets". However, this base topos, although we have restricted it to be QD, is not necessarily the category of abstract sets; part of the philosophical content of the work of Galois is that, for the Being of algebraic geometry over a non-algebraically closed base field, a much more accurate picture is achieved if the base is taken to be a well-determined Boolean topos of more-subtly B e c o m i n g sets

which is not of the purest abstract kind where the axiom of choice would hold. The base in fact seems in examples to be determined by the given category of Being itself, either as the latter's QD reflection with the extra localness condition supplying the right adjoint pure B e c o m i n g inclusion, or else (for example simplicial sets) as the d o u b l e - n e g a t i o n s h e a v e s with the extra e s s e n t i a l n e s s c o n d i t i o n s u p p l y i n g the left adjoint inclusion (in the latter case it is in Hegelian fashion always the smallest level for which both 0,1 are sheaves). Within the class of all levels over the base (of course it is a set in fact if the category of Being is a topos), the base itself is often further d i s t i n g u i s h e d by having a still further left adjoint to its discrete inclusion, this extra functor therefore assigning to every space in Being its set of components. The downward (non faithful) functor itself we regard, of course, as assigning to any space its set of points. The relation between the trivial level and the base level above it is only the first case of a possible strong relation between two levels which (hoping not to do too great an injustice to Hegel) I will call Aufhebung relative to the given category of Being: this is the relation between a lower level and a higher level whereby the first level is not only included (on the left and equivalently on the right, or simply that the l o n g e r d o w n w a r d f u n c t o r factors across the shorter one) in the higher, but moreover that the longer left adjoint inclusion factors across the shorter right adjoint inclusion; equivalently, the higher coskeleton functor fixes both the skeleta and the coskeleta in the sense of the lower level. A very simple picture (not involving toposes) involves taking the basic downward functor to be any given map from a seven-element totally ordered set onto a threeelement one, which just amounts to a partition of the big set into three non-empty closed intervals; an arbitrary intermediate level is simply a finer partition of each of these coarse intervals into finer subintervals, but an intermediate level is an Aufhebung of the lower one iff the following more stringent condition is satisfied: among the subintervals within each coarse interval, the left-most one is a singleton. In this simple example, as in some but not all examples involving toposes, every level has a smallest aufgehobenen level over it, which could reasonably be called "the" Auflaebung of it. U n i t y - a n d - i d e n t i t y - o f - o p p o s i t e s , the A u f h e b u n g relation between two such within a given unity: this is a second proposed philosophical guide. It is not limited to distributive categories, nor is the dual case of an inclusion which has both left and right adjoint retractions without interest; that dual relation holds for example between graded modules and chain complexes, and it is the image of the canonical map between the opposites which defines the h o m o l o g y functor.

Having described a basic framework, we can now return to the question of the intrinsic meaning of "one-dimensionality" of an object within such a framework. The basic idea is simply to identify dimensions with levels and then try to determine what the general dimensions are in particular examples. More precisely, a space may be said to have (less than or equal to) the dimension grasped by a given level if it belongs to the negative (left adjoint inclusion) incarnation o f that level. Thus a zero-dimensional space is just a discrete one (there are several answers, not gone into here, to the objection which general topologists may raise to that) and dimension one is the Aufhebung of dimension zero. Because of the special feature of dimension zero of having a components functor to it (usually there is no analogue of that functor in higher dimensions), the definition of dimension one is equivalent to the quite plausible condition: the smallest dimension such that the set of components of an arbitrary space is the same as the set of of components of the skeleton at that dimension of the space, or more pictorially: if two points of any space can be connected by anything, then they can be connected by a curve. Here of course by "curve" we mean any figure in (i.e. map to) the given space whose domain is one-dimensional. Continuing, two-dimensional spaces should be those negating a subtopos which itself contains both the one-dimensional spaces and the identical-but-opposite sheaves which the one-dimensional spaces negate. I f by " f u n c t i o n " we m e a n a map with o n e - d i m e n s i o n a l codomain, then any function naturally defined along each of the surfaces i n - a n arbitrary space uniquely extends to a smooth function on the space itself. That "surfaces" might even be replaced by "curves" is the basis of recent interesting work on infinite-dimensional differentiation (as it was the basis of the very first work 250 years ago on that subject); the possibility of using curves as test figures may not be the result of the somewhat restrictive category of spaces considered, but rather of a more refined property of the basic codomains of functions, such as the line and circle. These are not only one-dimensional but even belong also to the Aufhebung of a still smaller level, since they are retracts of map-spaces of infinitesimal spaces. The infinitesimal spaces, which contain the base topos in its n o n - B e c o m i n g aspect, are a c r u c i a l step toward determinate Becoming, but fall short of having among themselves enough connected objects, i.e. they do not in themselves constitute fully a "category of cohesive unifying Being." In examples the four adjoint functors relating their topos to the base topos coalesce into two (by the theorem that a finite-dimensional local algebra has a unique sec-

10

tion of its residue field) and the infinitesimal spaces may well negate the largest essential subtopos o f the ambient one which has that property. This level may be called "dimension E "; calling the levels (i.e. the subtoposes essential over the base) " d i m e n s i o n s " does not imply that they are linearly ordered nor that the Aufhebung process touches each of them. The infinitesimal spaces provide (in many ways) a good example of a non trivial unity-and-identity-of-opposites inside the ambient topos of Being: explicitly recognizing the t w o inclusions, as spaces which could be called infinitesimal and formal spaces respectively, may help clarify the c o n f u s i n g but powerful interplay between these two classes which are opposite but in themselves identical. The calculation of the E - s k e l e t o n and E - c o s k e l e t o n , of a space which is neither, needs to be carried out, and also the calculation of the Aufhebung of dimension E . The idea behind the identification of the levels in a category of Being with dimensions is that a higher level is a more determinate general Becoming, that is, it contains spaces having in them possib l y - m o r e - v a r i e d information for d e t e r m i n i n g processes. Thus one conjectures that dimX only depends on the category P(X) of particular Becoming associated to X (and not on the important structure sheaf which recalls for the little category the big environment in which it was born). In other words, if we have an equivalence of categories P(X) ~ P(Y), then X,Y should belong to the same class of UIO levels within the category of Being in which they are objects. Suitable hypotheses to make this conjecture true should begin to clarify the relationships between the two suggested philosophical guides. Ill. Why does the epsilon difference leave room for the triumph of geometry over narrow logicism? What might have led Grothendieck to p r o p o s e his (still u n p u b l i s h e d ) p r o g r a m for " t a m e t o p o l o g y " , wherein he arrived at roughly the same real analytic spaces as logicians working on a Tarski problem of "decidability"? It seems that all attempts to characterize continuity in a purely intensive logical way, such as the frame algebra, leave another kind of room in spite of their profound contribution to calculations - room for the obviously non-physical space-filling curves and nowhere-differentiable functions. Though we have been led to believe that this subjectivelygenerated Raum shows that our basic intuition of space is unreliable, still we have not been shown anything in the real world which could more than provisionally be modeled as a discrete infinity. Rather than such speculations about the unreliability of knowledge, i t seems that still more serious work is needed, marshalling all the achievements of subjective logic and objective logic, of geometry a n d algebra to hone still more realistic models of continuous space. As several

tl

have suggested, a guide is to consider figures, i.e. maps to the space, as fundamental in determining it, with intensive quantities (i.e. maps from the space) being derived by naturality rather than the other way around; this does not metaphysically mean of course that the nature of the domains of the figures is not derived by a careful ascending interplay between all four of the mentioned subjects. The above considerations are related, as suggested in a Como discussion, with old problems such as Fermat's. Diophantus probably considered natural numbers not in the abstract way which we habitually now do, but as born from actual objects. While the method of formally adjoining negatives and ascending to powerful cohomological calculations etc. leads to many results, we should not forget the objects themselves. Just as realizing c o h o m o l o g y classes by vector bundles via K-theory p e r m i t t e d powerful interplay b e t w e e n those calculations and directed manipulations of the objects by actual maps, so a similar possibility is opened by the Burnside rig of a distributive category, wherein polynomial equations satisfied by objects are revealed as specific structure on the objects themselves. For example, the equation x = 1 +2x arises from an object with a point and operated on by a 2-generator monoid, with an additional inverse map. But even the dangerous x = l+x does not lead to unbridled infinity. While the study of linear equations on distributive categories is packed w i t h surprising subtleties, higher-degree equations are also approachable with, for example, homogeneity retaining some of its usual properties when interpreted in this more demanding functorial manner. I was surprised to note that an isomorphism x = 1 + x 2 (leading to complex numbers as Euler characteristics if they d o n ' t collapse) always induces an isomorphism xT=x. The rough similarity between Grothendieck's tame spaces and the finitely sub-analytic [FSA] objects considered in logic is in fact a major difference; the same sort of difference exists between real algebraic geometry and semi-algebraic sets, as well as between the ordinary c o n t i n u o u s PL category and the polyhedral category o f negative sets. All of these are quite different from categories in which countable coproduct decompositions are c o m m o n . The difference within each of the three pairs mentioned may all exemplify a single general process which I'll call A r i s t o t e l i a n i n t e r v e n t i o n ; some such analysis seems also basic to attempts to hone a more physically realistic model for the p r o g r a m m i n g of "digital" computers. Aristotle pointed out that the continuum is divisible but not infinitely divided. One can break a stick. Repeating that and all which it led to for 40,000 years has created a lot of indispensable chairs, buildings, etc. but has not changed the fundamental continuity of space; neither will billions of times dividing possible current-levels into "yes and no".

12

Given a map of coherent toposes, there is not only the usual induced topology but there are also topologies in which only the inverted maps between c o h e r e n t objects are taken as covers. For example, to construct the generic solution of the equation x = l+3x+2x 2, consider first the classifying topos for the free algebraic theory with one constant, three unary, and two binary operations; the free algebra on no generators determines a point of this topos. If we were to consider the full induced topology, our topos would collapse to that of discrete sets; a coherently induced topos however might not only satisfy the equation but have a non-trivial Burnside ring, whereas just because it lacks the metaphysical "minimality of the fixed point", it may provide a more physical model of potential lists and trees. We don't yet know which presentations of rigs can be Burnside-realized from distributive categories, because the very concreteness of the non-isomorphisms in the latter may give rise also to unexpected isomorphisms. We also don't know whether finitely subanalytic sets can be obtained via such a uniform procedure from some sort of continuously tame ones. Another possibility would be to take not the base topos but the infinitesimal one as the domain of the topos map which is used to induce the Aristotelian intervention - is it possible that even after such an explosion, functions could still have a welldefined derivative at every point? Certainly the resulting sites need not be Boolean; for example, consider a half-open interval x: it should satisfy x = 2x but its endpoint is not a coproduct summand. Naturally, models like the polyhedra constructed from below from real space are more satisfying than those constructed from above by classifying abstract algebras, but as usual the goal is to be approached by pushing hard from both sides.

If the general program proposed above is correct at least in rough outline, it would serve both the advancement and the dissemination of the subject to have it clearly worked out. As clearly formulated in Grassmann's introduction, only a good philosophical preamble can orient the student toward what kind of applications of a purely mathematical development he should look out for; that theory of pedagogy is at least as deserving of trial as the pragmatist theory of teaching only skills, which as we have seen did not achieve its goal. In spite of temporary setbacks of all kinds, the many-sided and passionate advance of category theory has been on the whole remarkably steady. On the basis of all that work many questions of both

13

fundamental and applied nature future of our science is bright.

are

now

becoming

clear,

thus

the

Bibliography [ Q D C ] Johnstone, Peter, "Quotients of Decidable Objects in a Topos" in Math. Proc. Camb. Phil. Soc. 93, (1983), 409-419.

[QDB] Lawvere, F. William, "Qualitative Distinctions Between Some Toposes of Generalized 92, (1989), 261-299.

[ECD]

Graphs" in C o n t e m p o r a r y

Schanuel, Stephen, "Negative and Dimension", (this volume).

Sets

have

Euler

Mathematics

Characteristic

[FSA] Van den Dries, Lou, "A Generalization of the Tarski-Seidenberg Theorem and Some Nondefinability Results" Bull. Am. Math. Soc. 15, (1986), 189-193.

This paper is in final form and will not be published elswhere.

WHAT

ARE

LOCALLY

JIAf ADAMEI(

GENERATED

AND

CATEGORIES?

Jll~f ROSlCK9

ABSTRACT. The answer to the question in the title, as given by Gabriel and Ulmer, is well enough known: precisely the locally presentable categories. We prove that the same result holds for mono-locally generated categories, a concept we introduce since we find it more "natural" than that of locally generated categories.

Recall from [GU] that a category ]C is called locally ),-presentable iff it is cocomplete, and has, for some regular cardinal ),, a set of ),-presentable objects such that every K:-object is a ),-direct colimit of objects lying in that set. And K: is called locally ,k-generated iff (a) 1~ is cocomptete (b) /C has for some regular cardinal ), a set .4 of A-generated objects (recall that an object A is ),-presentable (),-generated) iff its hom-functor horn(A, - ) preserves ),-direct colimits (),-direct colimits of monomorphisms)) such that every L:object is a ),-direct colimit of objects lying in that set (c) each .A-object has only a set of strong quotients. The condition (c) is quite unpleasant, particularly in view of the fact that each locally presentable category is co-wellpowered (as proved directly in [GU] and very elegantly in IMP]). Now, Gabriel and Ulmer prove that a category is locally generated iff it is locally presentable (with unequal ),'s, in generM). This leads to the following O p e n p r o b l e m 1. Is each category, satisfying (a) and (b) above, locally presentable? That is, can (c) be omitted? R e m a r k 1. (i) This problem can be stated in a more suggestive form: Is each full reflective subcategory of a locally presentable category/~, closed in /3 under ),-direct colimits of monomorphisms, locally presentable? In fact, to see that the affirmative answer to this question is equivalent to the affirmative answer to Problem 1, let/C be a category satisfying (a), (b) above. The closure .A of .~i under all colimits of less than ), objects is dense in ]C and consists of h-generated objects (see [GU]). Thus, the Yoneda embedding t3 : ~ --~ Se~ A°p, E K = horn(-, K) is full and faithfull. Since K] is cocolnplete, E is a right adjoint (see [ G U ] ) a n d since A E .~ is ),-generated, 13 is easily seen to preserve ),-direct colimits of monomorphisms. Hence K; is (isomorphic to) a full reflective subcategory of a locally presentable category which is closed under ),-direct cotimits of monomorphisms. Conversely, let/C be a full reflective subcategory of a locally presentable category/~ which is closed in/~ under A-direct colimits of monomorphisms. Then ]C is cocomplete

15

and, taking A such that /2 is locally A-presentable, reflections of A-presentable objects of Z; form the desired set .4 from (b). (ii) The answer is either affirmative, or it depends on set theory because, as we have proved in [RTA], under Vop6nka's principle every limit-closed full subcategory of a locally presentable category is locally presentable. Thus, an absolute negative answer to the above question would imply that Vop6nka's principle is wrong (and thus imply the non-existence of huge cardinals, see [ART]). (iii) The value of the concept of locally generated category lies in the fact that Adirect unions of subobjects are more natural and more often used (e.g. in model theory) than general A-direct colimits. The trouble with the concept of Gabriel and Ulmer is that, whereas A-direct colimits of monomorphisms appear in the condition put upon ~4-objects, they do not appear in the representation of objects of K as colimits of Aobjects. We believe that the natural condition is the following: each object is a A-direct colimit of a diagram of ~4-objects and monomorphisms. The aim of this note is to prove that by changing (b) in this manner, (c) can be omitted. Since in model theory one works with submodels (=strong subobjects of E-structures) rather than just subobjects, whereas in algebra subobjects are sometimes more natural than strong subobjects, we formulate our result relatively w.r.t, a class 3.t of monomorphisms. By a factorization system in a category K we mean the classical concept of J. Isbelh compositive classes $ C_ Epi(K;) and 3d C_ Mono(/~) with $ VI 3.t = Iso(K;) such that each morphism has an essentially unique ($, Ad)-factorization. Definition. Let 34 be a class of monomorphisms in a category K., m~d let A be a regular cardinM. (1) An object A of lC is caiied A-generaAed w.r.t 34 provided that h o m ( A , - ) preserves coIimits of A-dlrect diagrams of 3,t-monomorphisms. (2) 1~ is said to be 34-locally generated if (a) K is cocomplete, and (b) K has, tbr some regular cardinM A, a set A of A-generated objects w.r.t. 34 such that every 1C-object is a A-direct colimit of a diagram of A-objects and 34-morphisms. T h e o r e m 1. For each category IC equivaient are: (i) h~ is Iocally presentable, (ii) K~ is mono-locally generated, (iii) K~ is 34-1ocally generated for some factorization system ($, 34). R e m a r k 2. We will actually prove that whenever K has a factorization system (to, ~M0) such that K; is Ad0-1ocally generated, then h~ is 34-locally generated for each factorization system (g, 34). Proof If K is 34-locally generated, then the set ,4 in the above definition is easily seen to be dense. Therefore/~ is equivalent to a full reflective subcategory of Set "~°p which is closed under A-direct colimits of 34-morphisms (via the Yoneda embedding, ef. Remark 1). It follows that ]~ is complete and wellpowered - thus, it has (strong epi, mono)-factorizations. Therefore, we see that given (g0,340) as in Remark 2, then (ii)

16

holds. T h e implication (ii) ---+ ( i i i ) is evident. Furthermore, (i) implies t h a t (C0,.Ad0) in Remark 2 exists: if K; is locally A-presentable then the (essentiMly smM1) collection .4 of all g-quotients of all )~-presentable objects satisfies (b) above. Thus, it is sufficient to prove that (iii) --~ (i). By the above observation, since £. = Se~ "a°P is locally w-presentable, it is only necessary to prove the following: Given a locMly w-presentable category !; and its full, reflective subcategory/C closed under ,X-direct colimits of JVl-morphisms where (g, 3/[) is a factorization system of ~ , then K7 is locally presentable. (1) We first observe that the set .4 of (b) above represents all ,~-generated objects w.r.t..Ad. Given a/~-generated A, let D be a k-direct diagram of JM-morphisms with di

the colimit ( D i ---+ A ) I E I . Since it is the A-direct colimit of monomorphisms in £ a n d / 2 is locally w-presentable, any di is a monomorphism. Since A is A-generated w.r.t. ¢t4, i d a factors through some diG. Thus dig is a split epimorphism and, as a monomorphism, it is an isomorphism. (2) Let P denote the collection of all reflection arrows of ±;-objects in K; which are either finite colimits in/2 of diagrams in .4 or codomains of multiple pushouts in/2 of g-morphisms with a domain in .4. Observe that P is essentially small: the case of finite colimits is clear, for the multiple pushout use the fact that each g-morphism with a domain A-generated w.r.t. 3 / / h a s a codomain A-generated w.r.t. ¢L4, and apply (1). Thus, the class P ± of all ±l-objects orthogonal to any arrow from P is a locally presentable category (see [GU]). T h e choice of P guarantees that K; is closed under finite colimits and g-cointersections in P ± . Besides, K; is reflective and, since any domain and eodomain of any P - m o r p h i s m is A-generated w.r.t. 3d, K; is closed under A-direct colimits of Ad-morphisms in Pro. Consider a morphism f : K -+ L such t h a t K E A and L E P ± . Let e I : K -+ H I be the cointersection in /C of a l l g morphisms K -+ K ' , K ' E K7 through which f factors. Since K; is closed in p_k under g-cointersections, there is ra I : H I -+ L such that f = m I • e I. Since g is closed under g-cointersections, e I belongs to g. We will show that rn I is g-extremal, i.e., given e : H I -+ Hi, in g such that m I factors through e, then e is an isomorphism. In fact, given rn I = ra 1I • e, e E g, then f factors through e . e I E g, and the definition of e I guarantees that e. e I factors through e I - thus, e is an isomorphism. Observe the following "functoriality" of the above factorization. Given f ' : K ' --+ L, K ' E `4, then for each h : K --+ K ' with f = f ' • h ~here exists an dM-morphism h* : H I --+ H I, with ra I = rn I, • h*. In fact, let us form a pushout (in/C or P ± ) of h and el: e!

~ Hy

K

K'

fi!

, f-I

Since r r ~ y . e I = f ' . h , there is q : H -+ L such that q - h = m f and 9 " e l = ft. Since ef 6 E, e p = d . e l for some d : H -* H I , . T h e n h* = d . ~'~fulfils m y , . h* • e I = m p . d . ~ I . h = f ' . h = m y . e l , thus, r a p .h* = m I. To prove that h* is an .&/-morphism,

17

factor it as h* = m - e0, e0 E $, then f factors through the g-morphism e0 • e~,, hence, e0 is an isomorphism. (3) /C is locatly presentable because for the (essentially small) collection Q of all reflection arrows of A-presentable objects of ~; in ]C we will prove t h a t / C = ( P U Q)±. In fact, K~ C_ ( P U Q)±, and to prove the reverse inclusion, consider any L E P ± N Q±. d~

Since/2 is locally w-presentable, we have a A-direct colimit (Di ~ L ) i e l with Di Apresentable in Z;. For each i form the reflection r i : Di "* D~ in K:. Since r i E Q , di factors through ri, say, di = d~ . r. It is easy to see t h a t D~ E .4 and we can take the factorization d~ = m i . ei from (2), mi = D~' --~ L. By the functoriality, i < j implies that there is a morphism d}} : D}s ~ D "1 with rni = r n j . d;~-. Each d~} is an Ad-morphism because given an (g, 2M)-factorization d~} = m . e, then mi ( = m j . m . e) factors through e, however, rni is g-extremal, thus e is an isomorphism. Thus, we have a A-direct diagram D " of 3,4-morphisms whose colimit obviously is (ml : D}I -+ L). Since )E is closed in l: under such colimits, we get L E K. R e m a r k 3. In the proof of Theorem 1 we made use of ideas presented in [GU]. Let us call a categoryAd-acce~ible if it has 3,-direct colimits of 3d-morphisms and satisfies the condition (b) in the above definition of Ad-locally generated category. (The concept of accessible category [MP] then precisely corresponds to .Ad = all morphisms.) By T h e o r e m 1., each cocomplete 3d-accessible category is locally presentable, we now prove the following related result: Theorem

2. A category 1~ is locMly presentable iff it is complete and Ad-accessible.

Proof As in the proof of T h e o r e m 1 we have a Yoneda embedding E : L: --~ Z; = Set A°p preserving 3,-direct colimits of 2¢/-morphisms. Moreover, since t3 preserves and reflects limits, we can consider ~ as an actual full subcategory of £ closed under limits and A-direct colimits of .Ad- morphisms. We will prove that K: is a reflective subcategory of /2. T h e n )E is cocomplete, and we can apply Theorem 1. Let 34 denote the class of all monomorphisms rn : L -* K in Z; such that in each commutative square A

e

,B

(*) L

,K

with A, B E K and e E g there exists a diagonal d : B --~ L with p = d- e and q = rn. d. We have Ad = did N K . A monomorphism m : L --~ K in Z; will be called A-pure w.r.t. Ad (shortly pure) if in each commutative square (*) with A A-generated i n / 2 , B )~-generated in K: w.r.t. Ad and q E 2P/ (e is an arbitrary K;-morphism) there exists d : B --. L which belongs to Ad and p = d . e (no condition on m • d!). We will prove that K: is closed in L; under pure subobjects. Let rn : L --~ K be pure and K E K:. Since Z; is locally w-presentable, there is a A-direct diagram D of A-presentable objects i n / : with the colimit (Di p~ L)i6I. For

18

any i E I, since Di is A-generated, there is qi : D i --* K from YP[, w i t h / ) i E ,4, such that m . p i factorizes through qi: Di

ei

P'1 L

, Di I q'

T~

....~' K

Since m is pure, there is di :/9i --+ L with di • el = Pi and di E ~ . Denote b y / 9 " t h e diagram whose objects are Di,i E I and whose morphisms are f : / ) i ~ / g j such that d j • f = di: Since di E A//, any such f belongs to dvt. Since any A E ¢4 is w-presentable (thus, A-presentable) in £ = S e t A°p, D is A-direct. Since L is the colimit o f / 9 (via & : Di --* L), L E E. For each morphism f : X --+ I( in Z: with K E K We will construct a pure monomorphism f* : X* --+ I ( through which f factors as follows: we define a chain m i : L i ---* K (i < A) of monomorphisms and put f* = m~,, where m0 is a monomorphism such that f = m0 . e0 for some epimorphism e0 i n / : , rni = U m y for each limit ordinal i, d-- V [Y = y']~(f x = y)&(f x' = y') y,yt

= V(fx

= y ) & ( f x ' = y)

Y

> [x = ~'1.

Proposition 1.8. Q-sets and their morphisms constitute a category. Proof : It is routine to check that the composition defined above is associative. The identity on a Q-set E is given by ( i d x = ~') = b = ~'1

which is easily seen to be a morphism. To prove that id is a unit for composition we use (M1) and (M2). "

35

P r o p o s i t i o n 1.9. The singleton {*} provided with the equality [* = *] = 1 is a terminal object in the category of Q-sets. For any Q-set E, the unique morphism f : E ~ 1 is given by (fx=*)= V [x=x']. x' EE

Proof: ({*}, [* = *] = 1) is clearly a Q-set. Let f : E --* {*} be a morphism of Q-sets, i.e. a mapping E x {*} -~ Q; x ~-} ( f x = *) satisfying in particular property (M5) V(fa = ,) = Vto = a'] ~t

Hence, if there is such a morphism f it must be given by

(fa=.)=

V[a=a'] QI

Now, all that remains to be done, is to prove that this formula defines a morphism of Q-sets. (M2) and (M3) are trivial. It is very easy to check (M1) and (M4). • Isomorphic Q-sets come in many sizes : Q itself, seen as a Q-set as in Example 1.4, is isomorphic to 1. More generally P r o p o s i t i o n 1.10. Given any Q-set E, the set E x Q equipped with the equality [(a,p) = (b,q)] = [a = b]&sp~q

is a Q-set isomorphic to E. Proof : Define f : E --* E x Q by ( f x = (x', q)) = [x = x']~q and define g : E x Q --4 E by (g(x, q) = x') =

=

It is straightforward, but long and tedious, to verify that E x Q is a Q-set, f and g are morphisms of Q-sets and finally that f g and g f are identities. • By a "global element" or a "global section" of a Q-set E, we mean as usual a morphism from the terminal object to E. We will denote the global section functor by F. P r o p o s i t i o n 1.11. The global section functor

F : Q-Set ~

Set ; E ~ { f : l ~ E [ f morphism of Q-sets}

has a left adjoint A. Proof : Let A be any set. The "constant" Q-set AA is the set A equipped with the Q-valued equality given by [a=a']

=

1 ifa=a' = 0 ifa~a'

36 Let E be a Q-set and f : AA --, E a morphism of Q-sets. For each element a E A, the mapping {*}xE~Q; xH(fa=x) defines a morphism f. : 1--~E Hence we obtain a mapping f : A--* F(E) Conversely, given any map ~ : A --+ F(E), the mapping AA × E ---* Q ; (a, x) H (~(a)(*) = x) defines a morphism of Q-sets ~0 : AA ~ E. Now

(f°a=x) = x)

= (7.*=x)=(fa=x) =

(V°(a)

= x) = (v(a)(*)

=

This yields a bijection HomQ_Set(AA , E) --~ Homset(A , F(E)). Verification of the naturality of these bijections is left to the reader who hates himself. (A is defined on mappings between sets in the only possible way : (Afa=b)= l iffa=b = 0 if f a ~ b . ) • A more detailled study of the properties of the category of Q-sets requires too much space to be included in this paper whose main aim is a representation theorem for rings. They will be treated in a later paper. We conclude part 1 by analysing briefly what happens in a change of base quantale. D e f i n i t i o n 1.12. A morphism of quantales is a mapping

f :Q-~Q' satisfying the following properties (1) (2) (3)

f(Vierai) f(a&b) f(1)

= Vlexf(ai) = fa&fb = 1

P r o p o s i t i o n 1.13. A morphism of quantales f : Q -~, Q' induces a functor

f* : Q-Set -~ Q'-Set.

37 Proof : Let E be a Q-set with equality [a = b] taking value in Q. It is easily verified that the equality < • = • > defined by < a = b >= f[a = b] makes E into a QLset. For any morphism of Q-sets g : E -~ E t, the composite

E x E' (~:=;) Q ~

Q'

defines a morphism y between the corresponding Q'-sets. And it is again routine to check that f* so defined is indeed a functor. •

2

Sheaf

representation

of rings

In this section R will denote an arbitrary commutative unital ring (as well as its underlying set) and Id (R) its quantale of ideals. For any element r 6 R, we will denote by Ann (r) the annihilator of r, i.e. the ideal {x E R [ xr = 0}. It is easy to see that Ann(r - s)- Ann (s - t) g Ann (r - t). From this and the fact that Ann (0) = R one verifies immediately P r o p o s i t i o n 2.1. The set R equipped with the equality given by

[r= s] = Ann (~ - ~) is a M (R)-set, denoted by Tt. It is just as easy to verify P r o p o s i t i o n 2.2. For any ro 6 R , the mapping {*}xR--*Id(R);

r ~ Ann ( r - r o )

defines a global d e m e n t pro : 1 -.~. 7~.

T h e o r e m 2.3. The mapping R --, r(7¢) ; ro ~ pro is a bijection of the elements of the ring R on the globM elements of the Id (R)-set 7£. Proof : The mapping is injective since for r0 ¢ rl (Pro(*) = to) = Ann (ro - to) = R ~4 (Prl(*) = ro) = Ann (rl - ro) ~ 1. Surjectivity is more difficult to prove. Let p : 1 ~ 7£ be a global element. By the last axiom (M4) for a morphism of Id (R)-sets, we obtain

38

This implies the existence of finitely many elements ri such that (P* = r l ) " (p* = " 1 ) + - . . + (p* = r , ) " (P* = r , ) = R Since (p* = r ) - ( p * = r) ~ (p. = r), we also obtain

(p, = ~,) + . . . + (p, = ~ ) = R . Therefore, there exist ei E (p* = ri) such that Cl+...+e~

= i.

(1)

Now take ro = e t r l + . . . + e , r n .

(2)

We will prove that p = Pro. It suffices to prove that (p* = to) = R since p satisfies (M2) and (M3). Thus, for each r E R (p* = ~) > (p* = r0)" A n n (~ - ~o)

and (p* = r ) - ( p * = r0)

+reIt(p*=r)'Ann(r-r0) (p* r l ) " Ann (r, - r0) + . . . + (p* = r,,)- Ann ( r , - r0)

> =

(p, = r l ) . (p, = ~1) + . . . + (p, = ~ ) . (p, = ~n) R

=

Observe that in the Id (R)-set 7~, [r = r] -- R for all r E R. This allows one to show that in any change of base Id (R) --~ Q, the full strength of a morphism of quantales is not needed to present R as a Q-set. More precisely

39 P r o p o s i t i o n 2.4. Let Q be a quantale and q : Id (R) -~ Q a functor fi.e. a morphism o£ posers) preserving the multiplication. Then the composite R×Rk~A

Id (R) £ Q

provides R with a Q-valued equality, making it into a Q-set denoted by T~q.

Choosing for q the various localic quotients producing the various classical spectra of R, we obtain the corresponding structural sheaves. In fact, any time such a quotient is sufficiently compatible with 0, 1, V and &, the Q-set obtained by composing with q is again a representation of R on Q. A precise set of conditions to be satisfied is given in the following T h e o r e m 2.5. Consider a commutative unital ring R , a quantale Q and a surjective morphism of posers q :Id (R) ~ Q. [f the folowing conditions are satisfied 1. q preserves the multiplication ; 2. q has a left or a right adjoint i ; 3. q reflects 1 ; 4. iq(O) = O ; 5. ~(~) . ~(~) 0)(v,, e R)(~" e ~ ~ ~ e I). This is equivedent to saying that I is the intersection of all prime ideals containing I. The corresponding quotient map q: Id(R) -* Rad Id(R) ~ O ( S p ( R ) ) is given by

q(I) = ~

= n { P I P prime ideM ; I C P}.

This quotient map admits the inclusion i as a right adjoint. The corresponding structural sheaf can thus be presented as the set R provided with the equality

[,.= ~1 =

~/Ann(~

-

,).

42 It represents the ring I:L as soon as I:L has a zero nilradical, i.e. when iq(O) = 0. This result, but in terms of etale spaces and global sections, can be found in [3]. We should remark that theorem 2.5, applied to this case, is rather trivial since in any semi-prime ring every Ann(r) is already a radical ideal. (Or one could say that this is a total demystification of the prime spectrum representation.) 3.2 Sp(R) = t h e m a x i m a l s p e c t r u m of t h e r i n g R Analogous situation with "prime" replaced by "maximal". 3 . ~ Sp(l:t) = t h e p u r e ( n e a t ) s p e c t r u m of t h e ring R

The locale of open subsets is isomorphic to the locale of pure ideals of 1R. [1]. We recall that an ideal I C R is said to be pure if

(v,.e z)(9 • The quotient map q: Id(R) ---* Pure Id(R) applies an ideal on its pure part (i.e. the biggest pure ideal contained in it). This time, the inclusion i of pure ideals in all ideals is a left adjoint to q. The corresponding structural sheaf is the set R equipped with the equality [r=s]=

Pure part of A n n ( r - s ) .

It always represents the ring R.

References [1] F. BORCEUX and G. VAN DEN BOSSCHE, Algebra in a localic topos with applications to ring theory, Springer LNM 1038, 1983. [2] D. HIGGS, Injectivity in the topos of complete Heyting valued sets, Can. J. Math. Vol. 36 n ° 3 (1984), 550-568. [3] K.H. HOFMANN, Representations of algebras by continuous sections, Bull. Amer. Math. Soc. 78 (1972), 291-373. [4] A. JOYAL and M. TIERNEY, An extension of the Galois theory of Grothendieck, Memoir of the AMS 309, 1984. [5] C. MULVEY, Representations of rings and modules, Springer LNM 753, 1980, 542587.

This paper is in final form and will not be published elsewhere.

NORMALIZATION EQUIVALENCE, KERNEL EQUIVALENCE AND ~ CATEGORIES Dominique Bourn Fac. de MathEmatiques, Universit6 de Picardie 33 rue St Leu, 80039 Amiens France.

In a recent paper [4], A. Carboni gave an interesting characterization of the categories of affine spaces, i. e. slices of additive categories, by means of a "modularity" condition, relating coproducts and puUbacks, which is a categorical version of the modularity condition for lattices, in the same way as the distributive categories are the categorical version of the distributive lattices. At the end of his paper, he shows that, for any modular category IE, the forgetful functor Grd IE --~ Grph IE from internal groupoids to internal reflexive graphs in IE is an equivalence of categories. F. W. Lawvere asked whether this equivalence could be part of a characterization of modular or of affine categories. P.J. Johnstone, studying internal Mat'cev operations in [6], pointed out that the classical category of affine modules satisfies the above equivalence, but is not modular. But for a left exact additive category A, there is a much more specific equivalence, namely the normalization equivalence N : Grd/A ---) C 1/A from internal groupoids to complexes of length 1. This equivalence still holds in any slice of additive categories and therefore in any modular category. Whence a natural question : would the normalization equivalence characterize the modular categories ? But it is easy to check that the normalization equivalence is the extension to categories of algebras of the kernel equivalence K : Spl/A ---)/A/ 0 associating to each split epimorphism its kernel and consequently follows from it. This kernel equivalence synthesizes the two first results the most often produced in a book on additive categories, namely : a) the (split) short five lemma which means that K is conservative, b) the decomposition of the domain of a split epimorphism into the direct sum of its kernel and its codomain, which means that K has a left adjoint right inverse. Now the kernel equivalence still holds not only in any modular category (slice of an additive category) but also in any coslice of an additive category. So, behind our rather anecdotal initial question, whose answer is no according to this last remark, lies the problem of the understanding of what is exactly the heart of additivity. It is shown here, that for a left exact category IE with an initial objet and O-valued sums, the kernel equivalence is equivalent to the following condition (called the essentially affine condition) : for any commutative square of split epimorphisms : w

0

~

0

the downward square is a pullback if and only if the upward square is a pushout. Now a left exact category is additive if and only if it essentially affine and pointed (0 = 1). It is modular if and only if

44 it is essentially affine and its terminal object satisfies a condition of modularity. It is equivalent to a coslice of additive category if and only if it is essentially affine and its initial object satisfies a certain condition of comodularity. On the other hand, it is well known that the short five lemma still holds in the category Grp of (non abelian) abstract groups. So it seems rather relevant to study the property following which the functor K is conservative or equivalently the property following which the following condition (called the "protomodularity" condition) does hold : given two commutative squares between split epimorphisms :

if the squares 1 ~ and C ) + C ) are pullbacks, then the square (~) is a pullback. Indeed this condition spreads some light on the categorical aspect of the exactness of the category Grp : in any protomodular category 1E, the monos are characterized by their kernels, the internal categories are always internal groupoi'ds and, when moreover IE is Barr-exact, the pullbacks along a regular epimorphism are always pushouts. It is not unexpected that, when IE is left exact, the category Grp IE of internal groups in IE is protomodular. What is more surprising is that not only Grp IE but also any fiber of the fibration ( )0 : Grd IE --~ IE associating to each internal groupo'fd its object of objects is again protomodular.

0. - Modular categories A. Carboni defines [4] a modular category IE as a left exact category with finite sums and such that : t~

for each slice category IE / U and each arrow f : X ~ Z in IE / U the canonical map : iyX

: X+(YxZ)

~

(X+Y)xZ

is invertible for each object Y in IE / U .

®

for each arrow f : X ~ Y in IE the following commutative square is a pullback : X

Y

ix

iy

",= X + O

~

+U

Then he shows that all possible modular categories are the slices of additive categories with kernels.

45 Remark 1. The axiom 1 says that the category • = IE / U which admits finite sums and products satisfies a certain condition (the modularity condition) which is, in fact, autodual in C , i. e. which holds in C if and only if it holds in IL-'oP.But obviously the axiom 1 is not autodual in IE. Remark 2. Categorically speaking, the axiom 1 is not very intuitive. Let us make it explicit : when the square in the left hand diagram is a pullback, then the right hand square is a pullback :

P

Y

(,1

~Z

P+X

,,.-Z

~ U

y + ,Z--T-'-~'-~U

't4t

Y

Actually there is an equivalent way of stating axiom 1 which is more connected to the axiom 2 : Prooosition 1 : In any left exact category a x i o m 1', for each arrow / : S ~

IE with finite sums, the axiom 1 is equivalent to the T , then each d e c o m p o s i t i o n of the arrow

I + U : S + U., m ~ K n -~ T + U , is such that K is isomorphic to J + U , where J is given by the pullback (*) : iS

S

,

l/

~S+U

J ,,~K

(,)

l+U

n~

T

T+U IT

with, furthermore, m = m + U and n = n + U , up to isomorphism. PrQgf. Apply the axiom t to the left hand square, then the right hand square is a pullback and K isomorphic to J + U : U S+U

j

T

J

J+U

iT

T+U

T+U

m.i

T

Conversely, let us suppose that the left hand square is a pullback :

~ K

T+U

46

X

P

P+X

('k'~

~Z

Y

Y

Y z.|

then the whole right hand square (being equal to the left hand one) is a pullback, Thus, if the square (~) is a pullback, the square C ) is again a pullback. Furthermore there is an arrow : P + X --~ 11 determining a decomposition of the arrow h + X (actually this arrow is exactly the one involved in Carboni's axiom 1). Following the axiom 1', this arrow is invertible.

•

It is the right place here to introduce the following definition which we shall need later on : D¢fini~iQn t : An object U, in any category IE, will be called modular when tile functor "sum with U" is defined and satisfies the previous axioms 1' and 2. Remark 3 : The kernel equivalence : Let f : Y --~ X be a given epimorphism split by a given morphism s. Considering now the following left hand diagram with the lower square (*) a pullback, the axiom 1' implies that the arrow in the middle of the right hand diagram is invertible : • 0

K

~

k

X:O+X

~y

~X=O+X

O+X

.y

,,"

K+X

X

(k, sly,

X

In other words a modular category satisfies the key result of an additive category with kernels following which the domain of a split epimorpism is canonically isomorphic to the sum of its codomain with its kernel.

Let us be a little more precise : let IE be any category. Let us denote by Pt IE the category whose objects are the split epimorphisms of IE with a given splitting and whose morphims are the commutative squares between such data. If IE admits pullbacks of split epimorphisms along any map (we shall say : IE admits split putlbacks) and an initial objet O, then there is a functor K (the kernel functor) :

47 K:PttE

---) I E / 0 x t E

defined by K(f, s) = (q0 : K ---)O, X) where cp is given by the pullback (*). If, moreover, for any O-valued objet K (i.e. an object such that there exists a map K ---) O) the functor "sum with K" is defined in IE (we shall say that E admits O-valued sums), then the functor K has a left adjoint L, def'med by L(tp : K --9 O, X) = (/p + X, ix). Our previous remark means exactly that the axiom 1' implies that the natural transformation L . K --4 1 is an isomorphism. The axiom 2 implies that the natural transformation 1 ---) K • L is an isomorphism. Thus we have the following result : Proposition 2 : : In any modular category, the kernel functor is an equivalence of categories. 1] The normalization functor Let IE now be any category with split pullbacks and let Grd IE denote the category of internal groupoids in IE. If moreover IE has an initial object, we shall denote by C lIE the category whose objects (called 1-complexes) are the spans [2] of the type : 0

Y----~X

{,

and morphisms the morphisms between such spans. The normalization functor N : Grd IE ---) C 1 IE associates to each internal groupoid d0

,.._

O0

v

ol

m

[

xo

the following 1-complex where the square (*) is a pullback : K(Xt)

k

d1 ....... ~ m X 1

i 0

.............. "~

Xo

When IE = /A is an addidive category with kernel this functor is well known to be an equivalence of categories [5, 7] . The same result holds for any slice of such categories (i.e. a category of the type/A / X for any object X of/A) and consequently for any modular categories. One of our aim is to investigate the following question : Would this property characterize the modular categories ?

48 2] The k~mel func~Qr gnd the norm~dizafion fon¢~0r, We shall show first that, when IE has O-valued sums, the normalization functor N is actually an extension to categories of algebras of the kernel functor K . Let us recall indeed, that, when IE has split pullbacks, the category Grd IE is monadic above the category Pt IE [3]. The endofunctor T of this monad is given by T(f, s) = (P0, So) where P0 is the projection of the following pullback and so is the diagonal map : Y ---) Yx x Y : Pl y ~ l - ...............

f

Yx Y

~LPox

I X,91- ...............

Y

the unit and the multiplication being given by the following diagram : Sl

P2

lTs, ollso olTso y

X

~

YXxY ~

v-,.- y

s

.A ~

Pl

Yx X Y x YX

YXxY

On the other hand, when IE has an initial object O and admits O-valued sums, the category C s IE is monadic above IE / 0 x IE. Indeed, we have the following monad on IE / 0 x IE, defined by : T'(q) : K---~ O, X) = (q) : K ---) O , K + X ) ~,'(q) : K -* O, X) = (1 K, ix) It'(q) : K --~ O, X) = (1 K, Sx) where i X and sX are the usual map : ix X

Sx ,~K+X(

K+K+X.

Now, the algebras of this monad are given by the maps : (h, (x) : (q~ : K ~ O , K + X)-----~ (q~ : K ~ O, X) satisfying the usual axioms. Necessarily h = 1K and c~ is an algebra for the monad (K + -) on IE and consequently is entirely determined by a map x : K ---) X . Therefore an algebra o f T ' is just a span : 0 ( cp K x ; X , that is an object of C 1 IE. Furthermore there is a natural transformation r : T ' . K ~ K . T which determines a morphism of monad : r(f, s ) = (1K, ( k ) ) : (¢p : K---~ O , K + X) ---) (q~ : K ---) O, Y).

49 ~roposition 3. When IE has O-valued sums, the normalization functor N is the extension to these categories of algebras of the kernel functor K . Proof, Straigtforward.

•

When IE is modular, the axiom 1' implies that the natural transformation r is a natural isomorphism. On the other hand, we saw that the functor K is then an equivalence. Therefore, the normalization equivalence is, for the modular categories, a consequence of the kernel equivalence. 31 The fibration of "pointed ob_iect~" in IE, Actually, as soon as the category IE admits split pullbacks the category Pt IE is fibered over IE along the forgetfull functor p : Pt IE --4 IE defined by p(f, s) = d1(f). We shall call this fibration p the fibration of pointed objects in IE. [So, what is denoted by Pt IE in Carboni's paper, is here only the fiber above the terminal object 1 and will be denoted by Pt IE [1]]. On the other hand the category IE / 0 x tE is trivially fibered over IE by the canonical projection rc. Furthermore the kernel functor K is clearly cartesian between these two fibrations (this is the obvious translation of the fact that the cartesian squares in Pt IE (i.e. the split pullbacks) have invertible images in IE / 0). So, when IE is modular, K determines a cartesian equivalence of fibrations. But all the fibers of the fibration ~ being isomorphic to IE / 0, all the fibers of the fibration p are consequently equivalent, more exactly each change of base functor of the fibration p is an equivalence of categories (Let us call trivial such a kind of fibration). In order to understand better this point, let us study more closely this fibration of internal points. Let us first establish the following fact : Prooosition 4. Let IE be a category with split pullbacks. If IE admits pushouts of split monos (we shall say that tE has split pushouts), then the change of base functors of the fibration p have a left adjoint (i.e. p is also a cofibration). Conversely if p is a cofibration and IE admits finite products, then the category IE has split pushouts. Pr0¢L Let h be a morphism : X ~ X' in IE. We must show that the change of base functor h* along h has a left adjoint h ! . When IE has split pushouts, then h ! (f, s) is given by the following pushout :

50 k

Y f

y,

f' I S'

s

I,I

X Conversely let us suppose that, for every h, the functor h* has a left adjoint h ! and let us assume that IE has finite products. Let us denote by (if, s') the pair h !(f, s). We must show that the following square is a pushout.

k

y

X

~Y'

T

~X'

h

So, let k and s be two maps with the same codomain Y satisfying k . s = s . h. Existence of ~ fa¢torization. The following diagram of split epimorphisms : (h f, k-') y

X

h

~X~x

~

determines, by adjunction, a unique morphism (cp,~) : Y' --) X' x Y such that Px' • (tp, ~ ) = if, (tp, V) " s' = (1, s) and (cp, ~ ) . k = (h f, k ) . Whence cp = f~, ~ . s' = s , ~ . k = k .

Unicitv of the factorizadon We must show that the pair (k, s') is jointly epic. So, let

l],l 2:Y~.-.-~Y

be such that Il . k = l

2.k(=X)

and 1 1 . s ' = l 2 . s ' ( = o ) .

The following diagram of split epimorphisms :

k Y

(f', 11) ~

Y'

~

~

iT

(]'~)

It

*| x ......... h

x' ~

X~x~

is such that the two upper composites are equal. That means that the adjoint maps of (t~, lz) and

51

(ft, 12) in the fiber above X are equal (they are both equal to (f, ~.) : Y----~X x Y). Consequently (ft, ll) and (re, 12) are equal and so 11 and l 2.

II

4] The essentially affine categories. Now to assume that the previous change of base functor h* is an equivalence of categories is to assume the following essentially affine condition : In any commutative square of split epimorphisms : k y ....... ~

X

h

~

"1('

X'

the downward square is a pullback if and only if the upward square is a pushout. D~finifion 2. We shall call essentially affine a category IE with split putlbacks and pushouts such that the essentially affine condition holds or equivalently such that the fibration p of pointed objects is trivial. Remark 4. Clearly, this definition is autodual. Examole 1. When IE is modular, IE has split pullbacks and finite products. Moreover the fibration p is trivial because of the kernel equivalence. Finally, following the proposition 4, IE has split puschouts. So the modular categories are essentially affine. The previous terminology is due to the following result : Prot~osition 5. If IE is essentially affine, the fibration p is additive (i.e. each fiber and each change of base functor is additive). PrQof. For each object X of IE, the fiber o f p above IE has a terminal object (1 x, lx) which is also an initial object (i.e. the fiber is pointed). Furthermore, IE having split pullbacks, each fiber admits finite products and each change of base functor preserves them. Let us show now that the products in the fibers are sums as well. Given two pointed objects

52 (f, s) and (f, s') above X, their products in the fiber (left hand square) determines by splitting a pullback (right hand square) which is actually a morphism in Pt IE :

YXxs° P

YxxY'

Y

Yx x Y,

,,_ y v

T~

Y'

~

Yxxs'

y

s'

X

'

Y'~

X

i

f,

Consequently, IE being essentially affine, the following diagram is a pushout in IE, that is a sum in the fiber : YXx Yf ..,, YXxS'

y

SXx~l yi ~

s'

X

For each (f, s), the codiagonal map Oy : Y x x Y --~ Y detemfines clearly a commutative monoid in the fiber. We must show that this monoid is a group. That will be the case if and only if the following left hand square is a pullback : yx x y

Pl

~

y

YxxY

P1

YXxST Y

f

X

y

~ y

s f

X

But the right hand square is a splitting of it and is a pushout since the square (*) is a pullback. So, following the essentially affine condition, the left hand square is a pullback. This natural abelian group structure for each split epi (f, s) determines a canonical structure of additive category on the fiber of p above X. Clearly this consu'uction is preserved by change of base.

"

Remark 5. This is an extended and fibered version of the Carboni's result following which, when IE is modular, the category Pt IE [1] = 1 / IE is additive. Remark 6. Let us suppose that IE is essentially affine. If IE has an initial object 0, then IE / 0 is nothing but Pt IE [0], the fiber of p above 0 (which, so, is additive). The kernel functor is d e f n e d and is an equivalence of categories, and therefore the normalization equivalence does hold in IE : N : Grd IE --~ C t IE. Conversely, if IE has split pullbacks, an initial object and admits 0-valued sums, then the kernel functor is defined and has a left adjoint L. If the kernel equivalence holds, the fibration p is trivial and it is easy to check that tE has split pushouts. Then IE is essentially affine.

53 Thus, in the presence of split pullbacks and 0-valued sums, the kernel equivalence is equivalent to the fact that IE is essentially affine. Finally, thanks to a dual of the proposition 4, it is also possible to show that if IE admits kernels of split epis and sums, then the kernel equivalence implies that IE is essentially aff'me. Remark 7. If IE is essentially affine and has a terminal object 1, there is a dual version of the kernel equivalence, namely that the following functor is an equivalence : Cok : Pt IE --~ (1 / IE) x IE where the first component is the pushout along the terminal map and the second one is just dl(f). In particular, the codomain of each split mono is the product of its domain and its cokernel. Remark 8. If IE is pointed (i.e. the unique map 0 --~ 1 is invertible) and essentially affine, then IE is itself additive since IE _= IE / 1 _= IE / 0. Conversely there is a characterization of additive categories with kernels of split epimorphisms. Corollary. The three following conditions are equivalent 1) ~k is additive with kernels of split epis. 2) ~ is pointed and essentially affine. 3) a) A is pointed, has sums and kernels of split epis b) the kernel equivalence holds. Proof. That 1) implies 3) is well known. That 3a) and 3b) implies 2) is our last observation in the remark 6. That 2) implies 1) is just the remark 8.

II

Definition 3. A functor between two essentially affine categories will be said essentially affine when it preserves split pullbacks. Corollary. A functor between two additive categories is additive if and only if it is essentially affine and it preserves the null object. Example 2. Let IE be a category with split pushouts and split pullbacks. Let F : IE ~ IE' be a functor preserving the two types of squares. If furthermore F is conservative, it reflects such types of squares. Thus, if IE' is essentially affine, then IE is essentially affine. Now, for any object X of IE, the forgetful functor IE / X ~ IE fulfils these properties, in the same way as the forgetful functor X / IE ~ IE. Therefore the essentially affine categories are stable by slices and coslices. In particular, if /A is additive with kernels of split epis, not only ]A / X but a l s o X//A is essentially affine.

54 Remark 9. The normalization equivalence does not characterize the modular categories. Indeed X //A ( ~ additive with kernels of split epis) has an initial object and is essentially affine, then the normalization equivalence holds (remark 6). But X / / k is not modular since the axiom 2 of Carboni does not hold any mor% in general, in X / Z~. Example 3. When IE is essentially affine, then Pt IE and Grd IE are essentially affine.

5] Essentially affine and modular categories. If IE is modular, then IE is essentially affine but the converse (remark 9) is not true. What kind of additional axiom has to satisfy an essentially affine category IE to be modular ? A careful inspection of the proof of the Carboni's theorem will give us the answer. Prooosition 6. A category IE is modular, if and only if : 1) IE is essentially affine 2) IE is left exact and has its terminal object modular (see definition 1). Proof. When IE is modular, IE is essentially affine (example 1) and any object is modular (proposition 1). Conversely the existence of the sum with 1 determines the following adjunction, where U is monadic : U ..,,,

1lIE

~

IE

F with U (x : ] -~ X ) = X and F ( X ) = 01 : ! '-) X + 1).

The comonad'yielded on 1 / IE is described by the following diagram :

X

~

X+l

,x+,

~

(X+1)+1

1

If, moreover, IE is essentially affine, let us show that this comonad on 1 / IE is simply the eomonad generated by the product in 1 / IE with (i t : 1 -~ 1 + 1). Indeed the left hand square being a pushout in IE, the right hand square is a pullback in IE and a product in 1 / tE :

55 X . ~

¢

~x.1

x

X

~ . . . . . .

i1,

#

X ~

(p+l

.

I

,

:

X+l

¢p

~'~"1 + 1

1

1

+1

1 ~

;

jgil i

1 + 1

o

Consequently, the category of coalgebras of this comonad is the slice category (1 / IE) / i 1 of the additive category 1 / IE. Let us show now that, when IE is left exact and 1 a modular object in IE, then the functor F is comonadic. Thus IE will be equivalent to (1 / IE) / i I and so modular. 1) The following square is a pullback, hence ixis the kernel of ix+ t and ix + 1 : ix

,xl l,x. X

~X+1

X+l

~X+1+1 ix+1

2) If f and g : X ~

Y are two maps such that f + I and g + 1 have a kernel K, then the left hand

pullback (*) determines the kernel of f and g :

J

J

f

~X

~Y

(.)

Y I+I

K

~X+1

F... Y + I

k

g+1

3) The functor (- + 1) preserves these kernels : indeed the morphism i x : 1 --> X + 1 equalizes f + 1 and g + 1 and so has a factorization i : 1 --~ K . Now let us consider the following diagram : f

0

~

J

~X

.......

,

(*)

~

o

Y

L

Y

f+1 I

i

m.~ ~

K

~ k

p.~ Y +

X +I

I

g+1

Let us denote by 0~x : 0 ~ X the initial map. Then k . i = i 1 = o~x + 1 . The square (*) being a pullback, K is isomorphic to J + 1 by the axiom 1' of the modularity of 1.

56

4) The functor (- + 1) reflects the kernels : f

Z

Z

"'X

'

k

r.._ y

o iz

(')

i

y

r

f+l

Z+l

k

r.._ Y + I

''-X+l~

z+l

g+l

If z + 1 is the kernel of f + 1 and g + 1, then, the square (*) being a pullback by the axiom 2 of the modularity of 1, z is the kernel of f and g.

•

Remark 10. This is a pleasant circumstance where, at the same time, U is monadic and F comonadic. Remark 11. It is possible to characterize the coslices of additive categories with cokernels as the essentially affine categories with pushouts and an initial object 0 which is comodular (i.e. satisfying the dual conditions of a modular object). The dual of Carboni's theorem would have only characterized the dual of such coslices. 6] Protomodular categories. The kernel functor K is an equivalence if and only if (as any equivalence) 1) it is conservative 2) its left adjoint L is a right inverse. When /A is additive, the condition 1 is known as the short five lemma for split exact sequences, the condition 2 means that 0---~ A - - ~ A ~ B

--~ B ---~ 0

is a split exact sequence. Now, the short five lemma still holds for non abelian groups and consequently, the kernel functor, for the category Grp of groups, is conservative K : PtGrp ~ G r p x G r p That implies that the change of base functors of the fibration p of pointed objects are again conservative. So it seems relevant to study such a situation. Definition 4. Let IE be a category with split pullbacks. It will be called protomodular when the fibration p : Pt IE --) IE of pointed objects has its change of base functor conservative. Examples. 1) Of course, the essentially affine categories are protomodular. 2) Following our initial remark, the category Grp is protomodular. 3) Let F : IE ~ IE' be a functor which preserves split pullbacks and is conservative, then if IE' is protomodular and IE has split pullbacks then tE is protomodular. 4) Thus protomodularity is stable by slices and costices.

57 5) If IE has split pullbacks, then the category Grp IE of internal groups in IE has split pullbacks. Furthermore the Yoneda embedding Y : Grp IE ~ Grp IE°p is conservative and left exact. Therefore Grp IE is protomomular. 6) What is more surprising is the following fact. Let IE be a left exact category and ( )0 be the forgetful functor from the category Grd IE of internal groupo'fds in IE to the category IE : ()0 : Grd IE --~ IE It is a fibration. The fibers of this fibration are protomodular, generalizing the property of Grp IE which is the fiber above the terminal object. The proof is given by checking first the result in the set theoretical context and then by using the Yoneda embedding. 7) The dual of the category of sets is protomodular. Let us state now an equivalent definition. Proposition 7. A category with split pullbacks is protomodular if and only if, when the squares C ) and C ) + 1~) are pullbacks in the following diagram, then the square (~) is a pullback:

iT ..oi. Proof. This condition implies protomodularity : take this kind of diagram with the lower map of the square (~) an identity. Then the square (~) being a pullback, the upper map is an isomorphism. Conversely, take the pullback ~ ) of the fight hand split epimorphism along the lower map of (~) : t

~

LTo 0

~ v

!;o 1 0

lb.._ i

~ w

e

it produces the dotted arrow, which, by the protomodularity condition is invertible. So (~) is a pullback,

n

In fact we have a slightly more general result : Proposition 8.. When IE is protomodular, if the following squares C ) and (D + (~) are ptillbacks, then (g) is a pullback :

58

o

~

I

~

e

(no splitting condition for the right hand map). Proof. Take the pullback ( ~

:

e

i~_ v

0

ii~_o v

it produces the dotted arrow which extends the splitting from the first edge to the second edge of the middle triangle. Then, apply protomodutarity.

•

The main interest of the protomodularity condition is to allow the recovery of some aspects of the exactness of the category Grp of groups. 1) The characterization of the monos. Proposition 9. The puUbacks reflect monomorphisms. P r ~ f , Let us suppose that the following square is a pullback with f* a mono : X'

y'

k

h

~v X

~

Y

In the left hand diagram, all the squares are pullbacks : X'

X'

1

f°

~

~

X'

k

~X

"1 1' Y'

h

~Y

Xo

X'

k

"--X ~

k

1

1'°

~X

f'"

"'- X ~,-

' "'-

So the square ( D + (~) is a pullback. Now, C ) is a split pullback and following proposition 8, ~ ) is a pullback and f a mono.

I

5g Corollary 1. A morphism f : X --->Y is a mono, if and only if there is a morphism h : Y' --->Y such that the pullback of f along h is an isomorphism. Proof. If f is a mono, take h = f, Conversely apply proposition 9.

•

C o r o l l ~ 2. If IE has an initial object, f is a mono if and only if ker f = 0. 2) The internal categories are internal m'ouodfds. Proposition 10. When IE is protomodular the internal categories are internal groupdfds. Proof. Let X 1 be an internal category :

rn'aXl

~o

._

d~

~

%

m x~

._

-.,~ So

d2

xo

dl

1) In the following diagram, the squares 1 ~ + (~) and C ) are pullbacks so C ) is a pullback m X1

s1

~ m2X 1

(9 X0

so

d2

mX 1

® ['Jo ~

m X1

X0

dl

2) the following squares C ) and ( ~ + ( ~ are pullbacks, then (~) is a pullback and X 1 is a groupo'fd (see [3], th 2, corollary 1) m X1

°°1 Xo

s1

m2X1 ,

dl

~

m X1

(9 [,Jo ® l'Jo so

m XI

do

-~

X0

Remark. The normalization functor N is also conservative, the kernel functor being conservative by definition. 31 The algebra of exact seouences. Prooosition 11. When IE is left exact and protomodular, then when the following diagram is a pullback, the pair (k, s) is jointly epic :

60

k

Y'

X'

Proof. Let t, t' : Y ~

h

~Y

~

X

Z be two maps such that t . k = t ' , k, t . s = t ' . s (= a). T h e n we have

the following diagram of split epic : (f,

y'

-

l)

y

~.. X x Z

1 "-

X'

X

N o w t k = t' k implies that h*(f, t) = h*(f, t') : Y' ~ X ' x Z . But h* is conservative and preserves kernels, so h* is faithful. So (f, t) is equal to (f, t') and t equal to t'.

•

Proposition 12. W h e n IE is left exact and protomodular, if the following square is a pullback, it is also a pushout : yi

,

k

~ v

f

X'

h

1' y

s

X

Proof. Let m : Y --->Z and l : X' ---) Z such that m k = I . t~ . 1) T h e n m . s . f - - m , Indeed. m.s.f.k=m.s.h.f~=m.k.s'.f~=/'.f~.s

'. f'=l.f'=m,

k

m.s.f.s=m.s 2) So m . s is the wanted factorization : m.s.f=m m.

s. h=m.

k. s'=l.

ft. s'=l.

The uniqueness is a consequence of the fact that f i s epic. Corollary. If IE has an initial object and f is a split epi then the sequence 0 ~

ker f ~

Y --~f'> X ~

0 is exact i.e. the following square is biexact :

61 Ker f

~ y

0

~X

When, moreover, IE is supposed to be Barr exact [1], that is such that every internal equivalence relation has a universal quotient, then we can extend the previous result to any regular epimorphism, through the two following steps : Prooosition 13. When IE is left exact, Barr-exact and protomodular, then if the following squares

C) and C) + (~) of regular epimorphisms are pullbacks, then (~) is a pullback: v

Proof. Take the associated equivalence relations (kernel pairs) e

1o1'o l v

~

; ®

e

v

o

v

1 ® v

w

Then C ) being a pullback, (~)

e

0

is a pullback. For the same reason (~) + (~) is a pullback. So

(~) and (~) + (~) are split pullbacks, then (~) is a pullback and, following [1] p. 7 3 , (~) is a pullback.

•

Prooosition 14. If the following square is a pullback of regular epimorphisms, then it is also a pushout: e

v

v

e

62

Proof. Extend the diagram by the kernel pairs : e

~

o

k g'

g

', t i

,

~

•

",

h

m

~'"

.......... /

-

i

_.I~Z

If m a n d l are such that m . k = l . g', then m . P0 = m . Pl since (the pair (k, 6) being jointly epic) : 1) m . p 0 . k = m . k . p ' o = l ,

g'. p'0=l.g'.p'l=m.k.p'l

=.'n.pl.k

2)m.p0.8=m=m.pl.8 whenceamap m:suchthat m . g = m . But m , h . g ' = m . g . k = m ,

Moreover m . h = l

k=l,q'.

if and only if m . h . g' = l .

g'.

•

Corollary. If IE has an initial object and f is a regular epi, then the sequence 0 ~

kerf~

1] M. Barr

Y ~

X ~

0 is exact.

Exact categories, in : M. Barr, P.A. Grillet, D.H. van Osdol : Exact categories and categories of sheaves. Springer L. N. in Math. 236 (1971) 1-120.

2] J. Benabou

Introduction to bicategories. Rep. Midw. Cat. Sem. I. Springer L. N. in Math. 47 (1967).

3] D. Bourn

The shift functor and the comprehensive factorization for internal groupoYds. Cab. Top. G6om. Diff. 28 (3) (1987) 197-226.

4] A. Carboni Categories of affine spaces. J. Pure Appl. Algebra 61 (1989) 243-250. 5] A. Dold - D. Puppe Homologie nicht additiver Funktoren. Ann. Inst. Fourier 11 (1961) 201-212. 6] P.T. Johnstone Affine categories and naturally Mal'cev categories. J. Pure Appl. Algebra 61 (1989), 251-256. 7] D.H. Kan 8] S. Mac Lane

Functor involving C.S.S. Complexes. Trans. A.M.S. 87 (1958), 330-346. Categories for the working mathematician. Springer Berlin (1971).

This paper is in final form and will not be published elsewhere.

COMPUTING QUOTIENTS OF ACTIONS OF A FREE CATEGOlZY

S. Carmody* R. F. C. Walters* Department of Pure Mathematics, University of Sydney Sydney, NSW.

In [1] we have described a procedure for computing left Kan extensions (Chapter 10 of [6]) of functors into Set which generalizes the traditional Todd-Coxeter procedure (described in [2],[4],[5]) for coset enumeration. In this paper we describe and prove the correctness of a key sub-procedure - the Quotient Procedure - which computes quotients of actions of a free category. The quotient procedure, being highly recursive, is the most subtle part of the left Kan extension procedure. In fact published proofs (as in [3],[7]) of the Todd-Coxeter procedure appear to tack a proof of the correctness of the quotient procedure for this special case. A further subtlety arises from the fact that infinite mathematical structures must, for computational purposes, be handled in terms of finite presentations. More specifically, we consider a finite directed graph G and ~'G, the free category on G, and define the notion of a partial description of a functor ~'G ~ Set, which we call a presentation. Given a presentation we describe its comple~ion to a functor P: .TG -~ Set. We then consider a finite system of equations S between the elements of a presentation P and develop the notion of the quo~ien~P/S which is the natural one to take so that P / S corresponds.to the usual quotient of the action by the congruence generated by S. The key result justifying computing with finite presentations in this paper, given in THEOREM 1, is that -fi/S is isomorphic to P/S. The typical situation then is that P is finite, whilst P is infinite, but in the procedure to compute a quotient of P we need only work with a quotient of P and hence a finite amount of data. The ideas developed in this paper are very naturally motivated through graphical interpretations and to this end we include several diagrams by way of illustrative example. In sections 1 and 2 we treat presentations of functors and quotients of presentations. The quotient procedure itself is described in section 3 and analysed in section 4. We finish in section 5 with a sketch of the application of this procedure to the calculation of left Kan extensions.

§1 Presentations of Functors We first make precise the idea of a to simply as a presentation.

presentation of a func~or, which we shall often refer

* The authors gratefully acknowledge the support of the Australi0ax Research Council.

64 DEFINITION 1.1

A presentation sets

o f a f u n c t o r , P, on a g r a p h G consists of

PA

for a l l A E G

and partial functions P g C__P A x

Notation: I f ( x , y ) E P g w e w r i t e

x

PB

for all g: A --* B in G.

g;yinP.

Given a p r e s e n t a t i o n P on a graph G, if x g-~ y in P for some y, we say P g ( x ) is d e f i n e d and t h a t P g ( x ) = y; otherwise it is u n d e f i n e d . T h e following d i a g r a m gives an example of a g r a p h G (later illustrations will be based on the same graph) and a presentation P on G:

P

G

A"

f

~*B P

DEFINITION 1.2 A presentation P on a graph G is called c o m p l e t e g: A --* B in G, the relations Pg are in fact functions.

B

if for each

Note: In this case P can can be considered as a functor 5VG --* S e t a n d in speaking of isomorphisms of complete presentations, we will m e a n isomorphisms of the corresponding functors. REMARK In the material that follows, given a presentation P , we will frequently be considering families of equivalence relations " ' A on P A indexed b y A E G. By a convenient abuse of terminology, we shall sometimes refer to ,~ as an equivalence relation on P , and b y an abuse of notation, we shall often d r o p the subscript from " A • We now t u r n to the question of obtaining a functor (complete presentation) from any given presentation P. We take two approaches, which t u r n out to be equivalent. T h e first, P , has a n obvious connection with the coend f o r m u l a for left K a n extensions, which will be discussed further in §5, while the second, P + pc~, suggests itself very

65

naturally from the d i a g r a m m a t i c representations of presentations, and is also convenient to work with as it does not involve a quotient by an equivalence relation. DEFINITION 1.3 Given a presentation P on a graph G, we define a complete presentation P - - the c o m p l e t i o n of P - - as follows: for each A E G, define

"BE.T'G where ,,~ is the smallest equivalence relation such that: • for g: A ---* B in G, f , fl: A --* B in .T'G, x,x ~ E PA, y C P B , (i) x " , y i n P ~ ( g , x ) ~ ( 1 B , y ) whenever d o m h = B in G. (ii) ( f , x ) ~ (f',x') =~ (hi, x ) ,-, (hf',x') We denote the equivalence class of ( f , x) by If, x]. • for each g: A -* B in G, define

P g : P A ~ P B : [h,x],

, [gh, z]

which is well-defined by (ii). DEFINITION 1.4 Given a presentation P on a graph G, we define a complete presenration P + P ~ as follows: • for each A C G,

P~A

~(g . . . . gl,Z) C E JZG(B,A) × P B I gi in G, PgI(x) undefined} BE..,~G

We denote (g . . . . gl,x), with Pgl(x) not defined, by g,~.., glz. Now define

(P + P ~ ) A = P A + P ~ A . * for each h: A ---* B in G, define

(P + P ~ ) h : (P + P ~ ) A ---* (P + P~°)B by if defined otherwise

f Ph(x) I hx gn

. . . gx x

~

~ hg.

. . . gl

x.

66 T h e following proposition makes the connection between these two completions: PROPOSITION 1.1

Given a presentation P on a graph G,

p~_p+ Proof

poo

For each A E G, define CA: (P + P ° ° ) A "--* P A by x,

, [1A,X]

g. . . . g l x ,

~ [gn...gl,x].

N o w b y (1.3) (i) a n d (ii) w e can express a n y e l e m e n t V of P A in the f o r m [g . . . . g l , z] where either P g x ( z ) is not defined or g . . . . gl = 1a. In the first case, y = eA([g . . . . glx]) and in the second, y = CA(x). Thus CA is surjective. To show t h a t CA is injective, note t h a t two elements u, v E P A are equivalent iff there is a chain U :

U 1 "~U

2 "~''"

~

U n

:-

I}

where each step of the chain is of the form ui = (g . . . .

g i g , x) ~ ( g n . . . g~, y) = ui+~

where x g ~ y in P , or the reverse of this step. We m a y assume that throughout such a chain, the length of the p a t h in the first c o m p o n e n t of each t e r m is either strictly increasing or strictly decreasing, since the only possible change f r o m decreasing to increasing occurs as (g . . . .

g i g , x) ~ (g . . . .

g l , y) ~ (g . . . .

g i g , x)

where x g, y in P since Pg is a partial function. Such a pair of steps m a y be removed from the chain. T h e same applies to changes from increasing to decreasing. T h u s (1, x) ~ (1, y) implies x = y and also if ( g n . . . gl, x) ~ (h . . . . h i , y ) with P g l ( x ) undefined then n < m. T h u s if Pg~(x) and P h i ( y ) are b o t h undefined, it clearly follows t h a t m = n a n d g i = hi f o r i = 1 , . . . , n a n d x = y . Hence t h e ¢ A axe injective and thus isomorphisms. We now check naturality of ¢. For each h: A --* B in G, ~heA(z)

= ~ h [ 1 A , ~] = [h, ~],

while e B ( P + P°°)h(x) = I e s P h ( x )

eB(hx)

if P h ( x ) defined otherwise

_ f [1A, Ph(x)] = [h,x] "[ [h, x]

-

if P h ( x ) defined otherwise

and

-Phe A(g . . . . gl x) = "Ph[gn . . . gl , x] = [hgn...gl,x]

67

while

¢ B ( P + P ° ° ) h ( g . . . . g l x ) = CBhg . . . . g l x = [h9 . . . .

~1, x].

[]

We can now illustrate P + poo and hence P with an example. We think of poo as adding necessary e x t r a elements to P so as to ensure it is complete. T h e dotted lines indicate the original P , and the points outside P constitute poo. P

/....'' ."

"., @

...----..

f

••

@

.•

•

}

" ........

"/~

>

.

....I.""

B

@

h/

PC T h e first simple p r o p e r t y of the completion process is the following:

PROPOSITION 1.2

Given a presentation P on a g r a p h G, P complete =¢- P ~ P.

Proof

P complete =~ P ° ° A = 0 for all A E G :. p + poo= p []

68

§2 Q u o t i e n t s o f P r e s e n t a t i o n s DEFINITION 2.1 relations ~'~A on x, # C PA,

PA

Given a presentation P on a graph G and a family of equivalence for each A E G, we say .-~ is c l o s e d under P if for all g in G and

~ x' ~ P g ( x ) ~ P g ( x ' )

when both P g ( x ) and P g ( x ' ) are defined.

DEFINITION 2.2 A presentation with e q u a t i o n s consists of a presentation P on a graph G and for each A E G sets SA c_ (PA) 2. We refer to these sets collectively as the equations S.

Notation: If (x,x') C SA,

we write x -

x ~ in S.

DEFINITION 2.3 Given a presentation P on a graph G with equations S, we define the q u o t i e n t p r e s e n t a t i o n P/S as follows: • for all A E G, define

(P/S)A : PAl "~A where .v is the smallest equivalence relations on P such that X--X

r ==~ X ' ~ X I

and ,,~ is closed under P. • for all g: A --~ B in G, define

(P/S)g = {([x],[y]) l x ~,,yinP} where [ ] denotes equivalence class under ~. We call ..~ the q u o t i e n t e q u i v a l e n c e r e l a t i o n induced b y S on P . We illustrate the quotient process diagrammatically. T h e equations S are indicated by a solid line, other induced equivalences by a dotted line.

p

p

B

(P/s)

f p ~ s (

)B "~a--~(p/S)C

69

PROPOSITION 2.1 Given a presentation P on a g r a p h G and equations S, if P is complete then ( P / S ) is complete.

Proof

Since the quotient equivalence relation -.- is closed under P ,

x ..~ x' =~ P g ( x ) ... P g ( x ' ) Hence for each g: A --* B in G,

( P / S ) g : ( P / S ) A --* ( P / S ) B : [x],

, [Pg(x)] []

is a well-defined function.

Our first main result justifies computing with finite presentations. THEOREM 1

Given a presentation P on a g r a p h G and equations S,

( P / S ) ~ (-p/s)

°

Proof We first construct a closed equivalence relation on P. Let [ ] denote equivalence class under the quotient equivalence relation induced on P b y S. We now describe a subdivision of p~o: * for each A E G, let Q A = {g . . . . glx e P°°A I ( P / S ) g . . . . (P/S)gl[x] defined} Q°°A = {g,~... glx E P°°A I ( P / S ) g . . . . ( P / S ) g l [x] not defined}. We now partition the sets P A as follows: • for each A E G, y C P A , let [y]~ = [y] t3 {g . . . . g~x ~ Q A [ ( P / S ) g . . . . (P/S)g1[x] = [y]} which partitions P A + Q A and now we partition Q°°A: Given gn . .. glx E Q°°A, if [y] = ( P / S ) g , . . . ( P / S ) g l Ix] is defined, but is not defined

we place g,~.., glx in a set denoted g , - . . g~+l [Y]e. This clearly partitions each set P A . We let ~ denote the equivalence relation corresponding to this partition. We claim this equivalence relation is closed under P . First we consider the effect of P g on elements of [y]~ C_ P A where g: A --+ B. This breaks into two cases: (i) (P/S)g[y] defined, then if z e [y]

70 (a) if Pg(z) defined then

-Pg(z) = Pg(z) c (P/S)g[v] c_ ((p/s)g[y]) (b) if Pg(z) not defined then -Pg(z) E QA and

-pg(z) c ((P/s)9[y]) °, whereas if z E [y]e \ [y], then

clearly -pg(z) e ((P/S)g[y])%

(ii) (P/S)g[y] not defined, then clearly for any z e [y]~,

-pg(z) C g[y]% Now considering g . . . . gxx E QO~A in g....gi+l[y] e, dearly -pg(g . . . . glx) = g g . . . , gxx C gg. . . . gi+1[y]%

Thus ~, is indeed closed under -P. It is clear elements equivalent under ~-~ equivalent under the closed equivalence relation induced by S on P and hence

(-Pl

are also

(-P/s).

There is, however an obvious isomorphism ¢A: (-ffA/w,) -.., (P/S)A given by g ....

g l I Y l ¢!

' gn..-gl[Y]

which is clearly natural. Thus ( P / S ) ~- (-P/S).

E]

§3 T h e Quotient Procedure We now describe a procedure to calculate quotients of the form -P/S. Given a presentation P on a graph G and equations S, the procedure specifies a way to modify both P and S which preserves P / S and hence -P/S by THEOREM 1. We begin with an equation u - - v in S with u ~ v and then the modification is carried out in two steps, producing P1, $1 and then P2, $2. The procedure terminates when every equation has the form u m u. When the graph G and the sets PA are finite, the procedure terminates after a finite number of steps. If we commence the procedure with P and S, terminating with Q and T, then clearly Q/T ~- -Q. Further, since the procedure preserves P / S (proved in §4), it follows that P / S ~- Q/T. Therefore -P/S = Q and the desired -P/S has effectively been calculated. We leave the analysis to section 4, and now simply describe the procedure.

71 The Procedure Begin with presentation P , equations S with u ~

v E P C and u # v.

Step 1 For all A E G and g in G, define

P1A = P A Pig(x) = (Pg(x)) °

if Pg(x) defined

( )°:PA---*PIA:a,

o, { a

where

Also

if a#12 if a = v .

u

ifA=C otherwise

S1A= { S°cuTcU{(u'v)} S°A U T A

where SOA =

{(a o, b°)I (a, b) e

SA}

and

T A = {(Pg(u) °, Pg(v) °) [if both defined for g: C --* A in G}. Step 2 For all A E G and g in G, define

P2A

f PIG \ {v} [ P1A

if A = C otherwise

P2g = PlgIP2A and

S2A=ISzC\{(u'v)} ColA

ifA--C otherwise.

We can illustrate this modification process graphica~y. If we begin with the presentation P and equations S as:

~

P

B

72 T h e n the modified presentations P1 and P2 can then be illustrated as:

P2

P1

~i, and there are natural bijections between M2(X,~) and [F.,G(~)], the set of homotopy classes of simplicial morphisms from a free simplicial resolution of an object X into the simplicial object associated to the crossed module ~, [14],[6],[7]. In the general context S ,the concept of crossed module has been explicitely given by Ellis, [16], who shows the equivalence between the category of crossed modules in S, XM(S), the category of internal groupoids in S, GPD(S), and the category of simplicial objects in S with trivial Moore complex at dimensions >I, which we will denote by I-HYPGD(S} and will call the category of l-hypergroupoids in S , because their objects are just the internal l-hypergroupoids in the sense of Duskin-Glenn,[17]. Using the fact that Simpl(S), the category of simplicial objects in S, is a closed simplicial model category, [28], we study the sets [~.(X),B.] of homotopy classes of simplicial morphisms from the standard cotriple resolution of an object X into l-hypergroupoid B., and then apply our results to the classical non-abelian cohomology theories. In fact, we will establish some facts about the sets [S.(X),B.], where B. is an n-hypergroupoid in S, [17], which is just a simplicial object in B whose Moore complex is trivial at dimensions >n, for an arbitrary nzl. The main reason to do it is that for certain n-hypergroupoids B. these sets will be used to provide an appropriate definition of non abelian ~n with coefficients in crossed modules in S. For nzl, we will denote n - ~ D ( S ) the full subcategory of Simpl(S) whose objects are the n-hypergroupoids.. Let us note too that the use of n-hypergroupoids in non-abelian cohomology appears in some recents papers [6], [7],[14],[15] ;and also in abelian cohomology since Duskin, [13], showed that the usual monadic cohomology of an object X in B with coefficients in an internal X-module A, Hn(X,A), is isomorphic to [G.(X),K(A~X,n)] I ,the subset of [6.(X),K(A~X,n)] x

whose elements are those f.:G.(X) .....) ... K(A~X,n) such

homotopy clases that ~0(f.)=Ix,

of simplicial morphisms where K(A~X,n) is the

n-hypergroupoid defined as the Eilenberg-Mac Lane complex associated to the X-module A (or to the internal abelian group in the comma category S/X

80 P

given by the semidirect product A~X K(A>~X,n)=

coskn+l(

r> X),

An+l>~X ..::--.'............[ A>~X ...............[ X ~

...

X

~ X).

Moreover, it turns out that abelian cohomology is strongly related to the structure of the sets [G.(X),B.].The first section of our paper is essentially dedicated to showing that for any n-hypergroupoid B., the abelian group in the comma category of Sets over the set (X,Ho(B.))=Homs(X,~o(B.)], [G.(X),K(Hn(B.)~Ho(B.),n)]> (X,Ho(B.)), acts in an "adequate form" on [G.(X),B.]

> (X,Uo(B.))-

In the second section we associate to any n-hypergroupoid B. an (n+l)-hypergroupoid C.(B.], in such a way that it generalizes, in a natural sense, the transition from the space K(A~B,n) to K(A~B,n+I), and using the actions of the abelian cohomology groups we construct a long exact sequence of sets with distinguished elements (neutral and null) f~ q~ 8 [G. (X),B. ] >[G. (X),B" ] n >[G. (X),K(C. (q.))] )[@. (X),C. (B. ]] > . q~ . .. [G. (X),C~I(B.)] >[G. (X),C~ i (B,,) ] n÷i> [G. (X) ,K(C~ I+i (q.) ) ] ... associated to any fibration of n-hypergroupoids q.:B. >B" for which Coskn(q.):Coskn(B.) >Coskn(B ") is a trivial fibration. In the last section , we define the (n+1)-cohomology set, hal, of an object X in ~ with coefficients in a crossed module @ in S by

~+~CX,~)=[G.CX),C~-~(~(~))] where @(@) is the l-hypergroupoid nerve of the groupoid associated to @. This cohomology reduces to the usual monadic abelian cohomology when one considers the crossed module M >0 defined by a trivial X-module , and ) • > ~" associated to a short exact sequence of crossed modules ~' we obtain m long exact sequence

~(x,~')- ~ ~2(x,~) ~2(x,¢") ~>~(x,~') 6 • ..~n(x, ~,, )

n-I >~+I (X, ~ , )

-~...

. )~ n + l [X,~)

>~n+l (X, ~,, ) .......

). . .

which extends the low dimensional exact sequences in non-abelian cohomology classically established , [I0], [12], [22].

I. ON THE STRUCTURE OF THE SETS [G.(X),B.].

Given B. canonical map

an n-hypergroupoid in B and X an object of B, there ~:[G.(X),B.] ) (X,Ho(B.)) given by x([f.])=Ho(f.).

map gives a decomposition of the set the

fibers

~-I(f)=[G.(X),B.]¢,

[G.(X),B.]

fe(X,Ho(B.)).

is a This

as the disjoint union of In

this

section

we

will

establish the following results:

£

H~(X,gn(B.)),

~

gn(B.)

/X~ X - m ~

~

f,

Ox~

an

[S.(X),B.] r.

~

81 >

B.

Coskn-l(B.)

Coskn(B.) ~

)

Coskn-l(B.)

~

taat df. a~I~r(X,N

B.

t~

d

~

and a

ia

~rioia£

> Coskn(B.)

aa.~

(B.))

1.1.The

a

~

~

attcA thai

dg.

~

, ~

g

~

~

,

~

~

[f.],[g.]~[G.(X),B.]f,

ana

o~

g

~

~

a~e ]large

o~

a[f.]=[g.]

structure of the homotopy groups.

We start recalling some facts about the homotopy groups of a simplicial object in the variety of m-groups ~ where ~---~otKlluQ2 is a set of operators (none of weight greater than 2) with only one operator of weight zero (denoted by 0), just two of weight two (denoted by + and ~, respectively), and at least one of weight one (denoted by -). The (Moore) homotopy groups of a simpllcial object Sn_ I

B. = . . in ~ a r e

. B

defined,

n

sI

:iii:ii::::'-"'"""":':":::":':-[ B d

•

n-1

B

d

o

as in the case of simplicial n

iO0 K e r (df : B n - 9

Bn_l

or equivalently

1

d

o

groups,

~B

0

o

by

)

II (B.) = n

I

[B

2

, n~-O dn+ I (i~0Ker ( d i =

:Bn+I---) Bn ))

the homology groups of the Moore complex

N(B. )=... > Nn+1 (B.) ) Nn (]3.) > Nn_l (B.) n-i where Nn(B. ) = iOoKer(di:Bn--~ Bn_ I ) and 6 is induced by d nII (B.) is an object

in S, and for all x E B

II

II

> . ..

and a e N (B.), II

n>--I, we

have: ix+a-x] = [s~d~(x) + a - sndn(x)] O0

ix*a] = [s~d~(x)-a]

where square brackets consequently

and

are used to denote

U (B.) is a ~ (B.)-module n

[Y][x]

n

+

0

n

= [So(Y) X-So(Y)]

[y]*[x] = [s~(y)*x]

x E N (B.) and y E N (B.) = B n

0

and

0

equivalence

classes

in Un(B.)

in S, for all nml, with actions:

, n

[x]*[y] = iX*So(Y)]

(see [5]).

for

and

82 1.2.

The

fibrations

~..

Let us now recall that an n-truncated simplicial object consists of objects Bo, BI, . . . . Bn and the usual face and degeneracy between them. If we designate by Trnsimpl(S) the category of n-truncated of S ,the functor "truncation at level n"

simplicial

objects

trn: Simpl (S) > Trnsimpl (3) admits a right adjoint coskn, called the n-coskeleton functor. A construction of the n-coskeleton of a truncated complex can be done by using simplicial kernels, as follows: Given a n-truncated simplicial object B. tr, nal, the n+l-simplicial kernel

of B. tr is an object,

dl:An+l(B, tr) d d I

j

=

d

J-1

denoted by An+1(B. tr), together with morphisms

> Bn, O- An+1(B.t),

all

O K(IIn(B. )>~II0(B. ),n+l) in which ~n+l:An+l(B. ) is given by ~n+1(Xo,Xl Moreover, in the sequence B. considering

> ]~n (B.)>~ 1]o (B.) ,Xn+1): (~(Xo,Xl, ,Xn+1},(10

> Coskn(B. )

Xo) )

> K(TIn(B. )~%(B. ),n+l)

K(IIo(B.),0) g K(IIn(B. )~IIo(B. ),n+l)

we have

dCB.) : ~.ICKCIIo(B.),O)). In p a r t l c u l a r , i f B. is an n-hypergroupoid, d therefore B. = ~.I(K(IIo(B.),O)).

is

injective

and

Proof. -

N(~.):N(Coskn(B.))

> N(K(IIn(B.)~IIo(B.),n+I)) is surjective because

it is the zero morphism in all dimensions ¢ O,n+l B0 > IIo(B.) in dimension 0 and it is

; it is the projection the projection from

Ker(6 :Nn(B. ) > Nn_I(B. ) ) onto II (B.)=Ker6 /Im~ in dimension n+1. n n n n+1 Therefore ~. is surjective and so ~. is a flbration, (see [28] ). On the other hand, in dimensions k-n+l

,

,we have that ~.1(K(iio(B .),0)) =

). If B.

is an n-hypergroupoid N (B.)=O for all m>n so m injective for all m>-O and then d is injective (see [28]).s Proposition 2 has a certain converse,

that N (d) m

is

PROPOSITION

object

~),

IIo(B. )=B

and

3.A

Let

tr B-module.

a

fibration with ~o:Bo is an isomorphism.

B.

be

a

n-truncated

Suppose

simplicial

~. :coskn(B. tr )

in

> K(A>~B,n+I)

a

> B the canonical projection and such that Nn+1(~.)

Then ~.I(K(B,O)I

is a

n-hypergroupoid

B-modules and the fibration associated to B. proposition identifies itself with the given one. Proof : --Nn÷I(~.)-INn+I(K(B,O)) = Nn+I(~.)-I(O) = 0

B., II (B.)=A as n according to the above

and

in

dimensions

m

higher than n+l is clear that Nm(B.)=O s n 1.3.- The action of Hr(X,~n(B.)) on [G.(X),B.]f. We now establish th~ announced main resuts of this section. Let B. be a n-hy~ergroupoid in ~; in what follows we will denote Hn(B.)=A and Ho(B.)=B and recall that A is, in a canonical way, a B-module. For any object X in ~ , the cotriple resolution of X, G.(X), cofibrant object in Simpl(~), consequently the fibration ~.:Coskn(B.) > K(A~B,n+I)

is a

84 induces another one between the function complex simplicial sets G. CX)

~.(X)

~.:Coskn(B.) for which

> K(A~B,n+I)

B.G. CX)= ~:~((KCB,O)~.(x) )

Observe that K(B,0) G'(X) is a constant simplicial set whose vertices are the simplicial morphisms in ~ from G. (X) to K(B,0) and these can be identified, via ~ , with the morphisms in ~ from X to B ; so we will identify K(B,0) ~'(~) with K((X,B),0), and B. s'(x) with ~[IK((X,B),0). Now, let f. be a simplicial morphism from G.(X) to B. and f=~o(f.):X .......... > B. Then the fibre ~ l ( f ) is isomorphic to the simplieial subset of B. ~'(X) whose simplices have morphisms g.:G.(X) > B. for which simplicial

set

by

(B.G'(X))£

and

let

all their O-faces the simplicial Uo(g.)=f. We will denote this [G.(X),B.]r=No((B.G'(X))£ ) which

is

just the set of homotopy classes of simplicial morphisms g. from ~.(X) to B. with ~o(g.]=f. Of course we can apply these facts to the n-hypergroupoid K(A~B,n), so the

fibration

identify

~.:Coskn(K(A~B,n)) ~'(x)

(K(A~B,n)~'(X))f

with

pro~ection Coskn(K(A~B,n))

(Coskn(K(A~B,n))G.[X)) f

)K(A~B,n+I) s'(X)

~[l(f);

but

in

this

case

allows the

us

to

canonical

> K(B,0) is a trivial fibration and therefore > (K(B,0) G.(X) )r

is

also

a

trivial

fibration,

consequently all the homotopy groups of (Coskn(K(A~B,n))G'(X))f are trivial and

there

are

canonical

isomorphisms

~I((K(A~B,n+I))G'(X),f)~

Ho(((K(A~B,n))~'(X))£) m [G.(X),K(A~B,n)]£ ~ H~(X,A)

the n-th cohomology

group of X with coefficients in the X-module, via f, A. Using the above results, we have PROPOSITION ~n(B.)=A.

4.-

Let B. be a

Associated

morphism f.:G.(X)

n-hypergroupoid

to any morphism

in ~, nzl. Let N0(B.)=B and

in S , f:X

>B, and any simplicial

> B. with ~0(f.)=f , the fibration ~.:Coskn(B.)

) K(AmB,n+I)

induces an exact sequence of groups and pointed sets ~l(Coskn(B.)G'(X),f.)

) Hf(X,A) ~ > [G,(X),B.]f

) [G.(X),CosknB.]

and an action of Hfn[X,A) on [~.(X),B.]f such that:

a) d.([f.])--d.([g.]) if and only if a[f.]=[g.]

, for some a~H~(X,A), and

b) w(a)=w(a ' ) if and only if a'= ~.(z)+a , for some z.m The following proposition gives sufficient conditions for the actions of the groups H~(X,A) on the sets [6,(X],B.]f to be principal.

85 PROPOSITION d:Coskn(B.)

5.d:B.

Let B. be an n-hypergroupoid in 3, nal, such that the > Coskn-l(B.) is a retraction and the morphism > Coskn-l(B.) is a trivial fibration. Then the homomorphism

~l(Coskn(B.)G'(X),f.) f.:S.(X)

> B.,

> Hn(X,A) is trivial for any simplicial morphism F U (f.)=f and the action of Hn(X,A) on [~.(X),B.] is o £ £

principal. Proof: Let s.:Coskn-l(B.) > B. be such that ds.=I. Then ~.s.:Cosn-1(B.) ) K(A~B,n+I) factors through K(B,0) since B.=~[IK(B,0). On the other hand, considering the induced morphisms COskn(B.)G. (X) s~d~ ) Coskn(B.) G* (X) d* >Coskn-l(B" )G.(X) so that s.d. is homotopic to the identity map since

we have that d.s.d.=d., d.

is a trivial

fibration.

Thus

through K(B,0)~'(X)=K((X,B),0)

~.

is homotopic

and consequently

to ~.s.d.

which

factors

gl(~.)=0.m

2.A LONG EXACT SEQUENCE FOR TIIE SETS OF HOMOTOPY CLASSES.

Classically, exact sequences in non-abelian cohomology appear associated to exact sequences of crossed modules, or more generally to quotient maps of gFoupoids in the sense of Higgins, [18]. In our context B of a variety of ~-groups, internal groupoids are equivalent to l-hypergroupoids and quotient maps of groupoids to morphisms of l-hypergroupoids B. >B" such that the induced morphism Cosk1(B.) , >Coskl(B ") is a trivial fibration. In this section we are going to show the existence of a long exact sequence [O.(X),B.]

>[~.(X),B"]

n>[~. (X),K(C.(q.))]

>[G.(X),C. (B.)]---)...

associated to any morphism of n-hypergroupoids q:B. >B" such that the induced morphism Coskn(B.)-----~Coskn(B" ) is a trivial fibration. This sequence will permit us in the subsequent section to obtain a long exact sequence in non abelian cohomology with coefficients in crossed modules, which extend the known classical ones and the 9-term exact sequences shown by the authors in [7]. Abelian cohomology is included in homotopy theory via the Eilenber~-Mac Lane complexes; with this point of view the coefficients for H n and H ~÷I are related, for any X-module A, by the equalizer diagram K(A~X,n)

> Coskn(K(A~X,n))

~ K(A~X,n+I) t.

where t. is the unique morphism which factors through K(X,0), corresponding to Ix; and IIi(K(A~X,n+I)) = IIi(K(A~X,n)) for i~n,n+l, IIn(K(A~X,n+I))=0 and II (K(A~X, n+l ) )=If (K(A~X,n))=A. n+l n In the same way, given any n-hypergroupoid

B. in •, we will show that

86 there exists an (n+l)-hypergroupoid

C.(B.) in S and an equalizer diagram r.

B.

~ C.(B.) t. for i~n,n+l, ~ n (C.(B.))=O ,and Un+I(C.(B.))=~ n (B.).

with ~I(C.(B.))=~I(B.) 2.1.The hypergroupoid

Coskn(B.)

C.(B.).

Given an n-hypergroupoid B., to build the (n+l)-hypergroupoid C,(B.) we use proposition 3: Let C. (B.) be the (n+l)-truncated simplicial object tr

in S which consists of the same n-truncation is the subobject of B n÷3 given by

as B. and in dimension n+l it

n

Cn+I(B" )={(b'o'b o ..... bn+ I) I dlbo=dlbo 'i=O ..... n and dlbj=dj_Ibi,i<j } with the face morphisms d :C I

and the degeneracies

p

n+l

(B.)

>B

given by

n

d i(b O,b 0,'" .,bn+ l)=b i, O-~[[o(B.),n+2),

dimension n+2 is defined as follows: Given an z=(z 0 ..... Zn+2)eAn+2(C. (B.)tr), z1=(b'io,blo,b11 ..... bln+l)ECn÷ I(B.),

which

in

element

n+ 2

~n+2(z)=(£(_ l)i (~(bio,bll ..... bln÷1)_~(b,lo,bll ..... bin+i)), do+Iboo ) I=0

It is plain

to see that

in the Moore complex,

Nn+2( ~. ) is an isomorphism

and therefore, applying proposition 3 , C. (B.)= ~I(K(IIo(B. ),0) is

an

canonical

(n+1)-hypergroupoid. morphism

Clearly

Cn+ i (B.)

) An+IC. (B.)

IT (C. (B.))=0, and by construction 17 n

IIi(C. (B.))=If i (B.)

n+1

surjective

i~n,n+l, the so

that

(C. (B,))=rfn(B. ).

We have the equalizer diagram B.

is

for

r.

) Coskn(B. )

I C. (B.) t. where r. and t. are respectively determined by tn+I (b° ..... bn+ 1 )=(bo, bo, b I ..... bn+ 1) , and r n+l ( (bO ..... bn+1 ) = ( ~ (bO' " " " ' bn+l ) +bo' bo' b l . . . . . bn+l )"

The following lemma shows how the above construction on the hypergroupoids K(A~B,n). LEMMA 6.- Let B be an object of ~ and A a B-module. fibration P.:C.(K(A~B,n)) >K(A~B,n+I). Proof: The fibration is determined by Pn÷1({a~,b),(ao,b),(al,b) ..... (an+1,b)l:(a~,b). •

C. (-) works

Then there is a trivial

87 2.2. Neutral and null elements. Given any n-hypergroupoid B. in ~, we observe the existence of a canonical inclusion j.:B. ) C.(B.) determined by Jn÷l:Bn+l )Cn+l(B.) given

by

Jn+l(b)=(do(b),do(b),dl(b) ..... dn÷l(b) ) This

simplicial

morphism

j. allow us to study the "neutral cocycles" with coefficients in C.(B.), that is, those simplicial morphisms from ~.(X) which factor through the simplicial subobject of C.(B.) generated by the degenerate (n+l)-simpllces. PROPOSITION 7.-The simplicial morphism j. is an isomorphism between B. and the simplicial subobject of C. (B.) generated by the degenerate (n+l)-simplices. Proof: Given b~Bn, sl(b)

=

(Si_ldo (b), si_ido (b) ..... sl_idl (b), b, b, sldi+l(b) ..... Sldn(b) )

=Jn+l(Si(b))

, i=O,l ..... n, and so any degenerate

belongs

the

to

n-hypergroupoid,

image

of

Nn+I(B. )=0

in÷l" On and

so

the

other

=

(n+l)-slmplex of Cn÷I(B.) hand,

because

((1-Sndn)" " " (l-Sodo)) (y)

Y~Bn÷ I and any element of Bn÷ 1 is a sum of degenerate

B. =0

is for

an any

(n+1)-simplices,m

COROLLARY ~.- The simplicial morphism j. induces a bijection between (G.(X),B.) and the set of neutral (n+l)-cocycles over X under C.(B.).m In order to establish the general to point out two larger subsets of (G. neutral cocycles. For this we observe two natural factorizations through determined by'the simplicial morphisms B.

long exact sequence it is necessary (X),C. (B.)) than that consisting of that the simplicial morphism j. has . . . . n the slmpllclal object Cosk (B.) r. and t° of 2.1. : )

C. (B.)

Coskn (B.) Define the "r-null" (n+1)-cocycles ( respectively the "t-null" (n+l)-cocycles) as those which factor through r. (resp. t. ). The respective classes of such cocycles in [S.(X),C.(B.)] will permit us to describe the exactness of the long exact sequence. n If we denote by CG(X,B.) the set of n-cochains (i.e., n-truncated slmplicial morphisms from TFn(S. (X)) to Trn(B. ) ) the universal adjunction property between the functors TF n and cosk n, (see 1.2), gives a bijectlon n C~(X,B. ) ~ (S. (X),Coskn(B.)) and now, because r. and t. are injective, we have PROPOSITION 9.- There is a canonical bijection between the set of r-null (respectively t-null) (n+l)-cocycles over X under C. (B.) and the set C~(X,B. ).l 2.3. The hypergroupoid kernel. Let q.:B. .........> B" be a morphism of n-hypergroupolds in S, nzl, and consider the induced morphism of (n+l)-hypergroupoids

88 C.(q.):C.(B.) >C.(B") We define K(C.(q.)) kernel" of C.(q.) by the cartesian square K(C.(q.))

......

"(n+l)-hypergroupoid

the

> Coskn(B '')

I c.(B.)

C.(q.)> C.(B )

Note that the existence of the commutative

square

n

Coskn(B. )

Cosk (q.) Coskn(B.)

r. i

I r.

C.(B.) implies diagram

r.

that

factors

C.(q.)> C.(B )

through

K(C.(q.)),

Coskn(B.)

r.

i e , there

is a commutative

> C. (B.)

K(C.(q.)) Thus, t h e n e u t r a l ( n + l ) - c o c y c l e s u n d e r K ( C . ( q . ) ) a r e t h e n e u t r a l ones under C.(B.). Note also that Ni(K(C.(q.)))=Ni(B.) f o r i ~ n , n + l , g ( K ( C . ( q . ) ) ) = O and n

~n+l (K (C* (q") ) ) =Ker (gn (B.) 2.4.

)Hn (B.) ) .

The c o n n e c t i n g map.

Let q.:B. > B" be a mor2hism of n-hypergroupoids in ~ . n a l , such that the induced morphism Cosk"(q.):Coskn(B.) )Coskn(B '') is a trivial fibration. There is map n

:[G.(X),B"]

defined as follows: Let f"e(8.(X),B") morphisms B. ........d...

q.

and

consider

> Coskn(B.)

the

following t.

diagram

of

simplicial

> C. (B.)

Coskn(q.) . ~./K(C. (q.))

"

@.(X] ...... f"

> [S.(X),K(C.(q.))]

C. (q.)

ro

> B"

d

) Coskn(B "]

~ C.(B") t.

Now, since Coskn(q.) is a trivial fibration and ~.(X) is a cofibrant object in Simpl(S), there exists a morphism g.:S.(X) > Coskn(B.) such that Coskn(q.)g.=df '' Moreover, the composition t.g.:~.(X) >C.(B.} factors through K(C.(q.)) because C.(q.)t.g.=t. Cosk n (q.)g.=t.df " =r. d f " and so it defines a simplicial morphism to K(C.(q.)). We define ~ [f"]=[t.g.]. n

That this definition is independent is an inmediate consequence of the "covering homotopy theorem", [28],applied to the cofibrant object ~.(X) and the fibration Coskn(q.).

89 Let

us

observe

that

(a) ~o(B. )=~o(B' )=~o(C. (S.))=~o(C. (B"))= Io(K(C. (q.))), (b) the sequence (C)

for

each

O-->TI

n÷l

(K(C. (q.)) )

f~(X,Ho(B. ))

morphism

~n: [•" (X),B'] f

respect

to

the

such

>17 (B")--+0 is exact,

n

that

)

classical

>17 (B.)

n

[~3. (X),17o(B.)]f

[~. (X),K(C. (q.))]~ connecting

The exact The

is

morphism

>H~+I (X,17n+I (K(C. (q.) ) ) and the corresponding 2.5.

is

not

and

empty,

the

equivariant ~n:

with

H~(X, l]n(B") )

>

actions.

sequence.

proof

of

the

exactness

of

the

promised

long

exact

stated at the first of this section will use the following about G-sets which is easy to prove. f'

sequence

general

we

result

f

LEMMA I0.- Let G' ) G > G" be an exact sequence of groups and consider a commutative diagram in Sets E' q' > E

q > E"

°Id X
[S.(X),K(C.(q.))]

i./ [S.(X),C.(B.)] is exact in the following sense: a)An element of [G.(X),B"] is in the mapped by 8

n

~ [S.(X],C.(B")] image

of q~

if and

only

if

it is

into a neutral class.

b]An element of [G.(X),K(C.(q.)]]

is in the image of 6

n

if and only if its

image by i~ is a t-null element. c)An element of [~.(X),C.(B.)]

is in the image of i~ if and only if it is

applied by q~ into an r-null class. Proof: c) is an elementary application of the covering homotopy theorem, since C(q.) is a fibration. To see a) and b), let us observe that for any f:X > B=[0(B.), we have the exact sequence of groups

90

> cx,A") where

A=En(B.),

diagram

A"=~ n (B")

and

q~

[G. CX),B.]

f

A' =~n+1(K(C.(q.))),

~

........ ) [~.CX),B"]

~

r

the

morphisms

d.

and

the

commutative

i~ n) [G. CX),KCC.(q.))] d.

[6.(X),Coskn(B")]< where

,.7'c×,:)

f

1

)[G. CX),C. CB.)]

f

d,

f

[G.(X),Coskn+IC.(B")]

in

the

triangle

are

the

compositions

of

the

canonical d~ with the bijections [S.(X),Coskn+IK(C.(q.))]S[G.(X),CoSkn+IC.(B.)]m[6.(X),CoSkn÷IC.(B")] which are induced by the weak equivalences Coskn+iK(C. (q.))

)Coskn+IC. (B.)

) Coskn+I(C.(B"))

(see proposition below). The diagram satisfies the conditions of the lemma I0 with the above exact sequence of groups and the respective actions showed in 1.3 ; thus only remains to show that: i) the image of ~nq ~ is the set of neutral elements ; ii) an element

in the image of i~ is t-null

if and only

if it is in

the image of i ~ n

It is clear that ~nq~[g.]

is a neutral element.

On the other hand,

[f.]~[S.(X),K(C.(q.))] is a neutral element [f.]=[r.dg.] g.E(G.(X),B.) and therefore ~nq~[g']=~n [q'g']=[t'dg']=[r'dg']= any neutral element in [~.(X),K(C.(q.))] Now image of i.6 n is clearly

for If')'

if

some then

is in the image of 6nq . .

included

in the set of t-null elements.

To complete the proof we first observe the existence of u.:C.(B") >K(A"~B,n+I} which in dimension n+l is given by

the

morphism

n÷1

Un+iCb;,bo ..... bn.l)=(~Cb~,b I .... bn.t)-~Cb 0 ..... bn÷1),d °

(bo)}

and has the property that u.r.=~.:Coskn(B" ) )KCA"~B,n+I) and v.t. factors through K(B,O). Then, if E " : S . ( X ) >Coskn(B '') has the property that r.g" is homotopic to t.g", ~.g" is homotopic to u.t.g" which factors through K(B,O) and so, by the covering homotopy theorem, there exists an f":S.(X) )B" such that dr" is homotopic to g " Now, let us suppose that i~[f.] is a t-null

element,

that

is

im[f.]

=

[t.g.]

for

some E.:G.(X) ......)CoskncB.); .

then C.(q.)t.g. represents a t-null class and also a r-null class,and therefore if g"=Coskn(q.)g., r.g" is homotopic to t.g" so that there exists f":S.(X) )B" such that dr" is homotopic to Coskn(q.)g. It now follows that i.[f.]= =i.~ [f"]. • n

The above proposition shows the exactness of the five first terms of the announced sequence. Now, in order to extend it to an infinite one we prove PROPOSITION 12 - Let q.:B. > B" be a morphism of n-hypergroupoids in n N n . nzl, such that the induced morphism Cosk (q.):Cosk (B.) )Cosk ( B ) is trivial fibration Then q. is a fibration and the induced morphlsm Coskn÷IcC.(q.)):Coskn+l(C.(B.)) ) Coskn÷I(c.(B")) is a trivial fibration. Proof: n Since Cosk (q.) is a trivial fibration it is surjective and so the

91 induced morphism N(Coskn(q. )) between the normal (Moore) complexes will be surjective too, but then N(q.) is surjective and consequently q. is a fibration. On the other hand, if we consider the Moore complex of the morphism Coskn÷l(C. (q.))we have the •orphis• of complexes ...0

) II (B.)

...... > ~ (B.)2

n

...0

> N (B.) .... >...

n

> g (B")

> IT (B")2

;%

) N (B.)

n

> N (B"))...

n

)B

1

0

; N (B")

n

) B" 0

I

which is clearly surjective and so Coskn÷l(C.(q.))is a fibration. Moreover, it is a trivial fibration because in both complexes the homology groups of the three non trivial first terms are themselves trivial and in the other n terms they are isomorphic, since Cosk (q.) is a trivial fibration. • Thus, according to the above propositions , we have finally THEOREM 13.- Let q. :B. ........>.. B" be a morphis• of n-hypergroupoids n n . ~ u c ~ that the induced morphism Coskn(q.):Cosk (B.) )Cosk ( B ) trivial fibration.Then there exists a long sequence

inS. isa

[•. (X),B. ]

>[@. (X),B" ] n)[o. (X),K(C. (q.)) ] >[~). (X) ,C. (B.) ] )... q. 6 ... [S. (X),C.(| (B.) ] )[•. (X),C.( i (B.) ] n+i> [G. (X), K(C.(I÷I (q.) ) ] ... in which C.(I(B.)=C.(C.(I-I(B.)), and which is exact in the sense that an element is in the image of qo if and only if its image by the corresponding is neutral; an element is in the image of 6 if and only if its image by i. is a t-null element; an element is in the image of i~ if and only if its image by q. is an r-null element.

3.

NON-ABELIAN

~n WITH

•

COEFFICIENTS

IN CROSSED

MODULES:

A

LONG

EXACT

SEQUENCE. Let • be a crossed module in S and consider the associated groupold • (@) and, via the functor Net(-), the associated 1-hypergroupold which we also denote ~(~). Now, applying the construction C.(-), we obtai~ a 2-hypergroupoid C.(@(~)) (this is isomorphic to the 2-hypergroupoid ~ {@) used in [7]), and by iterating this process we have an e•bedlng cn-1 XM(S) ~ GPD(S) " ) n-~D(S) of the category of crossed modules into the category of n-hypergroupoids in S. Note that, up to homotopical equivalence, this embeding, restricted t o the abelian category of singular objects in S (i.e. the internal abelian groups), is the same that the given by the Eilenberg-Mac Lane construction K(-,n):Ab(S) ) n-HYPGD(S), since @(M >O)=K(M,I) and C~-I(@(M---~O))=C?-2(K(M,2) ..... C.(K(M,n-I))= according to lemma 6. Now given XeS, we coefficients in # by

define

the

(n+l)-cohomology

K(M,n) set

of

X

wlth

92 ~n+lCx,~)=[~.Cx),c~-lCe(~))]

, n~l

S u p p o s e now t h a t ~ ' = ( B ' >A',~'), ~=(B ~ )A,g), crossed modules in ~ and recall that a sequence of

modules 0

~,(In,i)> ~

> B'

> B

surjective

and

Pl)

P=(Pl,Po )

B"

> ~"

) 0

is

is

Ker(P0)=Im(~').

This

called

~"=(B" ~ morphisms

a short

an

exact

last

condition

exact

sequence (which

~A",~") are of crossed

sequence Po

if

in

~,

is

is

sometimes

avoided in the definition of short exact sequences of crossed modules) links this concept of short exact sequence to the concept of short exact sequence of groupoids in the sense of Higgins, [18], via the equivalence of categories @(-). We state this correspondence in the following proposition, which also allow us to apply the results of section 2 to obtain the announced long exact non-abelian cohomology sequence associated to a short exact sequence of crossed modules (theorem 15 below) PROPOSITION

14.- Let p=(pl,po):#

> ~" be a morphism

of crossed modules

in S, and @(p):@(~) > ~(~") the induced morphism. Then the following conditions are equivalent: i) p is a quotient map of crossed modules, i.e., Ker(p) ) # > ~" is a short exact sequence of crossed modules. ii) @(p) is a quotient map of groupoids in the sense of Higgins. iii) Po is surjective, ~o(p) is an isomorphism and ~l(p) is surjective. iv) Coskl(@(p)):Coski(~(~)) >Coskl(~(~")) is a trivial fibration. Proof: i) ~ ii) ~ iii) is essentially proved in [7] and iii) ~ straightforward.m

iv)

is

P=(Pl,Po )

Given #,(in,l)) ~ > ~" a short exact sequence of crossed modules we can consider the induced morphism of n-hypergroupoids C. (~(p)):C. (@(~) (@(~")) and its n-hypergroupoid kernel K(C~-l(@(p))) (see .3) Then, for any X~S, we define the "relative (n+l)-cohomology set" as ~n+Icx,~'

) =

[S. CXl,K(C~-I(@(pl))]

Let us note that if ~'=(M' >0) > ~=(M >0) -p ) ~"=CM" >0) is the short exact sequence of crossed modules associated to a short exact sequence of trivial X-modules, it is easy to see that the homotoplcal equivalence C~-I(~(M--->O))= KCM,n) restricts to an equivalence between n+l K[M',n) and K(C~-I(@(p))) so that S~ ( X , ~ ) is the usual abelian cohomology with coefficients in M'. Finally we have P=(PI*Po )

THEOREM 15.- Let ~,!In,i)) ~ ) ~" be a short exact sequence of crossed modules in Z and X an object of S. There exists an exact sequence of sets with distinguished elements

M~(X,~' )

......

~CX,~) • • . H n (X, ~" )

~(x,~") ~>~(x,~')~ @ n-I

, ) )S~n + i [X,~

) . . .

)sn+1 (X, ~)

)~n+l

(X, #" )-'--'>.

..

93 Proof: Coskl(~(p)) is a trivial fibration by p r o p o s i t i o n 14 a n d t h e n , b y proposition 12, Coskn(C~-l(~(p)) is a trivial fibration for all nal. Theorem 13 gives now the exactness of the sequence in the corresponding sense, m REFERENCES I. M.Barr, J.Beck. ~oznebg9~ and ~ c ~ , L e c t u r e Math.80 , Springer-Verlag (1969),245-335. 2. R.Brown. ~onve a n n a&z//xm m2./Ao/a i n ~ t ~ and

Notes

in

,in ~ ~ ~ ~ Pr~ac. ~oz~. Ya6ede, OALe, 1983, ed. H.L.Bentley, et.al., Heldermann- Verlag, Berlin ( 1 9 8 4 ) , 108-146. 3. R.Brown. ~ ~ o~ ~ a a u ~ , J o u r n a l o f A l g e b r a 15 (1970), 103-132. 4. R.Brown, N . D . G i l b e r t . ~ & ~ & n z i c mo~leZa o~ 3-~q4aea a n d ~ ~ madu/2~, P r o c . London Math. S o c . (3) 59 ( 1 9 8 9 ) , 51-73. 5. A.M.Cegarra, M.Bullejos. ~ ~ and ~ d ~ ~aer~ £ n ~ , Journal of Algebra 132 (2), (1990),321-339. 6. A.M. Cegarra, M.Bullejos, A.R.Garz6n, ~£q]zeJ% d ~ o / ~ D% 0 2 / 7 ~ cateq~, Journal of Pure and Applied Algebra 49 (1987), 43-102. 7. A.M. Cegarra, A.R. Garzon, Non-a~ coAemoIx~ o~ a ~ ~gg~. ~]%e 9-te/un em:o2/ a e ~ , Cahiers Top. et Geom. Diff. Cat&goriques, vol XXX-4, (1989), 295-338. 8. E.B. Curtis, ~ / z n p 2 ~ Aenu2go~ gb2.x~, Advances in Mathe. 6 (1971) , 107-209. 9. P. Dedecker, ~u/~ ~a a - ~ non o ~ , C.R. Acad. Sc. Paris, t. 260, I (1965), 4137-4139. 10. P.Dedecker, ~ ~ rugn-a/~, Mimeographie, Fac. Sc. Lille, 1965. 11. P.Dedecker, ~]%n2~e d / ~ ~ 02~ ~ ~u% q ~ , Lecture Notes in Math. 92, Springer-Verlag (1969),32-64. 12. P. Dedecker, A.S.T. Lue, ~ a o n - a ~ 2 - d i n ~ ~ a ~ 02x7~, Bull. A.M.S. 72, (1966), 1044-1050. 13.J.Duskin, ~grnp~ oze6hod~ and ~ ~ o~ ~ . Memoir A.M.S. v o l 3 , i s s u e 2,163 ( 1 9 7 5 ) . 14. J.Duskin, N o n - ~ mnnad/~ ~ and geu~ o / ~ theo/~@. Math. Forschung Ins. Oberwolfach Tagungsbericht 33, (1976). 15. J.Duskin, ~ r u ~ ~ and ~ , Mimeographed notes . (1984). 16. G.J. Ellis, ~ a o ~ nzad~ and t]veg% ~ d £ ~ onabg~, Ph.D. thesis. Univ. of Wales (1984). 17. P. Glenn, ~eagiq~ ~ ~ c ~ £~ ~ e ~ ~ , Journal of Pure and Applied Algebra I , 33-107, (1982). 18. P.J.Higgins, ~ a t e q ~ and ~ , Van Nostrand Reinhold Math. Estudes 32 (1971) 19. P.J.Higgins, ~ u~ mu2/ip~ e p e ~ . Proc. London Math. Soe. 6, (1956), 366-416. 20. D. Kan, On A o / n o / ~ U ~ and c.a.a, q r u ~ , Annals of Math. 68, I (1958), 38-53. 21. F.Keune, ~metop~ 024~ and o / g e ~ ~-U~xyu4, Thesis Univ. of Amsterdam (1972). 22. R. Lavendhomme,J.R. Roisin, ~ ~ noz% o 2 ~ de . J. of Algebra 67 (1980),385-414.

94 23. J.-L. Loday, ~pacea adJA ~ ~ noa £rd2a/z/ h o ~ qazeupa, J. of Pure and Applied Algebra 2 4 (1982), 179-202. 24. A.S.T. Lue, N o n - c ~ ce/mme6~ o~ ~ ~ , J. Math. Oxford (2) 19, (1968), 159-180. 25. A. S.-T.Lue, ~ ~ o~ ~ r ~ /x~ a ~/d2/~, J. of Algebra 69 (1981), 155-174. 26. J.P. May, ~ ~ ~ /n ~ g ~ ff~, Van Nostrand, (1976). 27. T. P o r t e r , E a ~ , c ~ mnn~ and b ~ caZeq~ D% caZegen/ea ~ q ~ v~ a p ~ . Edinburgh Math. Soc. 30, (1987), 415-429. , 28. D.Quillen, ~ ~ ~ Lecture Notes in Math. 4 3 Springer-Verlag (1967). 29. J.P. Van Deuren, E/A/de de ~'oAa//u/c//oa au r ~ d'un ~ r u a r t a~. ~aaai d'une ~ d'ur~e 3 - ~ a on a ~ , Seminaire de mathematique pure, Univ. Catholique de Louvaine (1978). 30. J.H.C.Whitehead, ~ / r d ~ ~ ,If, Bull. A.M.S. 55, (1949), 496-543. 31. G.W. Whitehead, E E e z n ~ o ~ / ~ T u ~ x ~ ~ , Sprlnger-Verlag (1978).

ALGEBRAICALLY COMPLETE CATEGORIES by PETER FREYD Universityof Pennsylvania The title I used when I gave this talk was Computer Science Contradicts Mathematics. Many modem programming languages are inconsistent with standard mathematical foundations. The task of finding sound interpretations for what it is that computer scientists do strikes this writer as, perhaps, the highest type of applied mathematics. It is akin to the process that has been going on throughout the 20th Century with respect to physics. The interaction between the mathematicians and the practitioners in each case has resulted in the growth of both subjects. I wish here to report on just one side of this process: the effect of computer science on category theory, at least on my category theory. The effect has been pervasive and I can report here just one part of it, but a part for which I owe a lot to Italy and the Italians. I must particularly thank Pino Rosolini for the many conversations I had during a very productive visit to Parma. In many programming languages one may create new "data types" as "minimal fixed points" of "data constructors". In the broad sense of the word "complete" this is a completeness condition on the family of things that are considered to be data types. A computer scientist typically state this in terms of "domain equations", that is, given a "type constructor", T, one seeks a minimal solution to the equation D = TD. Let me relax the conditions a bit. If we consider the category of "data types" and specialize to the case that the "data constructor" is a covariant functor and weaken the equality to an isomorphism we arrive at the following condi-

The author was supported by the Office of Naval Research when this research was performed.

tion on a category:

lOP: For every covariant endo-functor, T, there is a T.INVARIANT OBJECT that is, an object D and an isomorphism TD.-g D. At the end of this lecture Adamek will tell me that in his recent book it is shown that this condition contradicts the usual categorical notion of completeness. To be precise, a category satisfying IOP and that has arbitrary limits must be a pre-ordered set, that is, there is never more. than one map from one object to another. Here is a proof, albeit not the proof in Adamek's book: suppose that the hom-set (A,B) has more than one element; let T be defined by TX =/-/.~rx-~,, that is, the S(X)-fotd cartesian power'i~'f" B where S is the set-valued contravariant functor that sends X to the power set of the set of maps (A,X). If D were an invariant object for T then the number of maps (A,D) would be the same as the number of maps (A, TD) .~ IIS(D/A,B) which is a clear contradiction in'the classical foundations. (The classical Tarski theorem, it should be recalled, says that any complete lattice satisfies IOP.) IOP does not catch the notion of "minimallity". We could strengthen it to the condition that the category of invariant objects has an initial object. But better, and closer to what is actually assumed in various programming languages, we consider the larger category of T-ALGEBRAS: tO wit, the category whose objects are maps of the form a:TA---~A, and whose maps look like:

96

Tx TA ---> TB aS Sb A >B x If f:TF --~ F is initial in this category it is called the INITIAL, or free, T-algebra. (The phrase "inductive type" is often used by computer scientists to describe initial algebras.) Lambek was the first to show that initiallity implies invariance, that is, f is necessarily an isomorphism. We are thus led to:

A category is called ALGEBRAICALLY COMPLETE if for every covariant endofunctor, T, there is an initial T-algebra. The most famous initial algebra is for the functor TX = i + X. A T-algebra structure on an object A may be viewed as a "point" 1--,A and an endomorphism A---~A. The Lawvere definition of NATURAL NUMBERS object may be restated as the initial algebra for this functor. (This works for any category with coproducts.) The dual property is of almost equal interest in computer science. A T-COALGEBRA is a map of the form a : A ~ T A . ThecoLambek lemma says that if f':F'--~TF" is a final T-coalgebra then f " is an isomorphism. In the category of sets a co-algebra structure on S for the functor TX = 1 + X, may be viewed as a partial endomorphism on S. The final coalgebra is a universal partial endomorphism. We may construct it as the "extended natural numbers" that is, the natural numbers with a point "at infinity" adjoined. The universal partial endomorphism is the predecessor function, understood to be undefined at zero and with infinity as a fixed point. Objects that arise in programming are rarely good only for input or only for output. I have called this the PRINCIPAL OF VERSALITY: every universal mapping definition is equivalent to a dual definition. This principal is quite untrue in mathematics and

is not claimed to be universally true in computer science. But it leads to:

A category is said to be ALGEBRAICALLY COMPACT if every covariant endofunctor has an initial algebra and a final coalgebra and they are canonically isomorphic. The canonical isomorphism is defined as follows: for a covariant endofunctor T let f:TF --* F be an initial T-algebra and f':F'---~TF" be a final T-coalgebra. Let g:F --~ F" be the unique map such that: g F )F" f'l $ S f" TF -'~ TF"

Tg If g is an isomorphism it is called the canonical isomorphism. (We used the fact that f is an isomorphism and f " is final. Alternatively we could have used the fact that f" is an isomorphism and f is initial. The same map g:F --* F" would be defined.) When the initial T-algebra and final T-coalgebra are canonically isomorphic we will, in practice, take the initial T-algebra structure f:TF --->F as primitive and use f-I: F--~ TF as the final T-coalgebra. The category of T-invariant objects appears as a full subeategory of both the categories ofT-algebras and T-coalgebras. Algebraic compactness thus says, in particular, that the category of T-invariant objects is a PUNCTUATED category, that is, a category in which the terminator and coterminator coincide. As in any such category, the (bi)terminator appears as a proper retract of every non-terminator and is thus distinguished by the fact that it has only one idempotent. For an algebraically compact category, therefore, we have that each covariant endofunctor has a distinguished invariant object D which is a proper retract--as an invariant object--~f every undistinguished invariant object. It came as a complete surprise to me that this

97

remains true (as we wiU see) even when the word covariant is replaced with contravariant.

Note in the case TX = 1 + X on the category of sets that F and F" are isomorphic (F" is a countably infinite set) but that g is not an isomorphism: that is, the principal of versality fails. In our LICS90 paper on Extensional PERS Mulry, Rosolini, D.Scott and I show that the realizable topos has a small full reflective subcategory that is algebraically compact in the relevant sense, that is, the condition holds for every endofunctor that is definable as a functor in the topos. This example is a rich one in terms of other structure. If one wants an impoverished example of an algebraically compact category take the full subcategory of the category of groups of all those groups that have at most two elements (or better, chose a skeletal subcategory with just two objects and five maps). Note that the only posets that as categories are algebraically compact are one-element posets. Let me go back to algebraic completeness. I will need some theorems true for any category. First, a standard lemma implicit in the proof of the Lambek lemma:

The uniqueness condition says that h;g is the identity map on F (the semicolon is the standard notation in computer science for the composition of operations in execution order, that is, h;g means first do h then do g). If we already know that g is an isomorphism then, of course, h is forced to be its inverse. But we may as well take the occasion to prove the Lambek lemma. Since h;g is the identity on F, Th;Tg is the identity map on TF. In the commutative diagram that defines h it says that g;h = Th;Tg, hence g;h is the identity map and g is seen to have a two-sided inverse. • A new theorem of surprisingly many uses:

THE ITERATED SQUARE THEOREM: For any category A and endofunctor T: A --~ A suppose that T 2 has an initial algebra. Then so does T. Let g:T2F-~F be an initial T2-~gebra. Let h be the unique map such that:

~F

g$

~

F

~F

$~

>TF h

LEMMA: If g:TF --+ F is an initial T-algebra then so is Tg:TeF---~ TF. Let h be the unique map such that:

Th TF --~ T2F g .l.], S Tg F )TF h We also have the diagram:

Tg

T2F -4 TF Tg 3, .l.g TF ...... " F

We may apply T to this diagram and paste to obtain: Th T2h T2F --4 T3F -.4 T'IF g .[. $ ST2g F >TF .-.-) T2F h Th The rightmost T2-algebra is T 2 applied to the initial T2-algebra. The lemma above says therefore that it is itself an initial T2-al gebra and the T2-algebra-map h;Th must be an isomorphism. Thus h has a right inverse and Th has a left inverse. But any functor preserves the existence of right inverses, hence Th has a right inverse. In any category any map with both inverses is, in fact, an isomorphism. Since both h;Th and Th are

98 isomorphisms it follows that h is an isomorphism. Note that it follows from the argument above that Tg:T3F---~TF is also an initial T2-algebra. Define f:TF --~ F to be the inverse of h. We wish to show it is an initial T-algebra. Note that from the construction of h that Tf;f = g. Given an arbitrary T-algebra a:TA-~A let x:F---~A be the unique map such that: T2x T2F ~ T2A Tf $ S Ta TF --~ TA f $ Sa F )A The middle horizontal map must exist since f is an isomorphism We temporarily denote it as y:TF---~TA. If we can show thaty = Tx we will be done. (The uniqueness condition is easy). If we apply T to the lower rectangle, remove it, and then paste it on top we obtain: T2y T3F --~ T3A T2f $ S T2a T2F ~ T2A Tf$ STa TF ) TA Y Since T2f;Tf = Tg we know that the left T2algebra is initial and y is unique. By applying T to the previous diagram we obtain:

r3x T3F T2fT2~F

-~ T3A $ T2a ._~ T2A

rf$

STa >TA

TF Tx

The uniqueness of y thus forces y = Tx. [] There is more in this proof than was recorded in the statement of the theorem. We

will need:

LEMMA: If f: TF -~ F is initial and if there is an initial T2-algebra then (Tf;f):T2F --~ F is initial. We showed that h;Th is an isomorphism from the initial T2-algebra to T2 applied to the initial T2-algebra. As in the last lemma, its inverse must be g. [] (Several comments about the theorem. First, the proof may be easily generalized to give THE ITERATED POWERS THEOREM: if T n has an initial algebra then so does T. The converses are not true. Given T let N be the set of positive integers { n / there is an initial T-algebra }. The theorem says that N is closed under divisors. For any such set of positive integers, N, we may construct a discrete category, A and a permutation T as follows: let A be the disjoint union of a fixed-point for T together with an m-cycle for T for each m not in N. ((A permutation on a discrete category has an initial algebra iff it has a unique fixed point.) (If A has products then the converses do hold. For the case n = 2 argue as follows: given an initial T-algebra f:TF ~ F show that (Tf;f):TeF ~ F is initial by taking any a:T2A~ A and considering the T-algebra :T(AxTA) ~ AxTA ((where l and r are the left and right projection maps from the product)); let <x,y> : F --~AxTA be the unique map of T-algebras. It is routine to verify that necessarily y =f'l;Tx and that x is the unique map of T2-algebras.) Back to algebraically complete categories:

THE PRODUCT THEOREM FOR ALGEBRAICALLY COMPLETE CATEGORIES: If A" and A are algebraically complete then so is A" × A Any endofunctor on the product may be resolved into its coordinate functors. Hence we suppose that we are given

99 T':A'xA ~A'andT':A'xA ~ A . We wish to construct an initial algebra for the functor that sends an object D" and d:TD'D -*D. We define d" to be the unique map such that:

T'(F'D)D

$

T'q "l -.~ T'D'D

$ d.

F'D

>

D"

q"

(We are using only that T'q 7 is an isomor-

phism.) d is defined similarly. Given : to be the unique pair such that: --* A"

We wish to show that the front face commutes. The bottom face commutes by definition of x;x; the left face by definition of d'; the back face by definition of F'x; the right face by definition of y " the top face because T'(F'x,3:)x = T'(q';x')x (the bottom face already says F'x,y" = q;'x'). Since T'q'l is an isomorphism we may

100

conclude that the front face commutes. The uniqueness of x , x is routinely verified from these same diagrams. [] As just one corollary we obtain the "dinaturality" of the two-functor that delivers initial algebras. That is, given a functor F : A --~B the induced diagram of categories commutes in the relevant sense: (A,A) >A

\ (B,B)

> B

The commutativity at the object level says: COROLLARY: Given F : A ---)B and G:B --~A, F applied to the initial (F;G)-algebra is the initial ( G ;F)-algebra. To prove it apply the last theorem to the endofunctor on A x B that sends to < GB,FA> . • Returning to algebraically compact categories it is just a matter of proof inspection to obtain: THE PRODUCT THEOREM FOR ALGEBRAICALLY COMPACTCATEGORIES: If A" and A are algebraically compact then so is A" x A The proof inspection consists of showing that if each of the isomorphisms that arise in the the product theorem for algebraically complete categories is inverted the result is the construction for the final coalgebra. • Algebraic compactness is a self-dual property, that is, if A is compact then so is A ° Hence: COROLLARY: If A is algebraically compact then so is A ° x A • Now suppose that T is a contravariant endofunctor on an algebraically compact catego-

ry A Consider the covariant endofunctor on A ° x A that sends to D. x

But, of course, the same can be said about the pair . All of which says that the contravariant functor itself has an invariant object and, in the relevant sense, an invariant object characterized by the fact that it has only one invariant idempotent. We may extend the argument to functors of mixed variance and, as usual, it suffices to consider the case of a bifunctor, T, contravariant on the ftrst variable, covariant on the second. By considering the covariant endofunctor that sends to we obtain: TIIEOREM: For any bifunctor, T, on an algebraically compact category there is a minimal invariant object, f:TDD _~ D, characterized by the fact that there is only one idempotent (indeed, only one endomorphism) such that: TDD f~. D

Zxx --) TDD Sf > D. x

[]

We look at an easy lemma: LEMMA: For any category the identity functor has an initial algebra iff the category has a coterminator.

101

Any functor that preserves the coterminator is easily seen to have the coterrninator as an initial algebra. Conversely, suppose that f: F---~F is an initial /-algebra. The uniqueness condition says that it has only one endomorphism as an algebra, hence f must be the identity map (because f commutes with f). Thus F is a coterminator in the full subcategory of those/-algebras for which the structure map is the identity map. But that category is, of course, isomorphic to the given category. [] As an immediate corollary:

LEMMA: Any algebraically compact category is punctuated. [] The tradition in mathematics is to denote the biterminator in a punctuated category as 0 and in computer science (and group theory) as 1. The unique map from 1 to an object A is denoted J.: 1-~A, pronounced "bottom" and thought of as the "undefined" or "never defined" state. Bottom-preserving maps are called STRICT. The purpose of the undefined state is often understood to allow partial maps to be encoded as maps. Total maps may be viewed as special cases of partial maps, to wit, those maps that not only preserve the bottom but reflect it, that is those maps for which 1--~1

±$

$±

A ---~ B is a pullback. In my paper on the well-ordering of choice objects I said that such maps "strictly preserve the base point" and referred to them as "strict maps." Because of the widespread use of the word "strict" to mean bottom-preservation I am now forced to escalate the language. I will say that such maps are "restrictive" and the category of such maps to be the category of RESTRICT MAPS.

In many of the categories that arise when one tries to find mathematical semantics for

programming languages it is the case that the lluf subcategory of restrict maps is coreflective. The coreflection of an object A is often denoted A± and pronounced "A lifted." In the extensional PERs example the category of restrict maps fails to be coreflective, indeed does not even have a terminator. There is, however, a maximal subterminator and it has been of interest for all sorts of reasons for many years and is traditionally denoted E. If we cut down to the category of restrict maps between objects that do have a restrict map to E we do obtain a coreflective subcategory. Indeed, the construction of the coreflection is not difficult: given A let Y.A be the maximal subobject of E x A such that Y.,4 ---~ x A--~E is a restrict map. As for any coreflective lluf subcategory the coreflection is an embedding functor and we may view the situation differently. Letting A denote the algebraically compact category and R the coreflective subcategory of restrict maps we may view the embedding A - - ) R as an enlargement of o,1.. Computer scientists call the objects of the ambient category "pre-domains." The objects that are isomorphic to objects coming from A are then called "domains" and their full subcategory, D, is called the "category of domains." In this context, A is viewed as the lluf subcategory of strict maps. I have taken A as primitive with R and D as constructs. If the truth be told, that is not the way it usually happens. Typically D is first defined then the notion of strict maps and then the notion of pre-domains (by "bottom deletion"). When viewed this way A , , or as I will prefer, Y_,A is the reflection of-a domain in the subcategory of strict maps. In a punctuated category 1 + X is naturally equivalent to the identity functor and the natural numbers object is just the biterminator. We consider some variations. We use the topologists notation for the coproduct in a punctuated category, the "wedge" A vB (called the "coalesced sum" in computer science). The variations on I vX to be

102

considered are the three functors ,F,v X , I v 2 X , F, v X X , where I write 2; for £,1. In the traditional models, the objects are "CPO's", that is, posets with suprema for directed sets, and the maps are maps that preserve such suprema. The variations on the natural numbers have all been studied and given names. If I may quote from myself: A Z v X algebra is an object A together with strict maps Zo--->A and Ao--->A. (The decoration on the horizontal arrows is the standard way of indicating strict maps.) Because X is the reflection of 1 into the subcategory of strict maps, we may rephrase: an object A together with an "initial point" 1 --gA and a strict "successor map" Ao-->A. The initial algebra is called the FLAT or HORIZONTAL NATURAL NUMBERS and is

usually pictured as

infinity:

I 3

I

2

I

1

I

0 The successor map is the up-shift, keeping the point at infinity fixed. A ,~vZX algebra--following similar arguments---may be rephrased to be an object A together with an initial point 1--9A and a successor map A--~A, neither of which need be strict. The initial algebra is called the LAZY or OBLIQUE NATURAL NUMBERS and is usually pictured as:

0 1 2 3 4 5 . . .

=i

4

3

IL

The successor map shifts the horizontal row to the right, keeping the bottom fixed. The left-most point on the horizontal row is the initial point. A lvXX algebra is an object A together with strict maps lo->A and £,Ao-cA. Of course, there is only one strict map of the form lo-.-)A (indeed, the functor in question is isomorphic to 2X ). Because 2:A is the reflection of A into the subcategory of strict maps we may rephrase: a lv,F,X algebra is an object A together with a (not necessarily strict) successor map A...~A. The initial algebra is called the ORDERED or VERTICAL NATURAL NUMBERS and is usually pictured as an ascending chain climbing forever to an invisible point at

2

\/

1

\/

0

\/

\/

\/ s(s(s(s_L)))

s(s(s.L))

s(s.L)

s.L

.L where it must be understood that there's a point at infinity above all the s-iterates of .1_. The points marked by numbers are each maximal elements. 0 is the initial point. The successor map is the north-east shift, again with the point at inf'mity fixed. Every endofunctor that arises on either of the ambient categories seems always to have the subcategory of strict maps as an invariant subcategory. Indeed, in the extensional PERs case, it is provably the case that all functors definable in the topos have this property. All of which gives interest to the easily verified:

103

THE REFLECTIVESUBCATEGORY LEMMA:

I f A is a reflective subcategory of B and if T is an endofunctor on B which has A as an invariant subcategory then a final Tcoalgebra for T viewed as an endofunctor on A remains a final T-coalgebra for T viewed as an endofunctor on ~3. []

Computer scientists have been using fixed points for endomorphisms longer than invariant objects for functors. To close this paper I wish to show that algebraic compactness entails much of the standard material on fLXedpoints. Viewing Z as a endofunctor on domains we note that the subcategory of strict maps is surely invariant and we will apply the reflective subcategory lemma to the vertical natural numbers. We define s : N ~ N as u,f where we are using u:N---~ZN to denote the reflector map. It is the universal endomorphism in the category of domains, that is for every a:A-->A there is a unique ~wict map Ja :N-~A such that : Ja No---> A s$ Sa No--~ A Ja

Recall that the decoration on the horizontal indicate strict maps. It is important that the vertical maps needn't be strict and that the horizontal map must be. Note that if a is strict then Ja is the map constantly equal to the bottom of A. Consider the Z-coalgebra u:l-->Z1. Using the reflective subcategory lemma we know that there exists a unique map y:l--~N such that: Y 1---> N

u $

$FI

Z1 ~ Z N .

Zy

Note that we also have:

Y 1 --~

u$

N

,l,u

Z1---> Z N .

From y f l = y;u we obtain y = y ; f - l j = y;u;f =y;s. That is, y is a fixed point of the universal endomorphism s. It is the unique fixed point: for any y the last square still commutes, from which one may infer that the previous square commutes iff y;f "1 = y;u hence i f f y = y;f-lgf= y ; u J =y;s. We will use the traditional name for the unique fixed point of the universal endomorphism ,o :I --~N. Since the universal endomorphism has a fixed point it is the case that every endomorphism between domains has a fixed point. By positing good invariant objects for endofunctors we have achieved fixed points for endomorphisms. (Note that it is a complete triviality that strict endomorphisms have fixed points. The computational usefulness of fixed points requires bottom-raising maps. From this point of view it is more useful to think of the bottom not as "never defined" but as "not yet defined.") We have more than the sheer existence of fixed points, we have a family of operators, traditionally denoted Y, one for each domain, which yield fixed points for endo-morphisms: given a:A--oA we define Ya:I-->A tO be y;Ja. PLOTKIN'S AXIOM for Y is now immediate, that is, if X

Bo--->A b$ Sa Bo--~A

then Yb;x = Ya. And since the universal endomorphism has a unique fixed point, the Plotkin axiom characterizes Y. (Which, albeit in a different context, was why Plotkin stated the axiom.)

104

LEMMA: Y(a 2) = Ya a:A--~A .

for any

endomorphism

First we prove it for the special case of s:N--~N. Not surprisingly we use the iterated square theorem. We will show that s 2 has only one fixed point, the one we akeadT Consider the functor Z ~. know, ~:I---~N. The initial zz-algebra may be taken to be Zf,f.'Z2N--~N. The reflective subcategory theorem says that there is at most one map y:l--)N such that : Y 1 ) N

Assuming that our algebraically compact category has products we can now easily prove the dinaturality of Y (just as for the initial-algebra operator), that is for any

f:A-)B:

(B,A)

Y

>N

,1. u;Xu

E21 -.~ Z2N

Hence the f'u'st square commutes iff We may follow each side with the isomorphism Zf;f to obtain: y

y ; f l ; Z f l = y;u;Zu.

= y;u;Zu;Zf;f = y;u;E(u;f)J = y;u;Z~;f = y;s;u;f =y;s 2, that is, the first square com-

/ \

(A, A)

(S, B) ~

X 2 fl';Zfl'V

But for any y we have:

u;Zu $

Ya.

$

u;Zu 221"1"~

1

The PlotkJn axiom says that Ja carries Y(s 2) to Y(a2). But we have just seen that Y(s 2) = ~. But by definition Ja carries oo to

Y

; (1, A)

(1, B).

The dinaturality of Y implies many things. It already implies that it is a fixed point operator: consider just the case that A =B and chase the diagram starting with the identity map in the left-most hom-set. It implies that isomorphisms are automatically strict. At a later time I will explore the many consequences of the condition that the subcategory of strict maps be multi-coreflective in the sense of Diers. It doesn't take too much else to get a representation theorem of algebraically compact categories into categories of CPO's.

mutes i f f y is a fixed point for s 2. (We used the equation u;Zs = s;u, the neutrality of u.) The uniqueness condition for final coalgebras thus forces y = ,,o. Given a:A--~A consider:

Ja N ~-~A s $ Sa N o-~A s $ ,l.a N o.-~ A Ja

This paper is in final form and will not be published elsewhere.

Order-Enriched Sketches for typed lambda calculi. John W. G r a y 1 University of Illinois at Urbana Champaign U r b a n a , I1, U.S.A.

Abstract: Order-enriched sketches are constructed for which the initial algebras are respectively, the simple typed ~-calculus, the PCF version of the ~.-calculus,and the polymorphictyped k-calculus. 1

Introduction

1.1 The context. There are three sources for this present work. The first is a general concern with the relationships between logical type theory and algebraic type theory discussed in [7]. As was shown there, if algebraic type theory is identified with the category of finite limit sketches, then it is in a sense a model of logical type theory; in particular, it admits constructions that resemble dependent types. (See Barr and Wells [I], or Gray [7] for basic information on sketches.) One purpose of the present paper is to show that there is a relation in the other direction. If logical type theory is identified with the typed X-calculus, then what is shown here is that the typed Xcalculus is an example of an algebraic data type; i.e., an initial algebra for a finite limit sketch. A second source is an attempt I made to implement the particular typed X-calculus, PCF, in the rewrite rule language Mathematica. I found this difficult to do, mainly because there was no clear theoretical basis for producing such an implementation. The third source is a remark in [7] to the effect that instead of viewing the collections Terms(o) of terms of type o in a typed Xcalculus as a family of sets indexed by the set Types of types, it might be fruitful to consider the collection of all terms of all types, Terms = ucr E Types Terms(o) as a set over Types via a function type : Terms ~ Types. This turned out to be the key idea in constructing a sketch whose initial algebra is the typed X-calculus. It also solved the second problem since, in [9] I had already shown how to implement the initial algebra for a sketch in Mathematica. Note there there are no special theoretical tools to determine when a given mathematical structure is the initial algebra for a finite limit sketch. There are tools for deciding if a given category is the category of all models of such a sketch, but one of the problems here is that it is not clear what should be taken as the category of all models of a typed X-calculus. The answer to this becomes clear once one has the appropriate sketch; namely, it is the category of all models of the sketch. A great deal is known about such categories. See, e.g., [1] or [13]. In this paper, explicit sketches are described in three cases: the pure typed X-calculus, PCF, and the polymorphic typed X-calculus F2. What is constructed in each case is an order-enriched sketch whose initial algebra in the category of preordered sets consists of the raw terms of the appropriate X-calculus together with preorder relations for terms of each sort corresponding to the rewrite rules for terms of that sort. The initial algebra in the category of sets consists of the lThis research was partially supported by the National ScienceFoundation.

106

normal forms of such terms. Each case presents its own difficulties. For the pure typed ~,calculus, the difficulty is to give a correct account of substitution. The most annoying problem is to make the choice of a new, or fresh, variable in the beta rule sketchable. This explains why we cannot use Stouten's treatment of substitution in [17]. PCF adds a mass of detail to this construction. Finally, the polymorphic typed )~-calculus adds four more basic constructions. In the last section of this paper there is an algorithm for converting such a sketch into a program implementing the initial algebra for the sketch. This algorithm is used in implementing PCF and F2 in Mathematica. See [10] and [11]. The relation of these sketches to the well-known sketch for cartesian closed categories has not yet been clarified. 1.2 Typed lambda calculi A typed 1-calculus L consists of types and terms of each type. i) The set Type of types is given recursively as follows: There is a finite or countable set B of basic types and if c and x are types then [ff --->x] is a type. ii) For each type x, there are a countable set Var(x) of variables of type x and a finite or countable set Const(x) of constants of type x. We set Atom(x) = Var(x) + Const(x). iii) Write f : x for "f is a term of type x". The set Terms(x) of terms of type x is described recursively as follows: a) Terms(x) D_Atom(x) b) If f : [~ --->x] and g : tJ, then (f g) : x; c) If g : x and x e Var(~), then ()~x : (r. g) : [(r --->I:]; 1.3 Operational semantics of typed lambda calculi. The operational semantics of a typed lambda calculus consists of some of the following rewrite rules. We use " ~ " here to mean the left hand side rewrites as the fight hand side. Type information is omitted, but it is assumed that all terms are well-typed. i) (ix - conversion) ()~x. f) ~ (Xy .[y / x] f) providing y ~ FV(f). ii) ([3 - conversion) (Q,x. f) g) ~ [g / x]f. (See below.) iii) (rI - conversion) ((~,x. f) x) ~ f (or the reverse rule) iv) (rewrite schemes) a) h 9k b) ......h ~ k c) h ~k (h g) ~ (k g) (f h) ~ (f ki (~,x. h) ~ (~,x. k) v) (5-conversion) Special rewrite rules for constants. Here, the operation [g / x] f of substituting g for x (where the type of g must be the same as the type of x) in f is defined recursively by the rules: [g / x] c = c i f c is a constant [g/x] x = g [g/x] y = y if y ~ x and y is a variable [g / x] (h k) = (([g / x] h) ([g t xl k)) [g / x] (~,x. f) = (~,x. f) (i.e., don't substitute for bound variables.) [g / x] (~,y. f) = (~,y. [g / x] f) if x # y and y ~ FV(g) [ g / x ] (~,y. f ) = (~,z. [ g / x ] [ z / y ] f) i f x # y a n d y ~ F V ( g ) , w h e r e z # x , z # y , z ~ FV[f] u FV[g] and type(z) = type(y). (Prevent the capture of free variables.) The set FV(g) of free variables of a term g is defined by the rules: FV(x) = {x} FV(c) = {} if c is a constant FV((h k)) = FV(h) ,, FV(k) FV(Xx. t) = FV(O- {x}

107

The following notation will be used throughout this work. Var = u {Vat(x) I x ~ Type}, Const = u {Const(x) I x ~ Type}. Terms = u {Terms(x) Ix ~ Type}. type : Terms --->Types denotes the function taking terms of type 'c to x. In particular, type(x : x) = x. 2 . T H E PURE TYPED LAMBDA CALCULUS. A pure typed lambda calculus is determined by a fixed set B of basic types, but no constant terms. We take it to have (t-conversion, 15-conversion and all three rewrite schemes in 1.3, iv). 2.1 The basic sketch for the pure typed lambda calculus. A sketch is a graph whose nodes represent the objects of a date type and whose arrows represent the operations, together with extra structure. Sketches were introduced in a somewhat different form by Ehresmann [2] as presentations of theories. We follow the usage in Barr and Wells [1] and Gray [7]. The basic graph for the pure typed K-calculus looks as fiDllows: Pf(var) appPullback var

lamProd terms let

"" letPullback

type bl one

:

= types -- fun

types × types

b. There an arrow from "one" to "types" for each element bi of the fixed set B. The names of the nodes are chosen to correspond to the intended interpretation of the values of the initial model at each node. In a sketch, certain nodes are specified as formal limits of one kind or another. Here lamProd is specified as a formal product and appPullback and letPullback are formal puUbacks, to be completely described below. In a sketch there are also certain diagrams which are specified as formal commutative diagrams. A model of a sketch in a category C assigns an object of C to each node and a morphism of C to each arrow in such a way that actual limits are assigned to formal limits and formal commutative diagrams are taken to actual commutative diagrams. Homomorphisms of models are families of morphisms, one for each node, which commute with the operations. The category of models of a sketch S in a category C is denoted by Modc(S). If C satisfies reasonable properties (e.g., is accessible) then Mode(S) has an initial object. If C is the category SET of sets, then the initial object is the data type presented by the sketch. 2.2 Formal 2-cells. A sketch contains more information about a data type than just the objects and operations. As described above, equations between composed operations in the graph are usually represented by diagrams of arrows in the graph which are declared to be formally commutative. However, in the present case, we are interested in representing the rewrite rules in the operational semantics of the pure typed ~.-calculus, which have a direction. Thus, instead of formal commutative diagrams, we introduce the idea of a formal 2-cell in a diagram. This looks like:

108

gl

g2

=

\

\

\ fk

Such a 2-cell represents a formal order relation; i.e., here it represents the formal order relation fl * f2 ° • • • ° fk < gl ° g2 * • • • * gin. More generally, a 2-cell can represent a "morphism" from its domain, fl * f2 ° • • • ° fk to its codomain, gl * g2 * • • • * gm. Note that identically equal paths are considered to have an identity 2-cell between them. Essentially, there is a formal 2-cell for each rewrite rule in the operational semantics, except that there are also extra formal 2-cells for auxiliary arrows. If we consider the rewrite rules to be named, then we use the more general notion of a "morphism", whereas if we are just concerned with the existence of a rewrite rule, then we consider the 2-cells as order relations.

2.3

Formal quasi limits

There is still a further aspect of data types that is represented in a sketch. As described above, an object can be declared to be a formal product of other objects or, in general, a formal finite limit of a diagram of other objects. For instance, there is a node "types x types", which means that this node' is declared to be a formal product of "types" with itself, and fun is a arrow from "types × types" to "types". In category theory, limits of diagrams are described in terms unique morphisms making certain triangles commutative. Here we have rejected commutative diagrams in favor o f 2-cells, so we are forced to make a corresponding change in the notion of limit. Recall that a 2-category, or a CAT-enriched category, C is an ordinary category enriched in the category o f small categories; i.e., the horn sets o f C are the sets o f objects of h o m categories and composition is extended to be a functor. (See Gray [6] or Kelly and Street [12].) Morphisms in such hom categories are called 2-cells in C. If these horn categories are all preordered categories (at most one morphism between any two objects), then C is called an order-enriched category.

A quasi-limit cone in a 2-category C over a diagram {A(i), a(i, j)l i • I} in C is an object X in C together with arrows p(i) : X ~ A(i) and 2-cells a(i, j) * p(i) ~ pfj) for all i and j, which are compatible with composition. Such a cone is called a quasi-limit of the diagram if given any other quasi-limit cone over the diagram with vertex Y, paths of arrows q(i) • Y ~ A(i), and 2cells a(i, j) • q(i) ~ q(j) which are compatible with composition, there is a unique arrow q : Y --~ X and 2-cells p(i) • q ~ q(i) for all i, satisfying a composition condition which is clear from the diagram, and such tha t given any other arrow f : Y --~ X and 2-cells p(i) • r ~ q(i) satisfying the same condition, then there is a unique 2-ceU r ~ q; i.e., q is the terminal such arrow.

109

Y

......

q- . . . . . ~

X

In the special case of an order-enriched category, all of the compatibility and commutativity conditions are automatically satisfied. Frequently q is written as and is called the canonical arrow determined by {q(i) : i e I}. Quasi-limits were introduced in Gray [61; a special case, called cartesian quasi-limits in [6], corresponds to what are called lax limits in Kelly and Street [12]. See 5.1 for an example. One can define a 2-graph, or CAT-enriched graph, as an ordinary graph in which the set of arrows between any two fixed vertices is the set of objects of a category. Morphisms in these categories are called 2-cells in the graph. If these categories are all preordered categories, then the graph is called order-enriched. However, this does not capture the situation here. As indicated above, we want to consider 2-cells between paths in the graph. There is a somewhat complicated interaction between the free category generated by the graph and the addition of such 2-ceUs which is described in Section 5. Here we shall call this a graph with added 2-cells. We can then describe a quasi-limit cone in a graph G with added 2-cells over a diagram {A(i), a(i, j)l i ~ I} in G exactly as in the 2-category case, except that there are no compatibility conditions. Such a quasi-limit cone can be declared to be a formal quasi-limit in G. The intention is that a model in a 2-category of this situation is required to take a formal quasi-limit in a sketch to an actual quasi limit. Given any other vertex Y, paths of arrows q(i) : Y ~ A(i), and 2-ceUs a(i, j) • q(i) ~ q(j), then an arrow q : Y ---) X is called the formal canonical arrow determined by this data if models are required to take it to an actual canonical arrow. Thus, we arrive at the following definition: 2.3.1 i) ii) iii) iv)

Definition. A CAT-enriched, finite quasi-limit sketch A consists of A (directed) graph IAt, called the underlying graph of the sketch. A set of formal 2-cells between (directed) paths in IAI. A (possibly empty) set of finite formal quasi-cones in IAI. A (possibly empty) set of formal canonical arrows in IAI.

2 . 3 . 2 Definition. A (2-category) model of a CAT-enriched, finite quasi-limit sketch A in a 2-category C is a function M taking i) vertices and arrows of IAI to objects and arrows in C, ii) a formal 2-cell in A to an actual 2-cell in C from the composition of the values of M on the domain of the formal 2-cell to the composition of the values of M on the codomain. iii) the chosen finite formal quasi-cones in A to quasi-limit cones in C, and iv) the formal canonical arrows in A to actual canonical arrows in C. In this paper, we are only concerned with the order-enriched case. The definition of an order-enriched sketch is exactly the same as the definition of a CAT-enriched sketch. The only difference is that we regard the 2-cells as order relations rather than as "morphisms". However, the intention is to restrict attention to models in an order-enriched category rather than a general 2-category. Such models are called order models of A. Note that the category SET of sets can

110

be regarded as a discrete order-enriched category; i.e., all horn categories are discrete categories (no non-identity 2-cells). In this case, quasi-limits coincide with ordinary limits, since the only 2-cells are identites. Thus, a model of an order-enriched sketch in SET is the same as a model of the ordinary sketch determined by replacing all formal 2-cells by formal commutative diagrams, which turns quasi-limits into ordinary limits. Standard techniques show that a CAT-enriched, finite quasi-limit sketch A generates a 2category with finite quasi-limits, which we call the 2-theory of A. As usual, there is a model of A in its 2-theory through which every other model factors by a quasi-limit preserving 2-functor. Similarly, A generates on order-enriched category with finite quasi-limits, which we call the ordered-theory of A. There is an order model of A in its ordered theory through which every other order model factors by a quasi-limit preserving order enriched functor. See 5.2 for an outline of the construction.

2.4 The formal quasi-limits for the pure typed iambda calculus. There are three important formal quasi limits in the sketch for the pure typed lambda calculus. 2 . 4 . 1 Application, which is written (f g) in the operational semantics, is realized by an operation "a" and written in the form a[f, g]. "a" is an arrow whose codomain (i.e., target) is the object "terms" but whose domain (i.e., source) is a certain formal quasi limit called appPullback. The picture for this formal quasi limit is: appPullback

p/ /pp

tet y r r r ' ~ N ~ / ~ / ~ u n Y p e s x t y p e ~ r l N ~ j e ~ s

types types Canonical arrows into appPullback will always be specified as triples of paths of arrows, , corresponding to the three arrows PPl, PP2, and PP3. The other two arrows into types will be taken to be fun and prl respectively following qq2, and the two inner 2-cells will be taken as identity 2-cells. A canonical arrow of this form exists exactly when there are 2-cells type • qql ~ fun ° qq2 and type • qq3 :::* prl * qq2. This describes appPullback, and a : appPullback ~ terms. In the picture, some objects and arrows are repeated for clarity. AppPullback can be thought of as a "triple quasi pullback". Read in terms of its intended interpretation by rewrite rules, this says that an element of appPullback is a triple (f, (o, x), g), where f and g are terms and ¢r and x are types, such that the type of f rewrite to the same type that fun(o, x) rewrites to, and the type of g and ff rewrite to the same type. This is the circumstance in which f can be applied to g. 2 . 4 . 2 Similarly, lambda abstraction, which is written ~.x. f in the operational semantics, is realized by an operation "lambda" and written in the form lambda[x, f]. "Lambda" is an arrow whose codomain is also the object "terms" but whose domain is the quasi product of the object of variables and the object of terms, called lamProd here. The picture for this formal quasi product is just: Prl pr2 var ~ lamProd ~ terms

111

so we can also write lamProd = var x terms when necessary. Canonical arrows into lamProd are of the form with no conditions. Note that there are then 2-cells prl • ql and pr2 • ~ q2. Again, these arrows are omitted from the basic graph.

2 . 4 . 3 Finally, 13-reduction is described in terms of a substitution function called "let". "Let" is an arrow whose codomain is also the object "terms" but whose domain is a formal quasi limit realizing the requirement that in a term let[x, exprl, expr2], the type of x has to match the type of expr2. Thus let : letPullback --->terms where letPullback is the following quasi-limit (where 2-cells are omitted for simplicity): Pl

letPullback P2 I

I

terms x terms 2.5

P var tjms

Prl

= terms

type

~

= types

R e w r i t e rules for type.

Each of the three constructions requires a rewrite rule to specify its type. These are given by the following three 2-cells. 2 . 5 . 1 The rule: if f : [~ --~ x] and g : or, then (f g) : x, says in the present terminology that type[a[f, g]] rewrites to "type(codomain(f))". Removing variables, this becomes the 2-ceU: a appPullback :- terms

types x types

pr2

,, types

2 . 5 . 2 The rule: if g : x and x E Var(~), then (~.x : o . g) : [~ -~ x], says in the present terminology that type[)~x : o . g] rewrites to [type(x) -~ type(g)]. Removing variables, this becomes the 2-cell: lambda lamProd = terms

"

Jl

var x terms

~

ivar x idterms

terms x terms

type x type

:

types x types

fun

type

= ,"ypes

2 . 5 . 3 Since we have introduced a "let" conslruction into the language, we must also specify its type, which is clearly the type of the last component of the let expression. This becomes the 2cell:

112

letPullback

let

"tenns

terms

--- types

2.6 Rewrite rules for terms: 13-reduction. There is only one specific rewrite rule for the general lambda calculus here: the rule for 13reduction. However, this makes use of the let operation which has several rewrite rules of its own which are rather intricate to represent in sketch form. Recall that ~ - reduction is the rule: (~.x. f) g ~ [g / x] f. Here this is written as a[lambda[x, f], g] ~ let[x, g, f]. To represent this as a 2-cell, consider the following confusing diagram: letPullback

sameType - ~

i .

var ~

..... . v

terms

~

-

-

~

',, terms

types x types

typeNNN~

~

types

n

Prl~

terms ~ype types

This illustrates a quasi cone over the diagram defining appPullback with vertex letPullback: , <m, o>] ~ eq(nat)[n, m] and eq(type)[x, ~7]. This rule is represented by the 2-cell vat x vat

i

=

,

~- (nat x type) x (nat × type)

w

(nat x nat) × (type × type) /~ [ eq(nat) x eq(type) bool x bool b ! ~ nd

Now, define two more formal pullbacks by cones giving the objects of equal variables and of unequal variables.

114

equalVar

one

ieqVar-

true

vat × vat

unequalVar

,,- bool

one

iuneqV~ ~ Var X v a r

false

,,

bool

2 . 7 . 2 First rule for let T o describe the 2-cell for the first rule, let[x, g, x] ~ g, consider the quasi pullback:

sameType

.

equalVar

var × var

var

var

=- terms

Here, sameTytre is the quasi pullback describing terms and variables o f the same type. Using the notation in the diagram, this gives a canonical arrow I.t(eq) = i(var) • pr2 • i(eqVar) • r2 : P(equal) --> terms and hence a canonical arrow : P(equal) --> letPullback. T h e desired 2-cell is the following: P(equal)

sameType

t. Pf(var) x terms Pf(Nat) x types

~ vat

sum x idtypes

119

Here, we use the definition, var = nat x type. Now, construct Q(e ) in the same way that Q(e ) was constructed, replacing false by true. The the last rule for let is represented by the following 2-ceU: Qe

q2 ~

sameType x larnProd "N~>

q2

sameType x vat x letPullback

sameType × lamProd

N,~dsamcType x idvar x let

sameType x lamProd / idsmaeType × lambda

idsameType x lambda

letPullback let

letPullback ........

3

/¢~ [let D, terms

T H E T Y P E D L A M B D A C A L C U L U S PCF.

3.1 T h e language PCF. An important example for theoretical work is the lambda calculus PCF (= Programming Computable Functions) which is part of the language LCF (= Logic o f Computable Functions) created by D. S c o t t , and described in Plotkin [15]. This lambda calculus has the following description. 3 . 1 . 1 Definition of PCF. The set of basic types is B = {bool, int}. There are the following constant terms: : bool tt, ff n : int, for n an integer bigger than or equal to O. suet : lint ---)int] pred : [int --> int]. : [int ~ bool], iszero if(bool) : bool --~ bool ---) bool --) bool if(int) : bool ---) int --~ int ---) int. fix(O : [1; --->x] ~ x for each "c ~ Type.

Note: We use the standard convention that --~ associates to the right; i.e., tl --->t2 - - > . . . --->tn = tl --> [t2 --> [ . . . ---> tn]. • • ]], as well as the standard convention that application associates to the left; i.e., fgh.., k = (((fg) h ) . . . k). 3 . 1 . 2 The operational s e m a n t i c s for P C F The rewrite rules for PCF consist of ii) ([3 - conversion) ((~,x. f) g) ==~[g / x]f. iv) a) h~k (h g) ~ (k g) b) h~ k only for f = succ, pred, or iszero, (f h) :=~ (f k)

120

v)

3.2

the following special rewrite rules for the constants: (succ n) ~ n + 1 for n bigger than or equal to 0 (predn+l) ~n for n bigger than or equal to 0 (iszero 0) ~ tt (iszero n + 1) ~ ff for n bigger than or equal to 0 if(o) tt f g ~ f where o = int or bool if(o) ff f g ~ g where o = int or bool (fix(x) f) ~ f (fix('c) f) for every type ~. The basic sketch for PCF.

The basic sketch for the typed lambda calculus is augmented by the addition of several new nodes and arrows. There are just two basic types, called int and bool. (The boolean basic type in spelled with an "e" to distinguish it from the node "bool" which is not included in the basic graph.) There are many constants, and in particular, the natural number object has an inclusion arrow into const. pi(var) appPullback var

suc

na(t) iNat

~..~-

0~i~'~it one

3.3

lamProd terms

l ~ ex int

let ~

letPullback

I type

g types ~

fun

types x types

Rewrite rules for type in PCF.

Next we determine the rewrite rules for the constants in PCF. They require many 2-cells, but are all quite simple. 3 . 3 . 1 The constants tt and ff have a type which rewrites to boole.

II, ff

one

iconst

= const -----------terms

3 . 3 . 2 The constants 0 and suc(n) have a type which rewrites to int. suc

one

0

Q

= nat

iNat

= const

iconst

~ terms

121

3 . 3 . 3 Succ and pred have a type which rewrites to fun[int, int]. SUCC,

one

pred

: eonst

lo'] I Vt. ~ . The set Term(G) of terms of type ~ described in 1.2 iii) is augmented by two extra rules: d) If t ~ Tv and g : ~ has the property that for all x : x E FV(g), t ~ fv(x), then

124

At. g : Vt. ~. Here, At. g is called a type abstraction of a term. e) If f : Vt. o and "ce T, then f[ x ] : [ z / t ] o. Here, f [ x ] is called a type instantiation. FV : Terms --->Pf(var) assigns to any term its set of free (ordinary) variables as in 1.3, and fv : Types --->Pf(TypeVar) assigns to any type its set of free type variables. (Pf(X) means the set of finite subsets of X.) Finally, [ x / t ] a means the result of substituting the type x for all occurrences of the type variable t in ~.

4 . 1 . 2 The operational semantics for the polymorphic lambda calculus. There is one additional rewrite rule for terms in the polymorphic lambda calculus involving the instantiation of a type abstraction. At. g[x] ~ [ x / t ] g where [ x / t ] g means substitute x for all occurrences o f t in g. Actually, substitution of a type for a type variable has to be defined both for types and for terms.

i) The definition of [ x / t ] o, where o is a type, is given by the rules: a) b) c) d) c) f)

[x/t]t~x I x / t ] s ~ s, i f s c t . ['c/t][o-)a']~[['c/t]c--->['c/t]a'] Ix/t] Vt.o~Vt.o [x/t]Vs.a=~Vs.([x/t]o)ifsc:tands~ fv['c]. [x/t]Vs.~Vu.([x/t][u/s]o)ifsctands~ fv[x], where u ~ fv[o] u fv[x], u ¢ t, and u ¢ s.

ii) The definition of [ "c/ t ] g, where g is a term, is given by the rules: a) b) c) d) e) f) g) h)

[ x / t ] (x : a) ~ x : [ x / t ] a, if x is a variable. [x/t](fg):=~(([x/t]f)([x/t]g)) [x/t](%x:o.g)~x:[x/t]o.[x/t]g [x/t]f[t]~f[x] ['¢]t]f[o]=:~([x/t]f)[['c/t]o] [x/t] (At.g) ~At. g [ "cI t ] (As. g) ~ As. ([ x / t ] g), provided s ~ t and s ~ fv(x) [ x / t ] (As. g) ~ [ x / t ] (Au. [ u / t ] g), provided s ¢ t and s e fv(x).

iii) There are also two additional rules for [g / x] f that have to be added to the rules in 1.3. a) [ g / x ] (At. f) ~ A t . [ g / x ] f b) [ g / x ] f i x ] ~ ( [ g / x ] f)[ z]. See [5] or [14] for examples of types and terms.

4.2 The basic sketch for the polymorphic lambda calculus. The basic sketch for the polymorphic lambda calculus looks as follows. There are no basic types and no constant terms. However, there are type variables, another type constructor, forall (corresponding to V), and several more term constructors; namely, LAMBDA, LET, typelet, and inst. The term constructors correspond to the operations A, [ x / t ]g, [ x / t ]o, and f[ a ] respectively. It is also necessary to consider finite sets of type variables and two free type variable operations, fv from types and FtV from terms.

125

Pf(var)

appPullback

lamProd

var

DA __ter

as

Pf(typevar) ~ y p e typevar .

-"

inst

instPullback

~

letPullback

:- ty ,es ~

tvoelet

1typevar forAll

LETPullback typeletProd

typevar × types

types × types

The actual skel~ch is much larger, of course. It contains all of the auxiliary structure of the sketch for the ordinaE.¢ typed lambda calculus as well as special auxiliary structure for the polymorphic lambda calculus that will be described below. 4.3 The formal quasi limits for the polymorphic lambda calculus. There are four new quasi limits in this sketch: LAMPullback, LETPullback, instProd, and typeletProd. 4 . 3 . 1 LAMPullback denotes the domain of the arrow LAMBDA,which corresponds to the nile: If t e Tv and g : ~ has the property that for all x : x E FV(g), t ~ fv(x) then At. g : Vt. ~. The condition on g can be described by an arrow FtV : terms ---> Pf(typevar) whose meaning assigns to each term g the set {t ¢- typevar I there is an x : x ~ FV(g) with t ~ fv(x)}. The rules for FtV are given below in 4.5.1. The construction of the diagram for LAMPullback requires an auxiliary structure for Pf(typevar) just like that for Pf(var) in 2.7.7. In particular, typevar has to have a decidable equality relation, so we assume that there is an initial typevar tO and a successor operation for typevar making typevar into a copy of the natural numbers. LAMPullback is the following formal quasi pullback: Lt~MPullback m typevar x terms

I idtypev~x FtV typevar x Pf(typevar)

false one

I isElem ,~ bool

4 . 3 . 2 LETPullback is easily described as a further formal quasi pullback from LAMPullback.

126

LETPullback

: typevar x types x terms <pr1, pr3>

LAMPullback

" typevar x terms

4 . 3 . 3 InstPullback is the object on which the instantiation of a LAMBDA terms at a type is defined. Such a term has a universally quantified type, so instPullback is the following formal quasi pullback: instPullback " terms × types I type x idtypes forAll x ida,pcs ~ types × types

typevar x types x types

4 . 3 . 4 TypeletProd is just the formal quasi product typevar x types × types. 4.4 Rewrite rules for type in the polymorphic lambda calculus. The types of a, lambda, and let are as in the pure typed lambda calculus. There are three new arrows whose composition with type has to be specified: LAMBDA, LET, and inst. Here there are the following rules: type[ LAMBDA[t_, expr_] ] ~ forAll[ t, type[expr] ] type[ LET[t_, ss_, expr_] ] ~ typelet[t, ss, type[expr] ] type[ i[expr_, t_.] ] ~ typelet[ type[expr][[1]], t, type[expr][[2]] ] Note that in the last case, expr has to be a universally quantified expression. These are represented by 2-cells as follows: LAMPullback = typevar × terms ~ idtypevar x type LAMBDA

typevar x types type

terms LETPullback

instPullback inst terms

~ forAll •" types

= typevar × types × terms I idtypevar x idtypes x type typevar × types x types //~ ~typelet

LET 1 terms

~

type

" types

= typevar x types x types ac typevar x types x types /~ ~typelet type

types

127

In the last diagram, ac is the associativity-commutativity arrow interchanging the last two factors. Also, type]et is given by recursive rules: typelet[t, x, t] typelet[t, x, s] s, provided s ¢ t typelet[t, x, fun[or, c']] fun[typelet[t, x, o], typelet[t, x, c']] typelet[t, x, forAll[t, ~]] forAll[t, a] typelet[t, % forAll[s, (~]] forAll[s, typelet[t, "c, c]] provided s ;~ t and s ~ fv(x) typelet[t, % forAll[s, or]] forAll[u, typelet[t, x, typelet[s, u, o]] provided s ~ t and s e fv(x) where u ~ fv[o] t_) fv[x], u ~ t, and u ~ s. It is evident that these can all be represented by appropriate 2-cells. 4.5

Rewrite rules for terms in the polymorphic lambda calculus.

4 . 5 . 1 In order to describe the rules for evaluation, the three free variables operations have to be described. Fv is as before, with two additional cases which pose no problems. For fv, everything is evident. For FtV, we need a special auxiliary arrow in the graph, fvSet : Pf(types) --->Pf(types)and the following two 2-cells for it: empty insert one :- Pf(types) ~types x Pf(types) [ fvSet e~mm pyt p t y ~ ~ ~ :~ fv x fvSet 1 Pf(types)

union

Pf(types) x Pf(types)

Then FtV : Terms ---) Pf(types) comes equipped with the following 2-celh FV Pf(type "iva r) terms :- Pf(var) .... = Pf(types) ~ ~ . ~ . . . . . . . . . . . . ~ / ~

l NSet Pf(types)

4 . 5 . 2 There is ordinary beta reduction plus the rule for instantiating a LAMBDA expression. i[LAMBDA[t, expr], s] ~ LET[t, s, expr]. The 2-cell for beta reduction is as in the pure typed lambda calculus. The 2-cell for instantiating a LAMBDA expression is as follows: The left hand side is constructed by taking the product of the 2-cell for type • LAMBDA with the object types. This is a pseudo cone on the diagram whose pseudo limit is instPullback, so there is a canonical arrow 'y : LAMPullback x types ---> instPullback. The left hand side is then inst • y. Similarly, there is an obvious canonical arrow 5 : LAMPullback x types --->LETPullback whose composition with LET represents the right hand side. Thus the desired 2-cell is inst • y ~ LET • 5. The rules for let have to be expanded to include the new terms and there are the many similar rules for LET. The new rules for let are the following: let[x[t], exprl, LAMBDA[s, expr2]] ~ LAMBDA[s, let[x[t], exprl, expr2]]; let[x[t], exprl, i[expr2, s_]] ~ i[let[x[t], exprl, expr2], s].

128

These just say that let goes inside a LAMBDA expression and inside an instantiation. They obviously correspond to 2-cells. There is a problem hidden here; the rules for let for the ordinary typed lambda calculus require that var has a decidable equality. This was arranged by declaring vat to be the product of nat and type and observing that both of these had decidable equalities. However, type is now considerably more complicated. In particular, it is no longer a free signature, but has non trivial two cells involving typelet. We assert without proof here that in the initial algebra for types, typelet does not occur in any normal form, and that these normal forms are the same as the initial algebra for the sketch for types without typelet. This is a free signature, based on the collection of type variables which is isomorphic to the natural numbers, and hence has a decidable equality. 5

T h e initial model o f an order e n r i c h e d sketch.

A model of a sketch assigns a set (or, more generally, an object in a category) to each node of the graph and a function (or, more generally, a morphism in a category) to each arrow in such a way that formal quasi limits are taken to actual quasi limits and formal commutative diagrams are taken to actual commutative diagrams. Actual quasi limits make sense in an order-enriched category. An ordinary category can be regarded as order-enriched by making each homset into a discrete poset. In this case, quasi limits coincide with ordinary limits. 5.1 Example. As an example, consider the category POSET of posets and orderpreserving functions. The quasi pullback of a pair of morphisms fl : A1 ~ B, and f2 : A2 --->B in POSET consists of the set { : fi(ai) < b for i = I, 2} with the pointwise ordering on such triples. Given a poset X and order-preserving functions Pi : X --->Ai, i = 1, 2, and q : X --->B, with order relations fi Pi < q, for i = 1, 2, then there is a unique order-preserving function r from X to the quasi pullback given by r(x) = (pl(x), q(x), p2(x)). Note that r need not be the only order-preserving function such that prAi r < Pi and prB r < q. However, it is the largest such function. The fact that r actually commutes with the projections shows that this quasi pullback is also a lax pullback. Other quasi limits in POSEr are constructed similarly. Note that P o s E r also has all ordinary limits 5.2 Construction of the initial model. The initial model, or term model, assigns to each node the set of directed paths of arrows in the graph from one to that node. The operation of an arrow is by juxtaposition of arrows. The 2-cells generate a preorder relation on such paths of arrows. A path of arrows is evaluated by finding a maximal element in this preorder bigger than it. In more detail, the initial model is constructed from the order-enriched theory generated by the sketch. Namely, start with any order-enriched sketch A; i.e., a graph IAI with specified formal quasi limits and 2-cells between specified paths of arrows. First construct the free ordinary category F(IAI) generated by the graph IAI. Its objects are the objects of IAI, its morphisms are (directed) paths of arrows in IAI and its composition is juxtaposition of paths of arrows. Make FOAI) into an order enriched category as follows: If Pl ° . . . °pn ~ ql ° . . . °qm is a 2-cell in A, then, for all paths h, k such that the following juxtapositions are defined, there is an order relation h°pl°...°pn°k -< h ° q l ° . . . ° q m ° k .

129

The reflexive, transitive closure of these relations on each homset F(IAI)(X, Y) makes F(1AI) into a preorder enriched category. Next, adjoin all finite quasi limits to F(IAI) with the required induced canonical morphisms and order relations together with isomorphisms between the objects of A that have been declared to be formal quasi limits and the appropriate ones of these new adjoined quasi limits. The formal canonical induced morphisms are identified with the new adjoined canonical morphisms. This is a countable completion process; i.e., it has to be repeated countably often. (Note: these quasi limits are not necessarily assumed to be enriched since that can lead to automatic divergence.) The resulting preorder enriched category with all finite quasi limits is called the preorder enriched theory, T(A), of the sketch A. The initial model of A is then given by the formula: IA(X) = T(A)(one, X); i.e., IA(X) consists of the paths of arrows in T(A) from one to X, regarded as a preordered set. The set of maximal elements of IA(X) is the set of normal forms of closed expressions of sort X. Nothing in general is guaranteed about these sets of normal forms. It may or may not happen that every expression is less than or equal to a maximal element. There may be loops (divergent terms) and Church-Rosser may or may not hold. This procedure, restricted to the basic sketch for PCF, yields the language for PCF and the order relations realize the operational semantics of PCF. Hence, in this case, each expression is the start of a unique chain terminating in a maximal element. 5.3 T h e o r e m . If I is the initial model for the pure typed k-calculus, PCF, or the polymorphic typed k-calculus in the category POSET, then I(terms) is the poset whose underlying set is the set of closed raw terms of the corresponding k-calculus and whose order relation corresponds to the rewrite ordering. Also, I(types) is the poset of all type expressions of the corresponding ~.-calculus with the rewrite ordering. Similarly, the initial model in SET has value at terms the set of normal forms of terms in the corresponding 7~-calculus and value at types the set of all types of the corresponding k-calculus.

6 Implementation of the initial algebra of a sketch in a conditional rewrite.rule programming language. We describe here an automatic compilation of a sketch into the rewrite-rule language Mathematica. The same principles would presumably work in other languages as well. An order-enriched sketch consists of nodes, arrows, formal finite quasi limits, and formal 2-cells. We describe in turn how each constituent is represented in Mathematica. 6.1 Nodes. For each node in the graph there is a predicate nodoQ on the universe of all Mathematica expressions. 6.2 Arrows. For each arrow of the form: arrow : dom ---> cod, in the graph, there is a rewrite rule codQ[arrow[x_]] 6.3

:=

True

/;

domQ[x]

==

True

Finite quasi limits. For each finite quasi limit in the sketch, there is a rewrite rule limitnodeQ[xl

,

nodelQ

[xl]

, ==

True

. . . . . . . . .

nodenQ [xn]

==

True

xn_]

:=

and and and

True

/;

130

comp[l, comp[l,

II [xl] 21 [xl]

== ==

xl x2

comp[i, j] [xi] . . . . . . . . .

==

xj

eomp[n,

== xn

and and and

n] [xnl

and and

6.4 2-cells. For each formal 2-cell fI " . . • " fm ~ gl " - .. " gn, where the two paths of arrows start at the node dom and end at the node cod, there is a rewrite rule f l [ . . . [fm[x__] ...]

:=

gl[... [gn[x] ...]/;

domQ[x]

==

True

These constructions are applied to the sketches for PCF and the polymorphic lambda calculus in the papers [10] and [11]. 7 References. [1] M. Barr and C. Wells, Categoriesfor Computer Scientists, Prentice Hall, 1990, [2] C. Ehresmann, Esquisses et types de structures algrbriques, Bull. Instit. Polit., Iasi XIV (1968), 1-14. [3] P. Gabriel and F. Ulmer, Lokal Prr.sentierbare Kategorien, Lecture Notes in Math., 221, Springer-Verlag, New York, 1971. [4] L-Y. Girard, Une extension de rinterpretation de Grdel a ranalyse, et son application a l'elimination des coupures dans ranalyse et la theorie des types. In J.E.Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, North-Holland Pub. Co., Amsterdam, London, 1971, 63-92. [5] J.-Y. Girard, Proofs and Types, Translated and with appendices by P. Taylor and Y. Lafont, Cambridge University Press, Cambridge, 1989. [6] J . W . Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Mathematics 391, Springer-Verlag, New York, 1974 [7] J.W. Gray, The Category of Sketches as a model for Algebraic Semantics, in Categories in Computer Science and Logic, J. W. Gray and A. Scedrov, Eds., Contemporary Mathematics 92, Amer. Math. Soc., Providence, R. I., 1989, 109 - 135. [8] J . W . Gray, The Integration of Logical and Algebraic Types, in Categorical Methods in Computer Science with Aspects from Topology, H. Ehrig et al, Eds., Lecture Notes in Computer Science 393, Springer-Verlag, New York, 1989, 16 - 35. [9] J.W. Gray, Executable specifications for data type constructors, in preparation. [10] J.W. Gray, PCF in Mathematica, in preparation. [11] J. W. Gray, Poly Lambda in Mathematica, in preparation. [12] G. M. Kelly and R. Street, Sydney Category Theory Seminar, Lecture Notes in Mathematics 420, Springer-Verlag, New York, 1974. [13] M. Makkai and R. Pare, Accessible Categories: The foundations of Categorical Model Theory, Contemporary Mathematics 104, Amer. Math. Soc., Providence RI, 1989. [14] B. Pierce, S. Dietzen, S. Michaylov, Programming in higher-order typed lambda-calculi, Technical Report CMU-CS-89-111, Carnegie Mellon University. [15] G. D. Plotkin, LCF considered as a programming language, Theoretical C o m p u t e r Science 5 (1977), 223-255. [16] J. C. Reynolds, Towards a theory of type structures, in Colloque sur la Programmation, Lecture Notes in Computer Science 19, Springer-Verlag 1974,408-425. [17] A. Stouten, Substitution Revisited, Theoretical Computer Science 59 (1988), 317 - 325.

This paper is in final form and will not be published elsewhere.

First Steps in Synthetic Domain Theory J. M. E. Hyland Department of Pure Mathematics and Mathematical Statistics 16 Mill Lane, Cambridge CB2 1SB (GB)

1 1.1

Introduction Aims of synthetic

domain

theory

Domain theory is the study of various concrete categories C typically of (directed) complete partial orders in which constructions fundamental to the analysis of computing can be performed. • One can take fixed points of endofunctions X --~ X, so as to give meaning to functions defined by general recursion. • One can find fixed points for various operations C -4 C in order to provide interpretations for recursively defined types. Domain theory is a well developed body of mathematics, in which the stress is on limits of increasing sequences. Continuity serves in the theory as a substitute for effectivity. This paper is concerned with an approach to programming semantics of a different flavour. The motivating slogan is "domains are sets". (More exactly but less memorably, "domains are certain kinds of constructive sets".) An investigation of the general kind presented here was proposed by Dana Scott in a talk at a meeting of the Peripatetic Seminar on Sheaves and Logic in Sussex in 1980. He had in mind the example of Synthetic Differential Geometry where generalized manifolds are treated as (special kinds of) sets with the result that the development of the basic theory becomes highly intuitive: and he asked for a treatment of domain theory in a similar spirit. Initial progress was slow, but it now appears that the major conceptual advance was made by Scott's student Rosolini in his thesis [10]. There he took the recursively enumerable subobject classifier E as the lynch-pin of the theory. Rosolini also described axioms in the internal logic which unified features of the effective and recursive toposes, and took the first steps towards formulating a synthetic theory of computation. (Mention should also be made in this context of Lawvere's student Mulry who had identified the recursively enumerable subobject classifier, in the course of his study of the recursive topos, see [6].) In a theory of computation, effectivity must be analyzed directly. The category theoretic approach is to consider intrinsic structure of objects within a category (see the discussion in [7]); and as envisaged by Scott and Rosolini, this exploits the analogy with continuity. The result is the development of a kind of coding-free recursion theory, and it is this that is stressed in this paper.

132

1.2

Scope

of the

paper

The study of Synthetic Differential Geometry has two complementary aspects: a synthetic (or axiomatic) aspect and a semantic aspect (involving categorical models), and so it is also with Synthetic Domain Theory. In this paper the emphasis is on the first of these two aspects. The approach taken is to attempt to axiomatize effectivity in a categorical framework by means of properties of a classifier E of 'semi-decidable' subsets. (One might regard E as playing the same central role in Synthetic Domain Theory as the synthetic line R does in Synthetic Differential Geometry.) The reader should perhaps be aware of limitations on the scope of the enquiry. We only reach the first aspect of domain theory: the existence of fixed points of endofunctions. Furthermore the word domain is used only in the general sense associated with Scott (though n o t as synonomous with what are usually called Scott domains) as distinct from the sense associated with Berry. This has a number of consequences. Computations may run in parallel and so the 'semi-decidable subsets' are closed under unions (Axiom 7). Also, for good objects the intrinsic order on function spaces is pointwise. Properties of Scott's topological notion of domain are reflected in all of the paper from section 3 onwards, though some aspects of the later sections are more widely applicable. Perhaps this is a drawback of the presentation; what we have is far from being an axiomatization of domains in the general sense. It remains a challenging problem to produce a synthetic theory of (the various flavours of) stable domains. 1.3

Background

assumptions

We shall work within a n o n - t r i v i a l category of sets S, and need to state what we assume about this category. Of course one can assume that it is a topos, but the full structure of a topos will not play any part in our axiomatization. Hence it seems important to give some more precise indication of the kind of situation which one should have in mind. We assume that we work in a category with properties enjoyed by (amongst others) the category of modest sets within the effective topos. In addition to the tacit assumption of non-triviality, the assumptions which we have in mind are as follows: 1. S is a locally cartesian closed subcategory of a topos E (in the internal sense); 2. S has a natural number object hi; 3. S is the category of separated objects for a topology j on a subpretopos of the topos; 4. S is a small category contained within and complete relative to the category of j-separated objects of the topos. A discussion of (1) for the particular case of the modest sets in the effective topos can be found in Hyland [2]. It seems best in view of some applications not to assume in (2) that N is the natural number object in the ambient topos. But we have the usual structure

I~N-~N satisfying Lawvere's 'initial algebra' universal property. And in view of (3) we may assume Freyd's formulation of a natural number object in terms of a coproduct 1+ N = hi IN -~ 1. As regards (3), Carboni and Mantovani have characterized and coequalizer N

133

the categories which arise as j-separated objects for a topology j on a pretopos. Such a category • is regular (in the logical sense - it has finite limits and stable (regular epi, mono) factorization); • has stable (epi, regular mono) factorization; • has finite stable coproducts; • has all 'quotients' of equivalence relations and these are quasi-effective (that is, the natural map from an equivalence relation to the kernel pair of its quotient is an epimorphism). There is some analysis of (4) in Hyland, Robinson, Rosolini [3] and a fuller discussion in Robinson [9]. T h e full force of this assumption does not seem necessary for the theory developed here; and it does not hold in some of our models. It is important however if we wish to model strongly polymorphic type systems. We give some examples of toposes in which we can find at least some of our background assumptions satisfied, and which support models for much of the synthetic domain theory which we shall describe. 1. Suitable realizability toposes. It seems that these may not all model the full theory which will be described below, but many do. For example * The effective topos. (Historically this was the motivating example.) • Toposes based on domain theoretic models for the lambda calculus. 2. Some toposes based on other notions of functional interpretation (modified realizability, Dialectica Interpretation). I have only made "back of an envelope" calculations in these cases, but they should be a good source of counter examples. 3. Topological toposes. These include • Johnstone's topological topos see [4]. • Scott toposes in the general sense indicated in [10]. 4. Recursive versions of topologicM toposes. The recursive topos of Mulry. (This was the first topos in which an r.e. subobject classifier was identified.) (The natural example is analogous to the simplest "well-adapted" models for synthetic differential geometry. 1.4

Conventions

and

notation

We shall use the usual set-theoretic language appropriate within (pre-)toposes, and refer to subobjects as subsets and to regular epis as surjections. A pretopos also provides good notions of (finite) intersections and unions of subobjects; and we suppose enough completeness to give us small (intea'nal) intersections and unions. Note however that the existence of two factorization systems already adds a nuance to the set theory which reflects the j-modality. The regular monomorphisms are j-closed subobjects, while the epimorphisms are j-dense maps. We mention a few standard category theoretic conventions. If A is an object of a category (with terminal object), then we shall write A: A --* 1 for the unique map to the terminal object 1. The identity map on A is denoted by 1A: A --* A, and the subscript will be dropped wherever possible. Also we use some non-standard notation. If

134

a: 1 --* A is a point or element of A, then I shall write k,: X --* A for the map of the form X x Jl a )A. Finally if u: A --~ C and v: B --+ C are maps, we write [u, v]: A + B ~ C for the induced map from the coproduct. 1.5

Acknowledgements

Recently many people have been taking an interest in aspects of Scott's challenging propose.Is to use domains in toposes as the semantics of programming languages. T h e work reflected in this paper was done as part of the EC Esprit BRA project 'Categorical Logic in Computer Science', and I have benefitted from the critical interest of Samson Abramsky, Eugenio Moggi, Andy Pitts, Pino Rosolini and Paul Taylor. Independently I owe intellectual debts on the one hand to Dana Scott and on the other to my student Wesley Phoa. Others from whom I have learnt in the course of the development of the effective topos as a context for programming semantics include Peter Freyd, Phil Mulry, Gordon Plotkin, and Edmund Robinson. Finally I acknowledge the use of Paul Taylor's useful diagram macros in the production of this paper.

2 2.1

Basic t h e o r y of partial m a p s Semi-decidable

or E-subsets

We follow Rosolini in making the notion of a E-subobject of an object the cornerstone of the theory. These E-subobjects should be thought of as recursively enumerable or (better) semi-decidable subsets, and we shall refer to them as E-subsets. Our first assumption is that this notion is classified. A s s u m p t i o n 1 We assume that we have an object E equipped with a subobject t: 1 --* E. The pullbacks o f t : 1 --~ E are called E-subsets, and we assume that t: 1 ~ E is a generic E-subset in the sense that any E-subset A --+ X appears in a pullback diagram

X

a

>E

T }

A

>1

f o r a unique map a: X ~ E. W e shall write A C_r, X f o r A is a E-subset of X .

This assumption has the obvious consequences. P r o p o s i t i o n 2.1.1 The collectiort o f E-subsets satisfies the basic closure properies: (i) For any object X , X C z X , that is the m a z i m a l subset 1: X ~ X is a E-subset. (ii) The pullback of a E - s u b s e t is a E-subset. Note that the object E x internalises the notion of a E-subset of X; its global sections correspond to the E-subsets.

135

2.2

The

lift functor

Now we can exploit the b a c k g r o u n d a s s u m p t i o n t h a t S is locally cartesian closed to obtain f r o m the classifying m a p 1 --~ E a further piece of structure, namely a E-partial m a p classifier. In the usual way, we let _l_(X) --~ E be II~(X --* 1) and obtain a pullback

±(x)

> E

X

>1

T h e m a p r / = r/x: X ~ .I_(X) classifies partial m a p s whose d o m a i n is a E-subset. We refer to these as E-partial maps; we let X ~r~ Y denote u )-],-partial m a p from X to Y. If A C r , X and u: A --~ Y is a E-partial m a p X ---'r~ Y, then there is a unique m a p fi: X ~ A_(Y) such t h a t

X

> A_(Y)

A >Y is a pullback. In particular, .1_(1) = E and ~: 1 --~ A_(1) corresponds to t: 1 --~ E. T h e object A_(X) is called the lift of X and is sometimes written X±. 3_ extends to a functor in a n a t u r a l way. If u: X ~ Y then _l_(u) = u± classifies the E-partial m a p X± --'r. Y defined on X C r. X± as u. F u r t h e r m o r e it is easy to see t h a t 2. is an S-functor. Finally note t h a t the lift functor has familiar preservation properties. P r o p o s i t i o n 2.2.1 The lift functor .1_ preserves connected limits in both the external and internal senses. We can of course iterate the lift functor, and so in particular can obtain a sequence of objects E~ defined by Eo = 1; E~+I = _l_(~,,). Clearly the sequences of m a p s 1

n3E n~E2...~,~_I n~E,~

classify sequences of subobjects A0 C~ A1 CE A2 ... A,,_I C_r X of an object X where each Ai is a E-subset of Ai+l. 2.3

The

lift monad

Our first axiom states the closure of (representatives for) E-subsets u n d e r composition. A x i o m 2 I f A is a T-subset of B and B is a E-subset of X , then A is a T-subset of X. T h e r e is an immediate consequence.

136

P r o p o s i t i o n 2.3.1 The collection of ~-subsets of an object is closed under (finite) intersection. C o r o l l a r y 2.3.2 If A C_~.B, then E A is a retract of E B. In the language of Rosolini [10], Axiom 2 says that the E-subsets form a dominance classified by g: 1 ~ ~. Axiom 2 has a useful alternative formulation in terms of the lift. P r o p o s i t i o n 2.3.3 Given the classifying map t: 1 ~ E for E-subsets, the following a~'e equivalent (i) E-subsets form a dominance, ~hat is Axiom 2 holds; (ii) the composite (1 ' , N " ,~±) is a E-subset; (iii) there is a (necessarily unique) natural transformation t~: (A-)2 --~ I such that (l,r/,/~) is a monad. Externally we have the usual 'subset ordering' or inclusion ordering on the Y].subsets of X. Suppose that a: X --* E and b: X --+ ~ classify A Cj: X and B C r. X respectively. The inclusion relation induces a relation on classifying maps: if A C B, then we write a C~ b. It is a consequence of Axiom 2 that we can identify in S a monic representing the subset order on ~. We denote such a subobject of E 2 (which automatically exists in an ambient topos) by D---* y],2 or C-~ ~2 as appropriate. (It is convenient notationally to distinguish between the subobjects C_ and _Dof ~2 which are isomorphic via the twist map.) We have two m a p s / ( N ) : ~± ~ Yl and /~I:E± --* E and these induce a map (#I,&Z(E)):Ej. --4 I~ x P,. It is a further consequence of Axom 2 that intersection is also represented in S. We let gl: P. x 1~ ~ 21 be the classifier of the E-subset (t, t): 1 --* ~ x E. P r o p o s i t i o n 2.3.4 (i) A ,nap (a, b): X ~ E 2 factors through (/zl, .I_(E)) if and only ira C_g b; (ii) (#1, &Z(~)): ~x ~ E 2 represents the 'subset order' C on ~; (iii) (#1, _1_(~)): E± ~ E 2 is the equalizer of the maps fst, N: E 2 --* E (iv) (E, T, C_,n) forms a (meet) ~emilattice. Finally it is easy to see that the monad (_L, r/,/z) is strong. Hence the second part of 2.3.5 follows from the first. P r o p o s i t i o n 2.3.5 (i) (E± " ,P,) is the free _L.algebra on 1. (ii) Each ~ x has the structure of a &z-algebra. 2.4

The

intrinsic

order

The subset ordering on ~(gx) gives rise to a preorder on any object X of S. D e f i n i t i o n 1 The intrinsic pre-order on an object X is the relation _< which appears in the pullback

T l


C ~x

137

Using the internal logic, the definition says that

S ~ (z < ~) ~

(VR e ~ x . z ~ R ~ v • R).

Generally the pointwise preorder on function spaces need not coincide with the intrinsic preorder (see the discussion in [8]). We define the pointwise preorder "~ on a function space B A in the internal logic by stipulating that S ~ ( f _ g) ¢==~ (Va • A.f(a) < g(a)). The intrinsic preorder on an object is analogous to the topological specialization order used in algebraic geometry. (Indeed the latter is a special case of the former.) Properties of the intrinsic preorder are most rapidly established using the internal logic. We now state a number of properties which we shall need and whose proofs are quite trivial in these terms. (I cannot resist remarking however that the intrinsic order seems to be a distraction. I would rather avoid reference to it where at all possible, and am distressed at my failure to do so more successfully.) We shall eventually have an axiom ensuring that the intrinsic preorder on an object of the form ~]A coincides with the inclusion ordering. One entailment is however both automatic and easy. P r o p o s i t i o n 2.4.1 R < S entails R C S in ~A. P r o p o s i t i o n 2.4.2 Internally any function preserves the intrinsic preorder:

S ~ V f • B AVa,c • A(a < c ~

f(a) < f(c)).

(And hence Externally functions preserve the intrinsic order.) P r o o f Essentially clear as f induces a map c ~ x ~ c ~r.

r3

C o r o l l a r y 2.4.3 The intrinsic preorder on a product is the product of the intrinsic preorders. Note that the pointwise preorder -< has a simple alternative characterization. P r o p o s i t i o n 2.4.4 We have the following in the internal logic:

S ~ Vf, g • BA(f ~_g ¢==~(VS • E~B)f-'(S) _Cg-'(S)). Preorders are rather a bore. It is convenient to work with objects for which the intrinsic preorder is in fact an order. D e f i n i t i o n 2 For any object X there is a natural map X --* ~ s x . An object X is a ~-spacc just when this map is a monic. X is extensional just when it is a regular monic. P r o p o s i t i o n 2.4.5 If X is a ~.space then the intrinsic preorder is an order. In what follows we shall often tacitly assume that we are working with ~-spaces.

138

2.5

Objects

with

bottom

There are clearly a n u m b e r of distinct notions of an object with a least or b o t t o m element in the intrinsic order. In conformity with the 'algebraic spirit' of the categorical axiomatization, it seems best to work with a rather strong notion of an object equipped with a bottom element. As a conceit I introduce the definition before Axioms 3 and 6 which ensure that there really are b o t t o m elements in these objects. D e f i n i t i o n 3 An object X (equipped) with bottom is an algebra l ( X ) ~ X for the lift monad. A strict map between such objects is a m a p of algebras. Note that as we have presented the definition~ having a b o t t o m element is an additional piece of structure. However if we restrict attention to E-spaces, there is at most one such stucture on any X , so we m a y regard having a b o t t o m element as a property. Since we m a y tacitly assume that the objects with which we are dealing are E-spaces, we shall not make any fuss over this. There is considerable interest in the free algebras for the lift monad, that is, in objects which are themselves lifts of some object. We say that such an object A± is a lift. We already know that maps intd such an object classify E-partial maps. But as .I_2(A) " ,A.(A) is a free .l_-algebra, we also know about strict maps from such an object into objects with bottom. In principle we want to regard these m a p s as m a p s preserving ±, but clearly one has to say this in a positive way as one cannot yet identify the bottom elements! One particular case of interest is that of maps to an object of the form E x. L e m m a 2.5.1 Suppose that W C_E X x A_(A). Then the diagram

X x I~(A)

1 X#A

> X x J_(A)

f

W

W

is a pullback if and only if W C_ X x A. P r o p o s i t i o n 2.5.2 Take a map u: A_(A) --* E x and let U C_r X x .l_(A) be the subobject classified by the transpose fi: X × A_(A) --~ ~ of u. Then u i~ strict if and only if UCXxA. 2.6

The

empty

set

As things stand it is possible that E = 1 and that the collection of E-subsets is trivial. The next axiom ensures that this is not the case. A x i o m 3 For any object X , the empty set is a E.subset of X . P r o p o s i t i o n 2.6.1 The following are equivalent: (i) For any object X , the empty set is a E-subset of X . (it) The empty set is a E-subset of 1.

139

(iii) There .i8 a map f: 1 --* ~ distinct f r o m t: 1 --* ~ in the sense that the diagram t

1

>~

T,

0

>1

is a pullback.

One simple consequence of Axiom 3 is that decidable subsets are semi-decidable. As usual a decidable subset of an object X is a subobject A C X which is complemented: there is an isomorphism A + B -~ X inducing A C_ X. P r o p o s i t i o n 2.6.2 Decidable ~ub~et~ are ~-sub~et~. P r o o f Decidable subsets are classified by maps into 2 = 1 + 1. As we have the pullback

f]

2

1

1 they are also classified by maps into ~].

>~

1 []

On the basis of Axiom 3 we start to get some recognizable structure. Note first that i.(0) = 1 and the m a p f: 1 -~ E is equal to 2 ( 0 -~ 1). The diagram ~]

A

> ~±

1 >~ is automatically a pullback. Thus we caal identify three distinct points of E l : i the point (TT~) • t of P,± which is the classifying m a p for 1 C_z 1 C z 1; • the point 77. f = ( f ± ) . t of :E~. which is the classifying m a p for 0 Cz 1 C~ 1; • the point .L(0 ~ P,) of ~ x which is the classifying map for 0 C~ 0 C~ 1. The two m a p s .l_(E): ~± ~ Yl, and #i: EL ---+ E are clearly distinct as they classify Ci: ~]± and 1 C~ ~ i respectively. W h a t ' s more, we can calculate the pullbacks of f along these maps. We get

~.t

/il

>~] and

1

77

140

where 1 --+ E± is _L(0 --+ E) and E --* E± is .1-2(0 --* 1). The left hand diagram is a pullback because _L preserves pullbacks. For the right hand one, note that X ~ ~E± ~' .~E factors through f if and only if u classifies subobjects 0 C_~ U C_g X; mad that this happens if and only if u = ( i f ) - (.I_E) • u. T h e reader will now readily see that the collection of objects E0, El, .... and maps between them constructed from the maps t, f , # by pullbacks and composition mirrors exactly the simplicial category of all n o n e m p t y finite ordinals and order-preserving maps.

3 3.1

Some dual structure Co-E-subsets

and

the

co-lift

functor

Further structure of E can be conveniently described in terms of a notion of co-T-subset dual to that of E-subset. We assume that f: 1 --~ E classifies these subsets which we think of as the complements of semi-decidable subsets. A s s u m p t i o n 4 Call the pullbacks of f: 1 ---* E co-E-subsets. We assume that f: 1 --~ E i~ a generic co-E-subset in the sense that any co-E-subset A of X appears in a pullback diagram X

a

>E

A >1 for a unique a: X --* E. We shall write A C~ X for A is a co-T-subset of X . Note that now E x also internalises the notion of co-E-subset. In fact more is true. P r o p o s i t i o n 3.1.1 There is a bijective correspondence between E-subsets and co-Tsubsets, where a co-E-subset of X corresponds to the E-subset with the same classifying map X ---* E. This correspondence reverses the inclusion orders on co-T-subsets and ~-~ubsets. Note that Axiom 3 enables us to 'see' the b o t t o m element in a lift A±: it is the m a p k.t: 1 --~ A± lying in the unique pullback of form 1

> A±

0

>A

As a consequence of Proposition 3.1.1 one can deduce a more intuitive form of Proposition 2.5.2. P r o p o s i t i o n 3.1.2 A map u: _L(A) --* E x is strict if and only if u. ks.: 1 --* E x names 0 C_E X . (That is, u is strict if and only if it preserves _L.)

141

From the classifying map f : 1 ~ E we can obtain structure dual to that of 2.2. Thus there is a co-~-partial map classifier. This consists of a map ~: X --* T ( X ) such that if A Cg X and u: A ~ Y, then there is a unique map fi: X --* T ( Y ) such that

x

A

> T(Y)

u

>Y

is a pullback. T h e object T ( X ) is called the co-lift of X and is sometimes written XT. The co-lift -F extends to an S-functor; and T enjoys the standard preservation property. P r o p o s i t i o n 3.1.3 The co-lift functor T preserves connected limits in both the external and internal senses. As things stand, there is some small interaction between the lift and co-lift. We have T(1) ~ E ~ 1(1). However if we start iterating T there is no reason yet to suppose that we shall obtain a sequence of objects (and maps) isomorphic to the Ens (and the face and degeneracy maps) of 2.6.

3.2

T h e co-lift m o n a d

Next we introduce an axiom dual to Axiom 2. A x i o m 5 If A i~ a co-E-subset of B and B is a co-E-subset of X , then A is a co-~subset of X . As in the case of Axiom 2 there is an immediate consequence. P r o p o s i t i o n 3.2.1 The collection of co-~-subsets of an object is closed under (finite) intersection. C o r o l l a r y 3.2.2 If A C~ B, then E a is a retract of E B. Furthermore as co-~-subsets form a dominance classified by f: 1 --* E, the duals of Propositions 2.3.3, 2.3.4 and 2.3.5 hold. P r o p o s i t i o n 3.2.3 Given that f: 1 ~ ~ classifies co-E-subsets, the following are equivalent (i) co-~-subsets form a dominance, that is, Axiom 5 holds; (ii) the composite ( I ~ / E ¢,ET) is a co-~-subset; (iii) there is a (necessarily unique) natural transformation v: (T) 2 --* T such that (T, (, v) is a monad. We have already seen in 3.1.1 that the bijective correspondence between ~-subsets and co-E-subsets reverses the (external) inclusion orderings. Let a: X ---* E and b: X E classify A C~ X and B C_g X respectively; then we have A _DB if and only if a C_~cb. Thus there is essentially just one inclusion ordering on E. Using Axiom 5 we can find a representation of the subset order on Y] dual to that of Proposition 2.3.4. We have two maps T(E): ~T ~ ~ and/I"1: ET --/" E and these induce a map ( T ( ~ ) , v,): ET ---* EX ~. It is a further consequence of Axiom 5 that intersection of co-~-subsets is also represented in S. However this does not necessarily correspond to union of E-subsets. Hence we write V: ~ x ~ ~ ~ for the classifier of the co-E-subset (f, f): 1 ---* E x E.

142

P r o p o s i t i o n 3.2.4 i/ a C~ b;

(i) A map (a, b): X ~ ~ /actors through (T(~), vl) if and only

(ii) (T(S), ~,): ~T -, r.~ repre,e.ts the 'subset order' C_ o. S; (iii) (T(S),~,,): ST --~ S 2 is the equalizer of the maps sad, V: }32 ~ S;

(iv) (s, J_, c_, v) [ o ~ , a (joi,~) ,e,~ilaUice. For completen~s note that the monad (T, ~, v) is strong, and we have the dual of 2.3.5. P r o p o s i t i o n 3.2.5 (i) (~T ~ '~) is the free T-algebra on 1. (ii) Each ~ x has the s~ructure of a T-algebra. As a consequence of 3.2.4 we have an isomorphism T(Y].) ~ A_(~). Indeed now if we start with 1 and iterate T we do get a sequence of objects (and maps) isomorphic to those of 2.6. A complete collect,_'on of identifications between the presentations in terms of / and T results. As an example note that • T(*): T 0 ) -~ T G ) ~orresponds to ~s: r, -~ X(E). P r o p o s i t i o n 3.2.6 (i) The co-lift functor T preserves E-subsets. (ii) The lift functor ± preserves co-~-subsets. P r o o f Suppose that u: X ---r ~ classifies U C_z X. Consider the diagram T(X)

T(u)> T(}]) ~ .1.(~)

T(U)

> T(1) -~ ~

> S

> 1

where we use some of the identifications just mentioned. The left hand square is a pullback since T preserves pullbacks; and the right hand square is the classifying pullback diagram for ~ C_z l ( ~ ) . Hence the composite T(X)

T(u) . > I ( S ) ~:E~

> S

classifies T(U) ___=T(X). This proves (1), and (2) is just the duN.

U

Finally note that we have the d u n definitions of an object equipped witt~ a ~op clement (that is, the algebras for the co-lift monad), and of the co-strict maps between them. Objects of the form AT (that is, the free algebras) are co-lifts. The d u n s of the result of 2.5 hold.

4 4.1

Finitary A higher

domain order

theory

axiom

After all the above axioms it still remains possible that t and f are the two distinct maps 1 ~ 2, so that the X~-subsets, the co-E-subsets and the decidable subsets all coincide. This degeneracy is avoided by the main finitary axiom which genuinely exploits the higher order structure in our (locally) cartesian closed category S. By 2.6 there is a (monic) map [t,f]:2 --* E.

143

A x i o m 6 The map l[t,ll:

E z __~ E 2

represents the inclusion order on E.

Let us spell out what this means in concrete terms. Write ev and e± for the evaluation m a p s lt: Ez --* E and 1I: E z --4 E respectively. T h e n e± C eT and there is a commutative diagram Ez

1

E
E snd

with the following universal property. Suppose that a C_ b: X -~ E; then there is a unique m a p (a; b): X ~ E z such that

/,l E
E

E~ eT

con-~-nu t e s .

Amongst m a n y automatic pullbacks EZ

ev

>E

(1; kT) E

>1

should be noted. This exhibits E as a E-subset of E z. As an application of the universal property, take/~ C A_(E): E2 ---* E; thus there is a m a p (#; A_(E)): E2 ---* Ez. It is easy to check that this m a p is inverse to the m a p E z -* E± which classifies 1 Cz E C~ Er~. Thus the content of Axiom 6 is given by isomorphisms E ~ : ~ E ± andduMty E ~ E T . Modulo these isomorphism we can identify m a n y maps. We give some examples: • r/~ o f: E --~ E± corresponds to (k±; kT): E --~ E:~; • r/z: E ~ E± corresponds to (1; kT): E --~ Ez; •

A.(f): E --* E± corresponds to (k±; 1): E --~ Ez;

• tz: E± ~ E corresponds to e±: E ~ ~ E; •

_k(E): E± --* E corresponds to

eT:

E z --~ E.

144

4.2

Properties

of the intrinsic order

One consequence of Axiom 6 is that it identifies the intrinsic order and the inclusion order. T h e o r e m 4.2.1 The intrinsic order coincides with the inclusion order on all objects of the form E x. P r o o f The diagram

(EA) 2

>

(~2)(~C~))

T clearly commutes. But by Axiom 6 (EA) z is the subset order on X]A. Hence it factors through the pullback of (E(E(~a))) 2 which is the intrinsic order. Thus the subset order entails the intrinsic order. But we already have the converse which was Proposition 2.4.1. o C o r o l l a r y 4.2.2 .L is the least element in Aj.. P r o o f In E, _k is the least element in the inclusion order and hence is least in the intrinsic order. Consider the map E x A_(A) --~ A.(A) appearing in the unique pullback of form E x _L(A)

> .L(A)

T

A A w~ h~ve f _< t in X: and so ( / x 1) _< (t x 1): _L(A) --* E x .1_(.4). Hence by composition []

k.t _< 1: A_(A) ---, .L(A).

4.3

E - s u b s e t s o f lifts a n d c o - l i f t s

We return to analyze the general behaviour of maps A_(A) --* E x. In fact it is simpler to consider the case of a map T(A) ---+ E x. We know by the dual of 4.2.2 that "1- is the greatest element of T(A). But maps preserve the intrinsic order; hence if we let E E x be the image of T in E x , s is the greatest element in the image in the intrinsic and hence in the subset order. It follows that if s: X --~ E classifies S C~ X , then T(A) ---+ E x factors through the standaxd split monomorphism X3s -4 E x. Clearly the resulting map q-(A) --* E s is co-strict. Thus we have bijective correspondences: T(A) ~ E x

S _C~ X

and a co-strict map

T(A) --~ E s

145

S C~ X

and

U Cz S x A.

As a consequence we derive the following connections between the lift and co-lift. P r o p o s i t i o n 4.3.1 S T(A) ~ -L(Y]~A) and dually y].l.(A) _~ phism~ being natural in A. Proof

T(~A), both

these isomor-

The first isomorphism is a consequence of the bijective correspondences: X -* ~-r(A)

T ( A ) .-e S x

S Cr. X

and

U G~ S x A

SC_I:X

and

SxA~E

S C~ X

and

S-~ N A

x --, ± ( s A) all natural in A. The second isomorphism is just the dual of the first,

rn

T h e attentive reader will have noticed that the argument from 4.2.2 onwards goes through simply under the assumption that the subset and intrinsic orders coincide. But Axiom 6 is a special case of Proposition 4.3.1; and so in fact it is equivalent to the coincidence of the two orders. 4.4

l:Lic&s Theorem

One can regard Axiom 6 as a weak version of the undecidability of the halting problem. Hence we can now derive a little bit of non-trivial recursion theory. P r o p o s i t i o n 4.4.1 For any objects A, B, and C, we have

(B + C) ~(A) ~- B ±(A) + C ±(A).

146

Proof

Let b: B + C -* E classify B Cz B + C. Consider the diagram

/

(B + C) ±(A) ,,,

B ±(n)

/ >

B

E ±(A)

1

> B+C

> E

>1

where the right hand face is the pullback exhibiting B C_z B + G', the left hand face is that pullback raised to the power ± ( A ) (and so is a pullback), and the horizontal maps are induced by ±: 1 --* ±(A). By the naturality in 4.3.1, the m a p E ±(A) ~ P, corresponds to T ( E A ~ 1). Hence as T preserves pullbacks, the b o t t o m face is a pullback. Ig follows at once that the top face is a pullback. T h e same argument works with C in place of B. So we have pullbacks (B + C) ±(A)

> B + C

T J

B ±(A)

> B

( B + C) ±(A)

C ~(A)

> B + C

,, > C

But coproducts are stable, so ( B + C) z(n) -~ B l(A) + C z(A).

C o r o l l a r y 4.4.2 I f X is an object with bottom, then for any B and C ( B + C) x ~ B x + C x. Proof

An immediate consequence is an abstract version of Rice's Theorem. P = (N±) N) for the object of E-part±a± functions from N to N. C o r o l l a r y 4.4.3 2 P Proof

[3

Since X is a retract of A = L ( X ) , this is obvious.

TM

Let us write

2.

P is a power of a lift and so is an object with bottom; and 2 = 1 + 1.

[]

147

4.5

Further

results

In this section we collect together some further consequences of Axiom 6, which we shall rigid. From 4.3.1 we can derive an intuitively plausible characterization of the intrinsic order on a lift. P r o p o s i t i o n 4.5.1 The intrinsic order on ± ( B ) is given in the infernal logic by

S ~cl

with S , = {(0, n), (1, n)} and {(1,n)} open but not closed in S,. B is the set {0, 1,2,...} which is topologized such that B - {0} is discrete, but 0 does not belong to any open set except B. It is easy to see that p : E ~ B with p(0, n) = 0, p(l, n) = n proves the claim. We also wish to point out that maps satisfying condition (iv) need not be open: consider the restriction of the first projection R x R ---* R to the subspace {(x,1)lx ___0} U {(0,y)10 < y < 1} U {(x,0)lx > 0} Therefore, the class of maps p for which p* reflects isomorphisms represents a wellcharacterized class Q of maps with { open surjective maps }

Q ~ { pseudo-open surjective maps }

By the Theorem, it is easy to see that Q enjoys a number of pleasant general properties: (a) p E Q , q E Q ~ q p E Q , (b) q p E Q =:-qEQ, (c) every map in Q is a universal Q - m a p , (d) p : E - - * B i n Q , p ' : E ' ~ B ' i n Q = = ~ p × p ' i n Q .

O

1.3 Recall that an epimorphism p in a category C is called extremal if p = m f with m monic holds only if rn is an isomorphism. (Obviously the latter property implies that p must be epic if C has equalizers.) The extremal epimorphisms in T o p are precisely the quotient maps. Now we can generalize the easy part of Theorem 1.1: PROPOSITION Statements (i), (ii), (iii) of the Theorem remain equivalent for any morphism p in a finitely complete category C if "quotient map" is read as "extremal epimorphism "

178

PROOF We first observe that p* : C / B --+ C / E has a left adjoint p! which assigns to an object ( C , v : C -~ E) the object (C, pT) • For ( A , a : A ~ B) E C / B , the co-unit ~(A,,) : p,p*(A,a) --~ ( A , a ) is simply the map 7r2 of (1), considered as a morphism over B. Now, for every category AM with equalizers and every functor U : A --~ X with a left adjoint, the following statements are equivalent (¢f.[4]): (i) U reflects isomorphisms, (ii) the co-units eA(A E AM) are extremal epimorphisras, (iii) U reflects extremat epimorphisms. It is easy to check that a morphism in C / B is an extremal epimorphism (or an isomorphism) if and only if it has the same property in C. Therefore the Proposition is an immediate consequence of the quoted more general fact applied to U = p* . []

2. Premonadicity and Monadicity 2.1 We briefly recall the general framework of monadicity. Let F -~ U : AM--~ 2( be a pair of adjoint functors with unit r/ : l x --* U F and co-unit e : F U --* IA . For the induced monad r = ( T , y , # ) = (UF, y,#) with # = UeF : T ~ ~ T one forms the Eilenberg-Moore category X ~ as follows: objects are r-algebras (X, ~) with an Xmorphism ~ : T X --* X such that ~ ' q x = l x and ~ . # x = ~ ' T ~ ; a m o r p h i s m h : (X,~) ~ ( Z , ( ) i n X * is a n X - m o r p h i s m h : X ~ Z with h.~ = ( . T h , and composition is as in X. There is the so-called comparison functor K : AM--* X ~ , A ~ ( U A, (YEA). U is called premonadic if K is full and faithful, and monadic if K is an equivalence of categories. If ,4 has (sufficiently many) coequalizers, K has a left adjoint L : for a r-algebra (X, ~), the AM-object Q = L ( X , ~) is given by the coequalizer

F~ FUFX~FX

q'Q.

CFX

The underlying X-morphism of the unit g(x.o : (X, ~) --* K L ( X , ~) is the composite

X ,Ix U F X Uq U Q . The definition of q easily yields

U ~ x(x,~) • ~ = Uq. ~lx • ~ = Uq ;

(3)

179

therefore, and since ( is the (split) coequalizer of the pair (T(, #x) = (UF(, UVFX), it is clear that a(x,¢) is an isomorphism if and only if U preserves the coequalizer diagram

(3). 2.2 PROPOSITION For a map p in Top, p* is premonadic if and only if p is a universal quotient map. This holds true if Top is replaced by any finitely complete category and if "quotient map" is read as "regular epimorphism". PROOF Regular epimorphism simply means coequalizer of any pair of morphisms. Very similarly to the proof of Prop. 1.3, the statement follows from the more general fact that a functor U : ¢4 ---* X with left adjoint is premonadic iff the co-units CA(A E .A) are regular epimorphisms, applied to U = p* . The statement for Top follows since regular epimorphism means quotient map here. [] 2.3 In order to examine monadicity of p" , we continue to work in a finitely complete category C. We described the left adjoint p! and the co-unit c in 1.3. The underlying C-morphism of the unit ,1(c,~) : (c,,~) ~ p*p,.(c,'7) is the morphism e = < % l c > rendering the diagram

C

E

P

(4)

1:

B

,

commutative. Let 7- be the monad induced by p, -~ p* . A T-algebra structure for (C, 7) is :E×oC--~C over E ( h e n c e T ( = T h ) s u c h t h a t ~ < 7 , 1 o > = l o and

E xB (E xs C)

1E xB ~

•

l '~ {

map

E xB C

IExs ~21 ExsC

a

,.

(5)

C

commutes. (This is the diagrammatic description of the equations given in the Introduction.)

180

2.4 PROPOSITION For a split-epimorphism (= retraction) p category, p* is monadic.

in a finitely complete

PROOF Let s be a section of p , hence ps = 1B • We give an ad-hoc construction of the left adjoint L of the comparison functor K : C / B ---+ ( C / E ) " , with T the monad induced by p: -~ p* , letting L be the composite

(C/E)" C/E

5"

C/B

with U" the forgetful functor. Hence L(C, 7; () is given by the middle vertical arrow in

E xB ( B XE C)

1 E

,

C

BXEC

(6) ................

P

E

Obviously, L K ~ 1 . Hence we need only to show that K L ~ 1 . This is done by giving an Mternative construction for the pullback (6). It is easy to check that

C

< %1c>

.ExBC

(sp) x B l c

. ExBC

17~)IIEXBT@I E

< 1E, l s > , E XB E

IExB7

(sp) ×B 1E , E xB E

(7) E xB C

< ")/9r2~~ :> , E x B C

7r$

..........

ExBE

.ExBE

zr2

,,

C

E

commutes and that each (D , @ and (3) is a pullback square. In order to show that a/so {D is a pullback square, it suffices to show t h a t < 7 r 2 , ~ > is an isomorphism since < ~r2, ~rl > is one. But ~rl < 7r2,~ > < 7~r2,~ > = 7 r 2 < 7~r2,~ > = 7~ = r l , and with (5) one has r2 < wr~,~ > < 7r2,~ > = ~ ( 1 E xB ~) < 7 r 2 , l s × B c > = ~(1E ×B 7r2) < 7~r2, 1E×BV >

: ~ < 7,1C > r2 = r2 ;

181

therefore, < 7tr2, ~ > < 77r2, ~ > = 1 . Since the composite of the lower horizontal arrows in (7) is, as in (6), s p , the pullback diagrams (6) and (7) must coincide up to isomorphism. If we denote the middle vertical arrow in (6) by 6 : Q --* B , this means that there is an isomorphism qo = ~{c,~;{} : C --* E xB Q in C such that rlqa = 7 and ~2~r2~ (with z'2 : E x s Q ~ Q and ~f2 : Q = B XE C --~ C) is the composite of the upper horizontal arrows of (7). Two facts remain to be shown in order to complete the proof: 1. ~ is a morphism of r-algebras, i.e. ~ : (C, 7; ~) --~ KL(C,7; {). This meaaas that we must show that

ExBC

1EXB'~

,,

ExB(ExBQ)

(8) C

,~,

,

E x~ Q

is commutative; this is done below. 2. qa is a natural transformation, i.e. for a T-algebra homomorphism h : (C,7;~) --* (C',7';~')

C

h

..............

C' (9)

E xB Q

1E Xs h

, E x s Q'

is commutative. This amounts to showing that diagram (7) is "natural in which is clear for parts O , ~), ~)- In case of {!), one must show that E x,s C

1

E xB C'

< 7 ~~'{ >

(6,7;~)

.... E x s C

(lo) < ~'~r~,(' >

, E xB C'

commutes; but this follows easily from the commutativity of

"

182

ExsC

~

l

E x s

.......

C

(11) C'

~'

...................

......

C'

Hence we are left with showing that (8) commutes. Trivially, ~1(1E XB 7r2)(1E XB ~) = 7rl = 3t~ = :rr,9)~ • Verifying that also ~c2rc2(1E XB 7r2)(1E XS ~) = #27r2~ holds amounts to showing that the composite c of the top horizontal arrows of (8) satisfies c7r2 = c~. In fact, since c=~<sw/,lc>, one has with (5) cTr2 = ~ < ~p~Tr2 , 71"2 > =

~ < 8pTrl,71" 2 >

= ~(1E XB zr2) < spTh, 1E×BC > = ((1E XB ~) < spTq, 1E×.C > =

< sp

l,

>=


=

c

[]

This completes the proof.

2.5 We describe the left adjoint L of the comparison functor K : C / B --* ( d / E ) ~ (with r the monad induced by p! -q p* ) in case of an arbitrary morphism p : E --* B in C . According to 2.1, for a r-algebra (C, 7 ; ( ) , one forms the coequalizer q of ( ( , r 2 ) in C and has an induced arrow 6 such that ExBC

~

7r2

"~

C

q

• Q (12)

E

P

.

B

commutes; now L ( C , 7 ; ~ ) = ( Q , 6 ) . By 2.1 and (4), the unit a = a(c,~;O : (C,7;~) ~ K ( Q , 6 ) is given by t¢ = (1E ×B q) < 7, lC > = < 7, q >: C ~ E XB Q , considered as a morphism in (C/E) ~ . 2.6 We call a map p : E --~ B in T o p locally monadic if for every b E B there is an open neighbourhood U in B such that p~r is monadic, with pu : p - I U --* U the restriction of p . PROPOSITION

p* is monadic if and only if p is locally monadic.

183

PROOF Trivially, p is locally monadic if p* is monadic. characterization (iv) in Theorem 1.1 it follows with 2.2 that locally monadic. In terms of the adjunction L -t K as given co-units are isomorphisms; it remains to be shown that the well.

Vice versa, from the local p* is premonadic if p is in 2.5, this means that the units are isomorphisms as

Certainly, as a Set-map, t~ = n(c.~;0 is bijective since p* : S e t / B -~ S e t / E preserves coequalizers and is therefore monadic (cf. 2.t). Hence it suffices to show that the continuous bijection a = < 7, q >: C -* E xB Q is a local homeomorphism, i.e. for x E C there is an open neighbourhood V of a(x) such that the restriction tc-l(V) --* V is a homeomorphism. For b =/rT(x ) E B one can choose U as in 2.6 and then consider V = p - l U x u 5-1U : p - I u XB 5-1U C E xB Q . Since p is assumed to be locally monadic, it suffices to check that diagram (12) can be restricted as follows: p-lU x U I~-lV

~l .......

qt ~

it-iV

: ~-1 u

(13) p-~U

pc~

,

U

with the top row a coequalizer diagram, and with (n-iV, 3"; ~') an algebra with respect to the monad induced by Pb • One easily verifies that the top row gives a coequMizer diagram in Set. Furthermore, q' is the restriction of q to open subspaces, with ~ - I V c_ q - l ( 5 - 1 U ) . Hence q' is a quotient map, and the top row of (13) yields a coequalizer diagram in Top. Checking that (g-lV~71;~') is an algebra is easy: the diagrams describing it as such are just restrictions of the diagrams describing the algebra 0 (C,7;~) • 2.7 A map p : E -* B in Top is a local split-epimorphism or locally sectionable if for every b E B there is a neighbourhood U in B such that the restriction ptr : p-IU "-* U is a split-epimorphism. Now Propositions 2.4 and 2.6 yield the key-result of the paper: THEOREM

For a locally sectionable map p in Top, p* is monadic.

n

COROLLARY p* is monadic if p is a surjective map with one of the following properties: p is a local homeomorphism; p is locally trivial; p is a covering map; p is the projection of a completely regular G-space onto its orbit space, with G a compact Lie group [6].

184

PROOF []

Each such map is surjective and locally sliceable [7], hence locally sectionable.

REMARKS (1) A map p in T o p with p* monadic does not need to be open: the projection of ( - 1 , 1 ] onto R/Z is easily seen to be locally sectionable but not open (cf. [7]). Other examples with such properties are given by surjective polynomial maps p:R--~ R.

U{b

(14)

/

(2) Since monadicity implies premonadicity, locally sectionable maps are universal quotient maps. The example of a non-open universal quotient map given in 1.2 also shows that such maps need not be locally sectionable. Hence at least one of the implications p locally sectionable ~ p* monadic =~ p universal quotient map must be proper. We don't know which one(s).

3. T h e M o n a d i c D e c o m p o s i t i o n o f p* 3.1 (Cf.[11],[1],[2]) If .A is (sufficiently) cocomplete and n is an ordinal or o o , a pair of adjoint functors F -t U : A ---* A' can be factored as

where Xi+, = X[' with ri the monad induced by the comparison functor Ui:.4--~Xi ( U 0 = U ) , and Xi = limjINI eNe ) MNN

*N* ) M®N

Similarly, I carries a unique commutative monoid structure. Moreover, there are canonical maps M = )M N I ~ MNN N .... = ) INN e n i d ) MNN and a codiagonal map from M®M to M given by * itself; it is easy to verify that these are monoid homomorphisms, and that they define the unit and counit of an adjuction between N (as a bifunctor from CMon(C) x CMon(C) to CMon(C)) and the diagonal functor.

204

Hence M®N is the coproduct of M and N in CMon(C); and a similar argument establishes that I is initial in this category. ] The best known application of this Lemma occurs when C is the category of Abelian groups. It then becomes the assertion that the coproduct of two commutative tings is (founded on) the tensor product of their additive groups. Interpreted when C is the category of sets (with its cartesian monoidal structure), it becomes the assertion that finite products and coproducts coincide in the category of commutative monoids (or in any full subcategory thereof that is closed under products, such as abelian groups or semilattices). Another application of Lemma 4.1 was exploited by Joyal and Tiemey [84]. There they considered frames as commutative monoids in the monoidal category SupLat of complete join semilattices (their suplattices), the monoid structure being given by finite meets. Of course, the general commutative monoid is a commutative quantale, not necessarily a frame; but, as Joyal and Tiemey observed, frames may be characterized as those commutative quantales for which the multiplication is idempotent and the unit is the top element. These conditions cannot be expressed by commutative diagrams in SupLat since the monoidal structure is not Cartesian, there is no diagonal map from A to A®A, which would be needed to express the idea of idempotency. However, they do suffice to recover the result (already known, of course, before the work of Joyal and Tiemey - cf. Wigner [79]) that the coproduct of two frames coincides with their tensor product in SupLat. Exactly the same arguments apply when we consider frames in relation to preframes. In fact, we have L e m m a 4.2 (i)

(ii) (iii)

Let A be a frame. Then the binary join map v: AxA -+ A is a preframe bihomomorphism, and the induced map from A®A to A gives A the structure of a commutative monoid in P r e F r m . A commutative monoid (A, *, e) in P r e F r m is (derived in this way from) a frame iff e(0) _u*0 = u for all u ~ A®B, and it suffices to verify that u*u = SupLat ( P (qua poset) I u < V X (whenever X covers u in C) > This facilitates the definition of sup-lattice homomorphisms out of frames, and is particularly useful in the work of Abramsky and Vickers, where functions are defined between frames and quantales. In practice, it is easy to work any frame presentation into the required form by putting in all finite meets of generators and any extra relations needed to give meet stability. This understanding is related to a well-known result from the theory of rings. If R is a ring and I is a subgroup (of R as an additive group), then, provided that I is an ideal, we have Ring ( R (qua ring) I r = 0 (whenever r ~ I) > _= Abelian Group ( R (qua Abelian Group) I r = 0 (whenever r E I) > The purpose of this section is to give an analogous result enabling one to present frames as preframes, instead of sup-lattices. T h e o r e m 5.1 The Preframe Version of the Coverage Theorem. Let P be a poset, and let C be a set of prefrarne relations of the form

AS< iASi where the sets S, Si are all finite subsets of P. Let A1 = PreFrm ( P (qua poset) 1C > (Every preframe presentation can be reduced to this form.) Suppose in addition that • P is a join semilattice, • C is join stclble, i.e. if A S < vl"i/kSi is a relation in C, and x e P, then the relation A{xvy: y ~ S} < V$i A{xvy: y ~ Si} is also in C. Then A1 is isomorphic to A2 = Frm ( P (qua v-semilattice) I C >, the generators corresponding under the isomorphism in the obvious way. Proof First, we show that A1 is a frame. 0 c P is bottom in AI, for {a e AI: a > 0} is a subprefmme containing the generators. (Hence there is a preframe homomorphism from A1 to this set which, when composed with the inclusion, gives the identity map on AI: so the inclusion is onto,) Next, if x E P, we can define a preframe endomorphism 0x of A1 by y ~ xvy. x 1--->0x is monotone and respects the relations in C, and so defmes a preframe homomorphism @ from A1 to [A1, A1]. Writing a@b for ¢(a)(b), @ is a preframe bihomomorphism and x@y = (xvy) for x, y ~ P. Now,

209

• b < a@b, i.e. Id < ~(a), i.e. y __ 1; if a@a = a and b@b = b, then (aAb)~(a^b) = a@a ^ b@b ^ a@b ^ b@a = a ^ b ^ a~b = a^b and if S is a directed set of such elements, then VTS ~9 VTS = VT{a~b: a, b ~ S} = VT{c~c: c ~ S} = VTS We have now shown that @ is a binary join in A1. We might as well write it as v. v in A1 distributes over ^, so A1 is a distributive lattice, and hence a frame; also, the injection of generators preserves finite joins. We now know that we can def-me a frame homomorphism c~: A2 -4 A1 and a preframe homomorphism 13:A1 --~ A2, both mapping generators to generators in the obvious way. c~ is a preframe homomorphism, so 13;cxis the identity on A1. It remains to show that 13 is a frame homomorphism, after which we know that c~;13is also the identity. Fixing y ~ P, the set {a E AI: 13(avy) = 13(a) v y} is a subprefmme containing the generators; and then fixing a, the set {b ~ AI: [3(avb) = 13(a) v 13(b)} is a subpreframe containing the generators. ] This result could in fact have been used in the proof of Theorems 4.3 and 4.4, though we preferred to put those results in a more general context. We give here another sample application concerning the upper and lower power locales. These, decomposing the Vietoris construction into two parts, were first studied as topologies in Michael [5 t]. They are also well-known in computer science following the work of Smyth [78, 83]; see, for instance, Vickers [89]. Suppose D is a locale. Following Vickers [89], we write Y2D for the corresponding frame "of opens", and if f: D ~ E is a continuous map between locales, we write f2f: f2E --~ f~D for its inverse image map. The upper power locale PoD is defined by f2PuD = Frm ( f2D (qua preframe) }. The lowerpower locale PLD is defined by f2PLD = Frm ( f2D (qua sup-lattice) ). If a ~ ¢2D, then we write I3 a and Oa for the corresponding generators of f2PuD and f2PLD; so [] preserves finite meets and directed joins, while ¢, preserves all joins.

Proposition 5.2 The upper and lower power locale functors commute. Let D be a locale. Then PuPLD = PLPuD.

Proof We define mutually inverse frame homomorphisms f20: ~ P L P u D --~ £2PuPLD

£2~: Y2PuPLD --~ f2PLPuD

for which f20 (O []a) = {30a and f2~ (O Oa) = O Oa.

210

~0: This must be equivalent to a sup-lattice homomorphism from f2PuD to OPuPLD taking Da to [] Oa. Johnstone's coverage theorem for frames, i.e. the "sup-lattice version", tells us that ~2PuD = Frm ( [] a (a e f2D qua ^-semilattice) I [] preserves V ? ) ---SupLat ( Da (a e D.D qua poset) I [] preserves V? } and • a ~ [] Oa does indeed preserve directed joins. f2~: This must be equivalent to a preframe homomorphism from f2PLD to f2PLPuD. The preframe version of the coverage theorem tells us that ~PLD -= Frm ( ,>a (a e ~ D qua v-semilattice) t preserves V t ) = PreFrm { Oa (a e f2D qua poset) I O preserves Vl" ) and then, as before, O a F--~[] a preserves directed joins.

]

6. Bibliography S. ABRAMSKY and S.J. VICKERS [90] Quantales,Observational Logic and Process Semantics, Department of Computing Report DOC 90/1, Imperial College of Science, Technology and Medicine, London, 1990. B. BANASCHEWSKI [88] "Another look at the Localic Tychonoff Theorem", pp. 647-656 in Commentationes Mathematicae Universitatis Carolinae 29(4), Charles University, Prague, 1988. N. BOURBAKI [70] [~ldmentsde Mathdmatique: AlgObre, Hermann, Paris, 1970. T. COQUAND [90] "An intuitionistic proof of Tychonoff's theorem", preprint (1990). M. COSTE [79] "Localization, spectra and sheaf representation", pp. 212-238 in M.P.Fourman, C.J.Mulvey and D.S.Scott (eds) Applications of Sheaves, Springer Lecture Notes in Mathematics 753, 1979. C.H. DOWKER and D. STRAUSS [76] "Sums in the category of frames", pp. 17-32 in Houston J. Math. 3 (1976). C. EHRESMANN [57] "Gattungen von lokalen Strukturen", pp. 59-77 in Jber. Deutsch. Math.-Verein. 60 (1957). P. FREYD [72] "Aspects of topoi", pp. 1-76 in Bull. Austral. Math. Soc. 7 (1972). P. FREYD and A. SCEDROV [90] Categories,Allegories, NoIXh-Holland 1990.

211

P. GABRIEL and F. ULMER [71] "Lokal pr~isentierbare Kategorien", Springer Lecture Notes in Mathematics 221, 1971. G. GIERZ, K.H. HOFMANN, K. KEIMEL, J.D. LAWSON, M. MISLOVE and D.S. SCOTT [80] A Compendium of Continuous Lattices, Springer-Verlag, Berlin,1980. KARL H. HOFMANN and MICHAEL W. MISLOVE [81] "Local compactness and continuous lattices", pp. 209-248 in B. Banaschewski and R.-E. Hoffmann (eds) Continuous Lattices: Proceedings, Bremen 1979, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin, 1981. J.R. ISBELL [72] "General functorial semantics I", pp. 535-596 in Amer. J. Math 94 (1972). P.T. JOHNSTONE [81] "'Tychonoff's theorem without the axiom of choice", pp. 21-35 in Fund. Math. 113 (1981). [82] Stone Spaces, Cambridge University Press, Cambridge, 1982. [85] "Vietoris locales and localic semilattices", pp. 155-180 in R.-E. Hoffmann and K.H. Hofmann (eds) Continuous Lattices and their Applications, Marcel Dekker 1985. P.T. JOHNSTONE and S.H. SUN [88] "Weak products and Hausdofff locales", pp. 173-193 in F. Borceux (ed.) Categorical Algebra and its Applications, Springer Lecture Notes in Mathematics 1348,1988. ANDRt~ JOYAL and MYLES TIERNEY [84] An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society 309, 1984.

i. K ~ 2 [85] "A constructive proof of the Tychonoff theorem for locales", pp. 619-630 in Commentationes Mathematicae Universitatis Carolinae 26, Charles University, Prague, 1985. F.E.J. LINTON [66] "Some aspects of equational categories", pp. 84-94 in S.Eilenberg et al. (eds) Proceedings of the Conference on Categorical Algebra, Springer-Verlag1966. [69] "Coequalizers in categories of algebras", pp. 75-90 in Seminar on Triples and Categorical Homology Theory, Springer Lecture Notes in Mathematics 80,1969. J.L. MacDONALD and A. STONE [82] "The tower and regular decomposition", pp. 197-213 in Cahiers Top. Groin. Diff. 23 (1982). C. McLARTY [86] "Left exact logic", pp. 63-66 in J. Pure Appl. Alg. 41 (1986). ERNEST MICHAEL [51] "Topologies on spaces of subsets", pp. 152-182 in Trans. Amer. Math. Soc. 71 (1951).

212

S. PAPERT [64] "An abstract theory of topological spaces", pp. 197-203 in Proc. Camb. Phil. Soc. 60 (1964). J. SLOMINSKI [59] "The theory of abstract algebras with infinitary operations", Rozprawy Mat.18 (1959). M. SMYTH [78] "Power domains", in JCSS 16, 1978. [83] "Powerdomains and predicate transformers: a topological view", pp. 662-675 in J.Diaz (ed.) Automata, Languages and Programming, Lecture Notes in Computer Science 154, Springer-Verlag, Berlin, 1983. J.J.C VERMEULEN [90] "Tychonoff's theorem in a topos", preprint (1990. STEVEN VICKERS [89] Topology via Logic, Cambridge University Press, Cambridge, 1989. D. WIGNER [79] "Two notes on frames", pp. 257-268 in J.Austral.Math.Soc. A28 (1979).

"l'his paper is in final form and will not be published elsewhere.

Strong Stacks and Classifying Spaces Andr6 Joyal* and M y l e s T i e m e y * * * Ddpartement de math6matiques,UQAM,C.P. 8888, Succ 'A', Montr6al, Canada, H3C 3P8 ** Dopartmentof Mathematics,RutgersUniversity,New Brunswick,NJ 08903

Introduction Let ~ be a Grothendieck topos, and denote by Cat(~), respectively Gpd(~), the category of categories, respectively groupoids in ~. In this paper we show there are Quillen homotopy structures on Cat(~) and Gpd(~), in which the weak equivalences are the (internal) categorical equivalences, the cofibrations are the functors injective on objects, and the fibrations have the right lifting property with respect to tim cofibration weak equivalences. The fibrant objects for these structures are called strong stacks, since they represent a strengthening of the notion of stack introduced by Grothendieck and Giraud [Giraud 1971]. When the topos ~ is the category of simplicial sheaves for a Grothendieck topology, strong stacks have an intimate connection with the theory of classifying spaces for sheaves of simplicial groups, or groupoids. The paper begins in section 1 with a general discussion of torsors and stacks for groupoids in ~. In section 2 we introduce the concept of a strong stack in Gpd(~), and treat the problem of strong stack completions by establishing the above-mentioned Quillen homotopy structure on Gpd(~). We finish the section with a number of examples. Section 3 is concerned with strong stacks in Cat(~), and the development is parallel to that of section 2. Some of the equivalences of Theorem 3 overlap with results of [Bunge 1979], though the treatment given here is quite different. Section 4 contains some applications of strong stacks to the problem of the existence of classifying spaces for sheaves of simplicial groupoids. For a full account of this topic, the reader should consult [Joyal-Tierney (to appear)].

1. Torsors and Stacks Let G be a group in a topos ~. A (fight) G-torsor in ~ is a non-empty object E (meaning E ~1 is surjectlve) equipped with a free (right) G-action a: EYG ~E, which is transitive. (Free and transitive is expressed by requiring that the map (x, g) J )(xg, x) from ExG to ExE be an isomorphism.) A mapping f: E'~

~E of G-torsors is a function f: E'

)E

compatible with the G-actions. It is always an isomorphism. Torsors solve the problem of finding all objects T locally isomorphic to a given object S of ~. Recall that T is locally isomorphic to S iff there exists a coveting K (meaning K is nonempty) and an isomorphism KxT ) KxS over K. Equivalently, T is locally isomorphic to S iff Iso(S, T) is non-empty. In fact, letting G = Aut(S), if T is locally isomorphic to S, then

214

E = Iso(S, T) is a right G-torsor, and T can be recovered from E since the evaluation mapping defines an isomorphism E®GS

>T. Moreover, the correspondence T i

> Iso(S, T) defines

a bijection between the set of isomorphism classes of objects T locally isomorphic to S and the set HI(G) of isomorphism classes of G-torsors in ~. The proofs are clear by localization, but can also be found in [Giraud 1971], which is a general reference for this section. For a general object X of ~, a (right) G-torsor overX is a G-torsor in ~/X. That is, it is an object E

>X over X provided with a free (right) G-action a: E×G---~E over X (free

meaning the map (x, g) i

>(xg, x) from E×G to E×E is injective), such that E/G

>X is an

isomorphism. When ~ is the category of simplicial sets, a G-torsor over X is a principal G-bundle over X [May 1967]. Let HI(X, G) be the set of isomorphism classes of G-torsors over X. When S is a fixed object of ~, and G = Aut(S), the above correspondence (interpreted in ~DO, yields a bijection between HI(X, G) and the set of isomorphism classes of objects T

>X locally isomorphic (in ~/X) to the projection X×S

>X. (Such an object in

simplicial sets is just a fibre bundle with fibre S.) In what follows, we shall have to consider not only groups, but also groupoids and their torsors. We recall some fundamental definitions. A groupoM G in ~ is a reflexive graph Go

u > G1~

Go

in ~, i.e. a diagram such that su = tu = id, provided with an associative composition c: GI×GoG1 ;G1 for which the elements of GO are units (via u), and each element of G1 is invertible. (These statements, of course, are interpreted in the internal language of ~.) If G and IH are groupoids, a functor f: G

>IH is a morphism of reflexive graphs, which respects composition, f is a

categorical equivalence if it is full, faithful, and representative in the sense that each object of I[-I is isomorphic to the image of an an object of G under f. f is a strong categorical equivalence if there is a functor g: II-I

>G such that gf-~ idQ3and fg-~ idlrt. (See [Bunge-Par6 1979] for a

discussion of various different notions of equivalence for internal categories.) Let G be a groupoid in ~. A (right) G-torsor is a non-empty object E over GO equipped with a free (contravarian0 action a: E×GoG1 which is transitive. A G-torsor E

;E

>X over X is a G-torsor in ~/X. The set of isomorphism

classes of G-torsors over X is denoted by HI(x, G) (we write just HI(E) when X is 1). HI(x, G) is contravariant in X and covariant in G. In fact, if g: Y E

)X is a map, and

>X is a G-torsor over X, then g*0~) is a G-torsor over Y, a n d i f f : 113...........; ~ is a functor

then E®~31t-I is an IB-torsor over X. Note that Hi(X, ~3) is invariant under categorical equivalence of groupoids.

215

Torsors for a groupoid solve the problem of finding aU objects T locally isomorphic to a member of a given family S k: K

......; I, by which we mean there is a cover K and a mapping

>I such that KxT -~ k*(S) in ~/K. In fact, let G be the groupoid Iso(S, S)

isomorphisms of the fibres of S ....... "I. Recall that Iso(S, S)

>IxI of

>IxI is the object Iso(SxI, IxS)

in ~/lxI. Since pullback preserves the construction of Iso, G has the property that given two mappings f,g: J

~I in ~, there is a mapping h: J

>Iso(S, S) such that

Iso(S, S)

/ J

(f, g)

Y IxI

commutes iff f*(S)-~ g*(S) in ~/J. Now if T is locally isomorphic to a member of S above, then E = Iso(S, T ) E®GS

)I as

>I is a G-torsor, and evaluation yields an isomorphism

>T. In this way we obtain a bijection between the set of isomorphism classes of

objects T locally isomorphic to a member of S

>I and HI(G). When X is an arbitrary object

of ~, we get a bijection between the set of isomorphism classes of objects T in ~/X locally isomorphic to a member of S >I and HI(X, G). Let G be a groupoid in ~. I f X is an object of ~, we denote by horn(X, G) the groupoid (in Sets) whose objects are maps f: X h: X >G1 such that

>Go. A morphism between f and g is a map

q

//~ ](s,O x

Oo Go

commutes. Let H(X, G) denote the category of G-torsors over X. (HI(X, G) is the set of connected components of H(X, G).) t: G1 ---->Go is a right G-torsor over GO, so there is a functor homOI, G) .... >H(X, G) defined on objects by f: X see that (s, t): G1

)G0 s

) f*(G1). It is easy to

)G0×G0 is canonicaUy isomorphic to IsoG( G1, G1)

)G0xG0, so this

function extends in an obvious way to a full and faithful functor. G is said to be a stack if this functor is an equivalence of categories. Clearly, G is a stack iff the functor is representative. That is, G is a stack iff for each ~;-torsor E

) X there is a map f: X

) GO such that

216

E "" f*(Gl). In the example above, ~ = Iso(S, S)

complete, i.e. for any f: X

T

>IxI is a stack iff the family S

>Iis

>X locally isomorphic to a member of S ....... >I, there is a map

~I such that T -~ f*(S).

Theorem 1 The following are equivalent for a groupoid G in ~. (i) G is a stack. (i.i) For each X in ~, every G-torsor E

>X has a section X

>E.

('hi) Every diagram of groupoids A -r

e

>IB A" t

/

G with e a categorical equivalence injective on objects has a dotted filler making the resulting triangle commute up to isomorphism. (iv) Every diagram of groupoids A

f

~IB j,

i'

¢

G with f a categorical equivalence has a dotted f~er making the resulting triangle commute up to isomorphism. Proof: (i) ¢ , (ii): t: G1 oft: G1

.......

>Go has the section u: G0

>G1, so if every torsor is a pullback

;G0, every torsor has a section. On the other hand, let E

X with a section s: X

>X be a G-torsor over

;E. Then ff f denotes the composite of s with the structure map

E .... >Go of E, f*(G1) " E. (ii) ~ (iii): Categorical equivalences injective on objects are stable under pushout, so to prove (iii) it suffices to show that if (ii) holds, then any categorical equivalence e: G ~ , [-t injective on objects has a retract up to isomorphism. For this, let

217

E

:-- H 1

J

G0xH 0

e x I -"- H0xH 0

be a puUback. We claim that E >H0 is a ¢3-torsor over H0. We argue with elements, which can be justified as in [Joyal-Tiemey 1984], and will be clearer for the reader. First, E = {(g, h, e(g) ---->h) I g e G0, h e H0, and e(g)---~hisan arrow inH1} E ) H0 is the function (g, h, e(g) ~ h ) ~ >h, and is surjective, since e: G, >]H is representative. The action a: E×GoG 1 >E of G on E ;Go is given by composition: a((g, h, e(g) ----~h), g'----~g) = (g', h, e(g') ----->e(g)----~h) It is flee by the faithfulness ofe. Finally, if (g', h, e(g')---~h) and (g, h, e(g)----~h) in E both project to h e H0, then since e is full and faithful, there is a unique g'---~g such that e(g')----~h is the composite e(g')----~e(g')---~ h. Hence, H0 is the quotient of E by the action of 113.By (ii), E ) H0 has a section s: H0 >E. Such an s is a choice, for each h • H0, of an element s(h) = (g, h, e ( g ) ~ h ) in E. Let r: H0 >Go be the composite H0 >E "G0. Thus, if h • H0 and s(h) = (g, h, e(g)---->h) then r(h) = g. Clearly, there is an isomorphism re = idc0. FinaUy, suppose h'---->his an arrow of H1. Since e: G, >]H is full and faithful, there is a unique arrow rh'---->rh in G1 such that h' s(h') I e(rh')

~ h I s(h) _~e(rh)

commutes. Setting r(h' ----~h)= rh'----~rh completes the definition of the retraction r. ('fii) ~ (iv): Let f: z~k >]B be a functor between groupoids in ~. Denote by II the groupoid in ~ with two objects 0 and 1, and one isomorphism between them. The cylinder on f is defined to be the pushout

218

z~xO

f

>lB

I There is a unique functor r: C f Z~kx]I, where re: z~kxlI Axl~

~AxE

Ii

>]13 given by the identity on ]13,and the composite fro on

>z~k is the projection. Let e: A ~,, >Efbe the composite

>El. Clearly, f = re, and e is injective on objects, r is a strong categorical

equivalence since ri = id and ir -~ id. It follows that e is a categorical equivalence if f is. Thus, a functorz~k

>G can be extended up to isomorphism over e: z~k'

~Cf iff it can be extended

up to isomorphism over f: A >lB. (iv) =~ (i): Let E ~X be a iG-torsor over X, with action a: EXGoG1 E

>E and structural map

~G0. The pair (a, ~1): EXGoG1

)ExE

is a groupoid tE with objects E (the categoryofelements of the action). It is as well the equivalence relation determined by the surjection E a E

G1

q is a functor ]E

~X. The diagram z-

>Go

t

) G. Letting disX denote the discrete groupoid in ~ determined by the object

X, we obtain a categorical equivalence IE .... >disX, and a diagram IE

~ disX S

J

s"

G which has a dotted filler up to isomorphism by

(iv).

The filler disX

) g is a map

219

f: X

;Go, and the isomorphism, on objects, is a map qo: E

E

qo r

G1

>G1 such that the diagrams E

qo

X

f

> G1

and

%

>-G 0

commute. The statement that qo is a natural transformation is equivalent to the statement that qo is compatible with the right E-actions. It follows that the second diagram above is a pullback, i.e. E "~ f*(G1), and G is a stack. • Let ~3 be a groupoid in 6. A stack completion of G is a categorical equivalence ~3

>G* with G* a stack. Stack completions are deffmed up to strong equivalence of

groupoids. We postpone the discussion of their existence until the next section, where a stronger existence result will be proved. Notice, however, that if G

> ~3" is a stack

completion of G, then since H(X, ~3) is equivalent to H(X, ~3"), it follows that H(X, ~3) is equivalent to hom(X, G*), i.e.G* "represents" H( , G).

2. Strong Stacks Definition 1 A groupoid G in 6 is a strong stack if condition (iii) of Theorem 1 holds on the nose. That is, if each diagram A¢

e

> IB ,e

/

G with e a categorical equivalence injective on objects has a dotted filler ma~ng the resulting triangle commute.

Definition 2 Let G be a groupoid in 6. A strong stack completion of ~3 is a categorical equivalence injective on objects •¢

>~3"such that G* is a strong stack.

220

When ~ is a Grothendieck topos, every groupoid I13 in ~ has a strong stack completion. In fact, we prove more: strong stacks are the fibrant objects for a QuiUen homotopy structure [QuiUen 1969] on the category Gpd(~) of groupoids in ~. Recall that a Quillen homotopy structure on a category ~ with finite limits and colimits consists of three classes of maps: weak equivalences, cofibrations and fibrations. These are required to satisfy the following axioms. QI:

(Composition) Given two morphisms f: A

)B and g: B

>C in ~3, if any two of f, g or

gf, are weak equivalences, so is the third. Q2 : (Retracts) Weak equivalences, cofibrations and fibrations are closed under retracts. More precisely, if

fJ

t

,--A'

A

-"-A

g

~B'

B

V

-~B

is a commutative diagram in 13 such that ts = ida and vu = idB, then if g is a weak equivalence, cofibration or fibration, so is f. Q3 : (Li~ng) A commutative square A

~X

J i /,

p

t

p

f

B

yY

in ~ such that i is a cofibration and p is a fibration has a dotted filler making both triangles commute if either i or p is a weak equivalence. Q4 : (Factorization) Any morphism f: X

) Y in ~ can be factored in two ways as

221

i

\/

X

~-E

Y where i is a cofibration and p is a fibration. One in which i is a weak equivalence, and one in which p is a weak equivalence. Theorem 2 There is a Quillen homotopy structure on Gpd(~), in which the weak equivalences are the categorical equivalences, the cofibrations are the functors injective on objects, and the fibrations have the right lifting property with respect to the cofibration weak equivalences. Before proving the theorem, we prove two lemmas. First, call p: IE

) 113 in Gpd(~) a

trivialfibration if p has the right lifting property with respect to all cofibmtions. L e m m a 1 If p: IE

) IB is a trivial fibration, then EO

>Bo is an injective object in ~/BO,

and p is a slrong categorical equivalence. Proof: Let p: IE

) I13 be a trivial fibmtion. A diagram

i

C

°

> B0

in ~ is equivalent to a diagram disA

> IE

disC

> IB

in Glxt(~), which has a dotted fiUer since disA~ ;disC is a cofibration. Thus the fh'st square

222

has a dotted filler, so EO.... ~B0 is an injective object in ~/~0- Let © denote the empty groupoid. ©

~It3 is a cofibration, so there is a dotted filler for the square O

-"-IE s/'

p

a' t

IB

id

~IB

In the commutative square (sp, id)

0Ex0)+0Ex 1)

ExlI

~E

~ IE

~- IB

~

P

where ~ is the first projection, the left-hand vertical mapping is a cofibration, so the square has a dotted filler h, which provides an isomorphism sp = id. • L e m m a 2 I f p : IE

)]B is such thatp is full and faithful, andEo

~/B0, then p is a trivial fibration. Proof: Let ~%*

) £ be a cofibration in Gpd(~), and suppose A

~E

f is a commutative square. The square

;B0is injectivein

223

A0

~ E0

I "1 g,," 4'

p

S

4, p

CO

has a dotted filler g since E0

f

> B0

>B0 is injective. Again, we indicate with elements how to

extend g to the arrows of C. Namely, suppose ct---)c is an arrow in C1. f(ct---~c) = f(c')---~f(c) = pg(c') ----~pg(c). But p is full and faithful, so there is a unique arrow g(c') ---~g(c) in E1 such that p(g(c') ---~g(c)) = f(c') ---~f(c). We set g(c~-~c) = g(c') ----~g(c). lU

Proof of T h e o r e m 2: The finite limits and colimits in Gpd(~) are clear, as are Q1 and Q2, whose verification we leave to the reader. Let f: G Embed GO

~H0 into an injective E0

>IH be an arbitrary functor in Gpd(~).

>H 0 over H 0. i G0c~

o

For example, we could take i to be the singleton mapping f~

~f~f in ~B-IO. Now pull back the

arrows of IH over EO to obtain a commutative triangle of groupoids

G¢,

i

>IR.

IN i is a Cofibration, and p is a trivial fibmtion by Lemma 2. By Lemma 1, a trivial fibmtion is a fibmtion and a categorical equivalence, so we have established half of Q4. Moreover, suppose f: G

>IH is a fibrafion and a categorical equivalence. Factor f as above into a cofibration i

followed by a trivial fibration p. By Q1, i is a cofibration weak equivalence, so the square

224

id

G i

rJ

f

.# J

IE

has a dotted filler r, making fi G

~G

i

J

P

~lg

) IH a retract of p: ]E

~IB. Since trivial fibrations are

closed under retracts, f is a trivial fibration. But this establishes Q3, for the first part holds by definition, and the second is just the statement that a fibration which is a weak equivalence, is a trivial fibration. It remains to show that an arbitrary f: G

>]H can be factored as a cofibration weak

equivalence followed by a fibration. However, cofibration weak equivalences are stable under pushout, and as in [Joyal (to appear)] they have a small set of generators, so we can use "the small object argument" [QuiUen 1967] to obtain the desired factorization by repeated pushouts and ordinal colimits. [] We consider some examples of fibrations, strong stacks and strong stack completions. First, recall that a functor f: IE

~I13 in Gpd(~) is called a discretefibration if the square

t

el

r B1

o

fo t

-~ B0

is a pullback. When this is the case, IE is always the category of elements (as defined in the proof of Theorem 1) for a (contravariant) action of I13 on EO

; B0 [Johnstone 1977]. We

claim that a discrete fibration is a fibration for the Quillen homotopy structure on GIXI(~). In fact, we prove more.

Proposition 1 A discrete fibration f: ]E

) ]]3 has the unique right lifting property with

respect to the cofibration weak equivalences in Gpd(~). Proof: Let f: ]E

~113 be a discrete fibration. We want to show that if A '

cofibration weak equivalence, then a commutative square

~£ is a

225

A

~E

,¢

C

~B

has a unique dotted filler. Since discrete fibrations are stable under pullback, it suffices to show that if i: A .... ; ]B is a categorical equivalence injective on objects, and if i*(E) has a section s over z~, then there is a unique section t of f over ]B, which extends s. But the category of discrete fibrations over ~ coincides with the category 61B°p of (contravariant) ]B-actions, and i*: 6 ~op

~S A°P is an (ordinary, external) equivalence of categories. In particular, it is full

and faithful. A section s of i*(E) is a map i d a there is a unique t: idlB

~i*(E) in 6 A°p. Since i d a = i*(id~),

>IE in ~ IB°psuch that i*(t) = s. •

Proposition 1 provides a number of examples of strong stacks. In fact, we know that a groupoid G is a strong stack iff G is fibrant in the Quillen structure on Gpd(6), i.e.iff G

~

is a fibration, where 1] is the terminal groupoid. Fibrations are stable under composition, so if ]E

"G is a discrete fibration and G is a strong stack, it follows that IE is a strong stack.

Thus, the category of elements of any G-action is a strong stack. In particular, the category of elements of a G-torsor is a strong stack. Also, for any object X of 6 disX

) ~ is a discrete

fibration, so disX is a strong stack with unique lifting. Another class of examples can be constructed as follows.

Proposition 2 Let p: E

~B be a surjective mapping in 6, and denote by IE the equivalence

relation EXBE~ ;ExE on E determined by p, considered as a groupoid in 6. Then ]E is a strong stack iff p: E

~B is injective in 6/B.

Proof." The categorical equivalence E

>disBinduced by p is a trivial fibration in Gpd(6) iff

p: E

>B is injective in 6/B by Lemmas 1 and 2. disB is always a strong stack, so if

p: E

>B is injective in 6/B, ]E is a strong stack. On the other hand, suppose IE is a strong

stack, z~k, >C is a cofibration weak equivalence, and we are given a commutative square

226

j-

,

A

>IE

f

C

~ disB

The square has a dotted filler making the upper triangle commute since IE is a strong stack. The two functors C

>disB agree on A so they agree on C by the unique lifting property of

disB. It follows that E

>disB is a fibration in Gpd(~). But it is also a weak equivalence, so

it is a trivial fibration and p: E

-B is injective in ~/B. []

As an example of Proposition 2, let X be a non-empty object of 8, i.e. X surjective. Then the full equivalence relation id: X×X

>1 is

~XxX, which we call X considered

as a groupoid of ~, is a strong stack iff X is an injective object of ~. In general, the strong stack completion of X is obtained by embedding X into an injective object Y, and taking the groupoid Y . If G is any groupoid in ~, a functor G

> X is just a mapping G0

>Xin ~,

and any two such are isomorphic. Thus, it follows from, say, condition (iv) of Theorem 1 that if X has an element 1

)X, then X is a stack, which, by the above, is not strong in general.

Proposition 3 A groupoid G in S is a strong stack iff for any X in ~, each G-torsor E is injective in ~/X, i.e. for any monomorphism A' A

I

't

B

>X

;B in ~, each commutative square wE

J

S S, ¢

J

"

:~X

has a dotted filler. Proof: If G is a strong stack, and E of the G-action on E

) X is a G-torsor over X, then the category of elements

)G0 is a strong stack by Proposition 1. But the category of elements is

also the equivalence relation on E determined by the surjection E

>X, so E

~X is injective

in ~/X by Proposition 2. In the other direction, suppose each G-torsor E

>X is injecfive in ~/X. As before,

since categorical weak equivalences injective on objects are stable under pushout, to show that

227

G is a strong stack, it suffices to show that each categorical weak equivalence injective on objects e: G'

>IH has a retract. As in Theorem 1, let E

H1

G0H0 is a G-torsor over H0. Furthermore, since e is full

and faithful, e*(E) = G1, i.e. we have a pullback diagram G"

~E

t] I

G~

e

>H0

Now t has the section u: GO >G1, and since E >H0 is injective over Ho it has a section s: H0 >E which extends u. As a result, the retraction r: ~3 >IFI constructed in Theorem 1 satisfies re = id, and we are done. • To finish this section, we give an example of a strong stack completion. Namely, let A be an abelian group in ~ considered as a groupoid in ~ with one object. Embed A in an injective abelian group I and denote the quotient I/A by B. Write p: I mapping. Let s,t: BxI (x, y)J >x+p(y).

>B for the quotient

;B denote, respectively, the projection, and the map

Proposition 4 The groupoid (s, t): BxI

>BxB, with composition given by addition, is a

strong stack completion of A. Proof: There is a functor from A to (s, 0: BxI -BxB given by sending the single object of A to the element 0 in B, and an arrow a in A to the arrow (0, a) in BxI. This is clearly a categorical equivalence injective on objects, so it remains to show that (s, t): BxI

~BxB is a

strong stack. To do this, recall that, for any abelian group, say C, in ~, there is an isomorphism Extl(7.X, C) = HI(X, C), where Z X is the free abelian group on the object X of ~. In fact,

228

the correspondence is given as follows: if 0 by C, then the pullback T

~X of E

)C

)E

; ~X

>0 is an extension of ~ X

>2~X along the inclusion X'

>Z X of the generators

is an C-torsor over X. We want to show first that the injective abelian group I is a strong stack considered as a groupoid with one object. (It is a stack by Theorem 1 since E x t l ( Z x , I) = 0.) We show that I-torsors satisfy the condition of Proposition 3. So, let T over X, which has a splitting s over Y, where Y'

)X. Let 0

>I

>X be an I-torsor >E

; ~.X .....~0, be

the extension corresponding to T

~X. The extension corresponding to the restriction of T to

Y is the pullback E'

~~ X along ~ Y '

>2~Y of E

~ZX. In the diagram

S

0

> I -----~E'-------~2~Y------~0

id4

0 ---~I the splitting s: E'

I

~E

~X----~O

>I can be extended to a splitting t: E

group. It follows that T

Proposition 3. Now, notice that I acts on B via p: I

t: B x I

>B, i.e. (x, y) i

(s, t): BxI

>I since I is injective as an abelian

>X is an injective object in ~/X, so I is a strong stack by >B. In fact, the action is just the map

>x+p(y). The category of elements of the action is the groupoid

>BxB, which is a strong stack by Proposition 1. •

3. Strong Stacks in Categories Until this point, we have only considered stacks and strong stacks for groupoids, since this is where all our applications lie. The theory has a perfectly good extension to arbitrary categories, however, which we now proceed to sketch. We begin with the classical definition of stack, rather than than the one adopted in section 1, so that the reader can more easily compare our treatment with that of [Giraud 1971] or [Bunge 1979]. Thus, let ~ be a Grothendieck topos, and suppose p: ~"

>~ is a categorical fibration

over ~. Then ~ is a stack ( for the canonical topology on ~ ) if (1) For each set I, and each I-indexed family { X i I i E I } of objects of ~, the canonical functor

igl

iml

is an equivalence of categories, and (2) For each surjcction q: X - - ~ Y of ~, the canonical functor ~ (Y)

~ d e s ~ (X) is an

equivalence of categories, where d e s ~ (X) is the category of objects of ~ (X) provided with descent data relative m the kernel pair of q.

229

A category C in ~ is called a stack if its externalization ~ (X) = hom(X, C) is a stack. Note that condition (1) is automatic in this case. To see more explicitly what condition (2) means, let q: X

"Y be a surjection of ~, and denote by ~ : the equivalence relation

XxyX, >XxX considered as a category in ~. q determines a categorical equivalence X >disY, and descent data on X with respect to q and ~: is simply a functor X ) £. Thus, C is a stack in the above sense iff for any q, each diagram ~ disY

##

~#j, ## J' £ has a dotted filler making the resulting triangle commute up to isomorphism. If C is a category in ~, let Iso(C) denote the groupoid in ~ whose objects are those of IE and whose morphisms are the isomorphisms of C. Let Cat(~) denote the category of categories in ~. For any surjection q: X---~Y, both ~ : and disY are groupoids, so the following proposition is immediate.

Proposition 5 ~E is a stack in Cat(~), iff Iso(£) is a stack(in the present sense) in Gpd(~).ll For any category A in ~, l e t T ( A y

>A0 be defined by

T ( A ) = {tE A o I V a E Ao3!a---~tin A1} T ( A ) is the collection of terminal objects of A . Passing from groupoids to categories in ~, torsors are replaced by locally representable functors. In fact, let £ be a category in ~, and F an internal, ~-valued, contravariant functor on C, with with structural map F >Co,and action a: FxcoC 1 >F. As before, the pair (a, ~I): FXCoC1

>FxF

is a category ][7 in ~ with objects F - the category of elements of the action. F is said to be

locally representable if T ( F ) is non-empty, i.e.T(]F)

~1 is a surjection. Note that when C is

a groupoid and the action is free and transitive, then T(]F) = F. Thus, if F is a torsor it is locally representable. On the other hand, again when C is a groupoid, if F is locally representable then the action is free and transitive, so F is a torsor. For a general C, F is locally representable iff T(]F) is an Iso(C)-torsor.

230

An action of C in Cat(~) on a family F >X indexed by X is an action of Xx~, considered as a category in ~/X, on F >X. Equivalently, this is an action of C on F, for which the map F

>X is constant. Given such an action, we say F

>X is a locally

representablefamily indexedby X if it is locally representable in ~/X. That is, we regard the collection T(]F)X of terminal objects of F computed fibrewise over X, and we ask that T(]F)x >X be a surjection. As above, this occurs iff T(]F)x >X is an Iso(C)-torsor over X. When T(]F)x >X is a surjection, it follows that X -~ rL0(]F), so that F is a locally representable family iff there exists a terminal object (in the internal sense) in each component of the category of elements of F. F is >X splits. said to be an X-indexedfamily of representablefunctors if T(]F)X When C is a category in ~, and X is an object of ~, the category hom(X, C) is def'med as in section 1. Let H(X, C) denote the category of locally representable families indexed by X. t:C1 ; C0 is a representable family indexed by C0, so (f: X ft~ and faithful functor hom(X, C) >H(X, E) as before.

>C0)l

ff*(C1) induces a

Theorem 3 The following are equivalent for a category C in ~. (i) C is a stack. (ii) Every locally representable family of functors is representable. (iii) For each X in ~, hom(X, C) >H(X, £) is an equivalence of categories. (iv) Every diagram of categories A c

e

>~

ss

] ~ SS J S £ with e a categorical equivalence injective on objects has a dotted filler making the resulting triangle commute up to isomorphism. (v) Every diagram of categories A

f

y, IB s

e

s

231

with f a categorical equivalence has a dotted filler making the resulting triangle commute up to isomorphism. Proof: (i) ~ (ii): Let F T(]F)X

)X be a locally representable family indexed by X. As above,

>X is an Iso(C)-torsor over X. If C is a stack, so is Iso(C) - in the sense defined

here. But this shows that any Iso(C)-torsor splits (see (iv) ~ (i) in the proof of theorem 1). Thus, T(]F)x

>X splits, and F

>X is a representable family.

(i.i) ¢ , (iii) is similar to (i) ¢=~(ii) of Theorem 1. ('ill) @ (iv) foUows the same pattern as (ii)~ (iii) of Theorem 1, except that the torsor E .....~H0 therein defined is now a locally representable family. The fact that this family is then representable allows us to define the retraction in the same way. (iv) @ (v) is the same as (iii) ~ (iv) of Theorem 1 - this argument does not depend on the fact that the categories involved are groupoids. (v) ~ (i) is obvious. • Let C be a category in S. A stack completion of C is a categorical equivalence C

.......>C*, such that C* is a stack. As before, stack completions are defined up to strong

equivalence of categories. To obtain the stack completion of a category C, it is enough to have the stack completion of its groupoid of isomorphisms. In fact, suppose Iso(C)

) Iso(C)* is a

stack completion of the groupoid Iso(C). Taking the pushout

Iso(~)

C provides a stack completion C

y ISO(C)*

> C*

>C* of C. The proof is immediate by Proposition 5.

Definition 3 A category E in ~ is a strong stack if condition (iv) of Theorem 3 holds on the nose. That is, if each diagram of categories

232

I

e

Ac

g

,e

-"- 113 ,t

,e

I ,e

C with e a categorical equivalence injective on objects has a dotted filler making the resulting triangle commute. Proposition 6 C is a strong stack in Cat(~) iff Iso(C) is a strong stack in Gpd(~). Proof: Clearly, if C is a strong stack in Cat(~), then Iso(E) is a strong stack in Gpd(~). The other direction follows immediately from the fact that if e: zK, ~IB is a categorical equivalence injective on objects, then Iso(A) c

I Ac

_'-- Iso0B)

e

~

I

is a pushout of categories, u Definition 4 A strong stack completion of a category C in ~ is a categorical equivalence C ' ~C* injective on objects, such that C* is a strong stack. In view of Proposition 6, the strong stack completion of a category C can be obtained from the strong stack completion of its groupoid of isomorphisms Iso(C) as above. As is the case for groupoids, strong stacks in Cat(~) are the fibrant objects for a Quillen homotopy structure on Cat(~). Namely, we have Theorem 4 There is a QuiUen homotopy structure on Cat(~), in which the weak equivalences are the categorical equivalences, the cofibrations are the functors injective on objects, and the fibrations have the right lifting property with respect to the cofibration weak equivalences. Proof: There are two ways to prove Theorem 4: either repeat verbatim the proof of Theorem 2,

233

which does not depend on groupoids, or repeat the first part, up to the point of showing that an arbitrary functor f: C

>ID in Cat(~) can be factored as a cofibration weak equivalence

followed by a fibration. At this point, factor Iso(f) in Gpd(~) as Iso(C) •

i

z- IE

IsoOD) where i is a cofibration weak equivalence in Gpd(~) and p is a fibration. Now take the pushout in Cat(~) i Iso(C) c ~ , -

£¢

IE

~IE*

and use the obvious generalization of Proposition 6, which states that f is a fibration in Cat(~) iff Iso(0 is a fibration in Gpd(~). []

4. Strong Stacks and Classifying Spaces Let ~ be a Grothendieck topos, and let S(~) denote the topos of simplicial objects in ~. Recall [Joyal (to appear)] that there is a Quillen homotopy structure on S(~), in which the weak equivalences are maps f: X

>Y inducing isomorphisms on the homotopy sheaves, the

cofibmtions are the monomorphisms, and the fibrations are maps having the right lifting property with respect to the cofibration weak equivalences, which we call anodyne extensions.

The homotopy category of S(~) is obtained by formally inverting the weak equivalences (or just the anodyne extensions) of S(~). If X and Y are objects of S(~), and Y is fibrant, then the set of maps from X to Y in the homotopy category is in 1-1 correspondence with the set IX, Y] of homotopy classes of maps from X to Y. (A homotopy is a mapping XxI >Y with I the constant simplicial sheaf on the 1-simplex A[1] of S.) We often use the principle of boolean localization to Wansferresults from the homotopy theory of simplicial sets to simplicial sheaves. Namely, for any Gmthendieck topos ~ there

234

exists a surjective geometric morphism p: ~B

>~ such that ~ is boolean and satisfies the

axiom of choice [Barr 1974]. The inverse image functor p* provides a faithful embedding of into ~B, whose logic is classical (i.e.ff5 is a boolean valued model of ZF set theory). Not all constructions are preserved by p*, but those using only colimits and finite limits are (the socalled geometric constructions). For example, p* preserves the construction of the homotopy groups of a simplicial sheaf. Thus, a mapping f: X ...... ;Y of S(~) is a weak equivalence iff p*(f) is. As a result, geometric constructions yielding weak equivalences in simplicial sets, yield weak equivalenees in S(~). The nerve of a groupoid E in ~ is a simplicial sheaf NG described as follows. (NE)n is the object Gn of composable strings of length n, G = Xn'-->xn-1 --->...xl ---->x0, of arrows of E . Faces and degeneracies are given by dOG = Xn--->Xn-l---->...--->Xl, dnG= Xn-1---->...-"->x0, and dig = Xn--->...Xi+l--->xi.1...---~x0 for 0< i < n, where Xi+l-->xi-I is the composite of the pair Xi+l-->xi--->xi. 1. siG = Xn-----~...---~xi---->xi--~...----~x0, where xi---->xi is the identity on xi. In particular, (NE)0 = GO, the objects of E, (NE)I = G1, the morphisms of E , and dO( c~: Xl----~x0) = Xl = so~,dl( ~: xl--->x0 ) = x0 = tc~, and s0x = id. x----->x= ux. When E is a groupoid in S(~), NE is a double simplicial object of ~, whose n-th column is the simplicial object Gn. Letting d: $2(~)

>S(~) denote the diagonal complex

defined by d(X)n = Xn n, we write BE for dCNE). In arguments involving B E , we often use the following, fundamental property of the diagonal complex. Namely, let f: X---->Y be a mapping in $2(~). We call f a vertical weak equivalence if fn * : Xn • >Yn * is a weak equivalence for each n_>0, and a horizontal weak equivalence if f, m : X, m

"Y* m is a weak

equivalence for each m_>0.Then it foUows by boolean localization, and the corresponding fact about double simplicial sets, that if f: X equivalence, df: dX

>Y is either a vertical or horizontal weak

>dY is a weak equivalence.

Proposition 7 Let E be a groupoid in S(~). If E ..... ~ E * is a strong stack completion of E, then G*0 is wcaldy equivalent (in S(~) ) to BE.

Proof: Proposition 3 says that each G* torsor is a ~vial fibmtion ( in S(~) ) over its base. In particular, t: G*I u: G*0

>G*0(and therefore also s) is a trivial fibration. Thus, the inclusion

>G*I of the units of E* is a weak equivalence. In fact, u defines a functor

disG*0 ........>E*, such that N(disG*0) ...... >N(E*) is a vertical weak equivalence. Taking the diagonal complex yields a weak equivalence G*0 ~ B(disG*0) is a categorical equivalence, NE t localization, so B E

,BE*. But since E '

>NE* is a horizontal weak equivalence by boolean

>BE* is also a weak equivalence, giving the result. I

>E*

235

Definition 5 A groupoid G in S(~) is said m be amenable if the functor H I ( , G) inverts anodyne extensions, i.e. passes to the homotopy category. Note that amenability is invariant under categorical equivalence of groupoids. Proposition 8 If G is an amenable strong stack in S(~), then G O is fibrant in the QuiUen structure on S(~). Proof'. Let G be an amenable strong stack in S(S), i: A ' f: A

>G0 a map. The pullback of t: G1

)B an anodyne extension, and

>GOalong f is a G-torsor T

)A over A. Since

113is amenable, there is a G-torsor S

) B whose restriction to A is isomorphic to T. Since G

is a strong stack, there is a map g: B

)G0 such that g*(G1) "" S. Thus, we obtain a mapping

h: A

)GI such that sh = gi and th = f. As above, s: G1

>GO is a trivial fibration in S(~),

so the commutative square h

A

i

rG 1

I "1 ks"

I

s

f

B

g

>G O

has a dotted filler k. The map tk is the desired extension of f to B. • Theorem 5 Let G be an amenable groupoid in S(~), and G,

>G* a strong stack completion

of G. Then for any X in S(~), HI(x, G) -~ [X, G*0]. That is, G*0 is a classifying space for G-torsors. Proof: Since G '

) G* is a categorical equivalence, HI(x, G) -~ HI(x, ~*), G* is amenable,

and G*0 is fibrant by Proposition 8. Since G* is a strong stack, HI(X, G*) "" x0(hom(x,G*)). Thus, we are left with showing that x0(hom(X, G*)) -~ IX, G*0]. More precisely, we want to show that for two maps f,g: X lifts to G*I, i.e. the pullback of t: G*I

)G*0, the map (f, g): X

>G*0×G*0

)G*0 along f is isomorphic to its pullback along g,

iff f is homotopic to g. For this, notice that the projection X×I

>X is a weak equivalence.

Hence, any G*-torsor over X×I is constant in I, by the amenability of G*. It follows that if f is homotopic to g, then the pullback of t: G*I g. In the other direction, consider the diagram

)G*0 along f is isomorphic to its pullback along

236

G* o

U

~ G *I 0

i 11 S ~'

s'

G~

(s, t) -'-G~}xG;

The right-hand vertical map is a fibration since G*0 is fibrant, u is a cofibration weak equivalence as above, so the diagram has a dotted filler. As a result, if (f, g): X

) G*0xG*0

lifts to G*I, then fis homotopic to g, and the theorem is proved. • We remark that in the category of simplicial sets, a locally transitive groupoid (ISis amenable, where locally transitive means that (s,t): G I ----~G0xG 0 is a Karl fibration. In fact, for simplicial sets the two concepts are equivalent.Thus, Theorem 5 provides a classifying space for any locally transitive simplicial groupoid, e.g. any simplicial group. See [3oyal-Tierney (to appear)] for a full discussion of amenability, local transitivity and classifying spaces. References

[1] M. Barr, Toposes without points, J. Pare App. Alg. 5, 1974, pp 265-280. [2] M. Bunge, Stack completions and Morita equivalence for categories in a topos, Cahiers Top. Geom. Diff. xx-4, 1979, pp 401-435. [3] M. Bunge and R. Pare, Stacks and equivalence of indexed categories, Cahiers Top. Geom. Diff. xx-4, 1979, pp 373-399. [4] J. Giraud, Cohomologie non abdlienne, Grundlehren #179, Springer Verlag, 1971. [5] P. T. Johnstone, Topos Theory, L.M.S. Monographs #10, Academic Press, London, 1977. [6] A. Joyal, Homotopy theory of simplicial sheaves, to appear. [7] A. Joyal and M. Tiemey, An extension of the galois theory of Grothendieck, Memoirs of the AMS vol. 51 No. 309, 1984 [8]

, Classifying spaces for sheaves of simplicial groupoids, JPAA, to appear.

[9] J. P. May, Simplicial Objects in Algebraic Topology, Van Nostrand Math.Studies #11, 1967. [10] D. G. Quillen, HomotopicalAlgebra, Springer Lecture Notes in Mathematics #43, 1967. [11]

., Rational homotopy theory, Annals of Math.90, 1969, pp205-295.

This paper is in final form and will not be published elsewhere.

T r e e s in D i s t r i b u t i v e C a t e g o r i e s S. K a s a n g i a n * S. Vigna* D i p a r t i m e n t o di M a t e m a t i c a , U n i v e r s i t £ di M i l a n o , V i a S a l d i n i 50, 1-20133 M i l a n o M I , I t a l y email: [email protected]

Dedicated to Max Kelly in the occasion of his 60th birthday.

1

Introduction

This note presents a brief account of one aspect of the talk given at the conference, and its aim is to relate work done by the authors on a labelled version of B6nabou's motors with distributive eatevories, viewed as a conceptual framework (as well as a practical tool) for the foundations of Computer Science (see [Law90, Coo90, Wa189, Wal]). Motors, introduced by Jean B~nabou in his talk at this conference (and also in lectures held in Rome and Milan [B~n90]) have been shown to provide a deep algebraic insight on the inductive nature of forests and also of labelled trees [KV91]. Hence they represent a highly valuable tool for manifold applications, specially in Computer Science. We would llke to mention, for instance, the algebraic characterization of Milner's observational equivalence shown in [DKV]. The relevance and ubiquity of distributive categories in many fields of Mathematics has been stressed by Lawvere and Schanuel (see e.g. [Law90]). They promise to become the conceptual, unifying framework for several aspects of Computer Science, as shown, e.g., in [Wal] and [Wal89]. In particular the Theory of Concurrency seems to benefit from the contrast between the "petit" and the "gros" aspects in distributive categories. In analogy with the classical formula for the free monoid [Mac71] we give a characterization of the initial X-motor in a distributive category ~D (where X is an object of ~D) as a power series in X with Catalan numbers as coefficients. Further, we extend B6nabou's unique decomposition theorem [B~n90, KV91] to distributive categories. We would like to thank Jean B~nabou, who taught us motors and gave us advice and support, and Bob Walters, since the idea behind this work arose in conversations with him, as witnessed (at least) by the flavour of his "apparently illegitimate calculations" [Wa189]. We would like also to thank the referee for his useful suggestions. *Workpartially supported by the italian M.U.R.S.T.and C.N.R., in particular by Progetto Finaliz~.ato Informatiea e CaleoloParallelo, obiettivo LAMBP~USCO.This paper is in final form and will not appear elsewhere.

238

2

X-Motors

Before introducing the general setting in distributive categories, we want to recall briefly the definition of X-motor as given in terms of set with operators. D e f i n i t i o n 2.1 Given a set X (to be thought of as an alphabet1), an X-motor is a tuple

(M, @, O, f ) , where (M, @, O) is a monoid, and f :X ×M---* M is a map. The definition above generalizes B~nabou's notion of motor, which we recover when X = {*}, so that f becomes an endoarrow of M. If a, fl, % . . . are elements of X , and t is an element of M, we will write a(x) (or even ax) for f ( a , x), according to the tradition of f~-group and R-module theory. By cartesian closedness, an alternative definition could be given using an X-indexed family of endoarrows of M. The resulting structure (which is basically a one-sorted algebra given by a monoid and by IxI unary operators) is soundly established in Computer Science as the basic tool for the study of synchronization trees (see, for instance, [Mil80, Mi189]). Indeed, this line of thought leads to a powerful calculus of trees which is basically shaped as the classical one, but is far more general (see [KV91]). Here, however, we are concerned with distributive categories, so we prefer Definition 2.1. As for the morphisms between X-motors, let us give the following D e f i n i t i o n 2.2 A morphism between two X-motors (M, @,0, f) and (M',(~', 0', f'), is an arrow ¢ : M ~ M' which is a monoid morphism and satisfies ¢(at) = ere(t), for all

t E M and a E X . Denote by Mx the initial object of X - M o t , the category of X-motors. In terms of the alternative definition (see above), the condition ¢(crt) = ere(t) simply means that ¢ is equivariant with respect to all the endofunctions. This viewpoint is certainly more intuitive, but Definition 2.2 generalizes to distributive categories (see Definition 3.2). Further, the axioms for an X-motor as given in Definition 2.1 are, after all, a weakening of the axioms for a left R-module. New we want to sketch some of the reasons justifying the use of X-motors to manage trees. Indeed, at first glance one could think that Mx is "too simple": after all, the initial object of M o n (the category of monoids) is the trivial monoid. But recall that 0 E M x , and that necessarily a(0) ~ 0 for all a E X, by the initiality of Mx. Further, still by initiality, we have that a(0) ~ fl(0) is a "new" element. Repeating these considerations one sees that Mx has actually many elements, but perhaps one does not have an intuitive insight of its structure. So we start with some heuristic remarks, which are essentially an adaptation of B~nabou's account of unlabelled forests. Let us consider an "object" T x whose members are to be thought of as trees ordered and labelled on their arcs. This object could have been defined in many different ways, and should also have been equipped with some operations. However, all we need to know 1Elsewhere we used :~ to denote the alphabet, according to the tradition of Computer Science. Here, however, ~ will be widely used to denote sums.

239

is that we can generate a new tree either by joining the roots of a pair of trees (preserving the order), or by adding a new labelled arc at the top of a tree. This second operation (left-prefix) can of course be performed with each label. The following picture illustrates these basic operations:

°(±)=% Of course, the "join" operation should be associative, and the "root-only" tree should play the r61e of the identity. So we would end up with a monoid. On the other hand, the "arc-creating" operation would be an X-indexed set of endofunctions, i.e., a map from X x M to M, so we would get a X-motor. Thus, we would have a unique arrow in X - M o t Mx

Pict Tx

since Mx is initial. Now, let us show some of the basic properties of labelled trees. For instance, we might wish to count how many arcs of a tree t E T x are labelled by a given label a. This means a m a p n o : T x .... ~ N. But i f t = t l @ t 2 then no(t) = n ~ ( t l ) + n ~ ( t 2 ) , and when one creates a new tree by adding a labelled arc at the top of a tree, n~(a(t)) = no(t) + 1, while n~(fl(t)) = ha(t) for every fl # a. This means that n : Tx

i< N,+,O,s~ >,

where s~(a,n) = n + 1 and s,~(fl, n) = n for all fl # a, is an X-motor map. But then, the unique map vc, : Mx ---~< N, +, O, s~ > must factor through Pitt, since by the initiality of Mx the diagram Mx

Pitt

, Tx

N necessarily commutes. Hence the slogan: "count the labels in Mx rather thaJa in T x ' . This process can be carried on again with entirely different X-motors. many other counting maps can be created this way (see [KV91D. It is now obvious in which sense Mx is, indeed, the object of X-labelled trees. The operation f creates new labelled arcs, (9 joins the roots and 0 is the root-only tree. Thus, o~(0) E) 8(0) and "),(a(O) • 8(0)) represent just:

240

It is worth observing that we disregard the real nature of Tx. We suggest that most (if not all) of the interesting properties of trees can be studied in M x , since most (if not all) of them should factor through Pier. Many interesting theorems can be proved about Mx. The fundamental one is that every element t E Mx has a unique decomposition a:(zl) @ a2(x2) @ " " $ an(x,~): this property is the link between the completely invaxiant definition and a more descriptive calculus which can be easily developed (see [KV91]).

3

Distributive categories

There is no established terminology about distributive categories. We give here a few definitions to encompass the cases which are of interest for us. D e f i n i t i o n 3.1 A category 29 is called distributive (in the sense of Waiters) iff it has finite products and coproducts, and the finite products distribute with respect to the finite coproducts, i.e., the arrow

A×B+A×C

,A×(B+C),

arising from the two arrows 1A × #B and 1A x ItV, is a natural isomorphism (ttB and tzc denote the injections). Since we want to be able to write power series, we will consider countably (resp. infinitely) distributive categories, i.e., distributive categories which have countable (resp. infinite) coproducts, and finite products which distribute over them. Further, a (countably, infinitely) distributive category will be said to be monoidal if it satisfies Definition 3.1 when the product is weakened to a tensor product. Extremely !mportant are also distributive categories in the sense of Lawvere-Schanuel. Recall that in their definition every slice category is required to be distributive: the relevance to Computer Science is due to the fact that in order to model labelled transition systems, one needs a category distributive in this sense. From now on we will often omit the product symbol. Integer indices without bounds axe assumed to range over N. When only natural isomorphisms are involved, we will denote them by equality. 3.1

X-Motors

in a distributive

category

Let us fix a countably distributive category 29 (actually, our results hold even in a monoidal countably distributive category). We axe interested in the most proper lifting of Definition 2.1 to this setting. It turns out that the appropriate definition is the following one:

241

Definition 3.2 A motor on the object (data type) X , or simply an X-motor, in a distributive category 79 is a tuple (M, It, ~?,f ) , where (M, It, ~) is a monoid and f :XM----* M is an arrow ofT). A morphism between two X-motors (M,#,r/,f) and (M',it',r/',f') is an arrow ¢ : M ~ M ' in 79 such that the following diagrams commute: I

'M

71

M' MxM

#

.M

¢

¢x¢ M' × M' XM

#1 • M ' f

¢

lx x ¢ ]

XM'

M

f,

M,

The reason for using in this context the term data type can be found in [Wa189]: an immediate naive motivation, however, is that in practical applications trees are "labelled" exactly in the sense that they carry data on each arc. This definition is completely internal, and allows us to speak of X-motors in 79: the main result we will prove is the existence of the initial X-motor: T h e o r e m 3.1 Given an object X of a distributive category 79, the initial X - m o t o r in 7) is M x = i + X + 2 X 2 + 5X 3 + 14X 4 + . . . .

~ C,,X" n>O

together with suitable #, 71 and f ; C, is here the n-th Catalan number, i.e.,

n+l

242

Before proving the theorem, we would like to give an insight. Catalan numbers are wellknown in combinatorics, because C,, is the number of binary trees with n nodes. However, we began with a heuristic discussion of trees, which are different from binary trees, even if the two concepts are known to be equivalent: indeed, in practical programming trees are always managed as binary trees 2. This correspondence is important, because (as we will see in the proof) the (apparently) complicated monoid operation of M x can be really understood only keeping binary trees in mind. To define p, r/and f we need a simple property of Catalan numbers, namely that if n > 0 then C. =

~_~

CkC,,-1-k = CoC.-1 + C , C . - 2 + . . . + C,,-1Co.

o_O

and we will denote with p.,,n the (n, m)th component of #. Since we want to use induction, we note that P.,m will land in C,~+~X "+'' and eventually in M x via the suitable injection, that will be understood with a slight abuse of notation. The map po,o is just the canonical isomorphism 1 x 1 = 1. Now, let us define pk,.-k for each integer 0 < k < n, and a fixed n. If k = O, let g0,. : C o X ° x C . X " ~ C . X " be the canonical isomorphism. If k > 1, given the t e r m C ~ X k x C . _ k X "-k, we know that CkX k x C._kX "-k = r-, X n - k + C, Ck_2X k x C . _ k X ~-k + . . . + Ck_~CoX k x C ~ _ k X ~-k CoCk_aX k x ,~.-k p

so we will break again I.tk,n_k in k maps ,uk,,_k, 0 < p < k - 1, from the last sum, and since

c.x~=

~

0 PlPt ° = 1B-

(3.5)

256

Suppose conversely that P has a right adjoint Q : B --~ A .

Since 1A < Q P ,

it follows, on taking

a = c = 1A in L e m m a 3.2, that for some e e E and some b we have b(P)e ; in other words, e = pit and b = P2 t for some t ;

so Pl like e lies in E . It remains to show that Pl is monomorphic.

Let x,y : K ----* P with pl x = p l y .

Using 1a < QP again, but now taking a = c = p:x in L e m m a

3.2, we get some e ~ g and some b with pÂxe(Q)b ; and since trivially we have p2xe(P)p:xe, Lemma 3.2 gives P2xe(PQ)b, whence p2xe = b since PQ d'S

respectively with square commutative, and componentwise composition. For each object a of a category C an instance of this construction is given by the category (C4/a) of objects over a. This is also called the slice category C/a (cf.[2]). It has objects f: c--->a and morphisms h: f --->f' with hf' =h. In this case T = 1 C, the identity functor on C, and S: 1 ---> C is the functor with value a on the category 1 which has one object and one morphism. The comma category construction for 2-categories is described by Gray in [5], page 29. We will be working with an instance of this construction, namely, that of the 2-category ([email protected]_.) of objects over E. Given an object E of a 2-category (3, the category (C,,ltlE) has objects U: C--->E and 1-cells (D,I~) with D: C--->D__ and p.: U===>DV a 2-cell of C. The 2-cells 13 of (C4/E) are 2-cells of C with ~'..13;V =1~.

284

In the 1-category case there are projection functors P, R: ( C ~ a ) ---> C defined on objects f: c-°->a by fP = c and fR = a and on morphisms h: f--->f' by hP = h:

c--->c' and hR = 1: a--->a.

Proposition 1.2: Each functor T: D ---> (C~I/a) determines a natural transformation ~: TP ===> TR. This transformation ~ in turn may be regarded as the same thing as a functor ~: D ---> (TP,,~,TR) with :~P' = =R' = 1 for P', R' the projections of ( T P 4 / T R ) . Proof:

A transformation 0: P ===> R is defined on objects f: c--->a of (C,J,,a)

by f = fo: fP = c---> fR = a. Given h: f--->f' in ( C ~ a ) we have hP.f'o = h.f' = f = f . l a = fe.hR and naturality follows. Given T: D ---> (Cqza) the natural transformation ~: TP ===> TR is given by = = To. The functor ~: D ---> (TPq/TR) is given on objects d by d~ = where dT: dTP .... > dTR as in (1.1)(cf. Mac L a n e [ I l l , page 48). Results from Proposition 1.2 can be generalized to (C,J/E) provided we replace the notion of natural transformation by lax natural transformation(cf. Gray [5]). Such a transformation ~ : F ===\ G between (strict) 2-functors F, G: C ---> D consists of the following data: a. For each object C of C there is a 1-cell C ~ : CF ---> QG of D. b. For each 1-cell f: C - - > D of C there is a 2-cell f~: C ~ ; f G ===>fF;D~ in D The following axioms hold: 1. The 2-cell f ~ is the identity if f = 10 . 2. The 2-cell f g ~ is the horizontal composite of f ~ and g ~ , that is, f g ~ = f~;gG..fF;g~. 3. Given B: f ===> f' in C, then f~.. 13F;D~ = C~;13G..f'~. As in (1.2) there are projection functors A, E: (Cq/E_) ---> C. defined on objects U: C ---> E by UA = C_C_,UE = E, on 1-cells (D,#) by (D,#)A = D and (D,I~)E = 1E and on 2-cells 2-cell on 1E.

13: (D',I~') ===> (D,I~) by 13A = 13: D' ===> D and BE = EA, the identity

Proposition 1.3: Each strict 2-functor T: K ---> (C,,ItE) determines a lax natural transformation ~: T A ===\ E. This transformation is the composite m

K ........ >

................. A

(C,,~,.~.)

u IJ" E

,,~

(1.4)

~ C

Proof: The value of ~ will be ~ = Tu for suitable u: A ===\ E. We define u on the object U: C ---> E of (C'4,'EL) by UU = U: C = UA ---> E = UE. The value of u on

285 the 1-cell (D,~) of (C~kE) must be a 2-cell D=(D,~)A C ........... > VA = D__

UA=

uu.oi

(1.5)

o, ,j vu.v

UE=E__. ............ > VE = E 1E=(D,li)E from the definition. Let (D,p.)u = I~. The lax naturality axioms for u hold as a consequence of the definition of 1-cell composition in ( C ~ E ) and the commutativity in C required of 2-cells in ( C , ~ E ) . An ordered category is a 1-category O whose hom sets are ordered with order preserved by composition (cf.Jay[7]). It is thus a simple kind of 2-category. If E is an ordered category and C is C a t then (C~,E) inherits an order.

2,

Program

functors.

Let E = S e t s * be the category of pointed sets. Its objects are pairs (X,*) where X is a set and * is a fixed element of X. A morphism f: (X,*) ---> (Y,*) in E is a function f: X ---> Y with f(*) = *. Such a morphism corresponds to a partially defined function fp from X \ {*} to Y \ {*} where fp(X) is defined if f(x) ~ * and if defined fp(X) = f(x). Suppose C = Cat.

Then the objects of (CUE_) are functors of codomain E and

1-cells (D,p.) are of the form (2.1) C . . . . . D . . . . >D__

where D is a functor and p.: U ===> DV a natural transformation. For each object c of C we have a morphism Cll: cU ---> cDV from the underlying pointed set of c to the underlying pointed set of its image cD under D. Let a program be a functor P: K ---> (CxPE) where the domain is a 2-category called the control category. The 2-category K may have only identity 2-cells. The 2-cell structure might be used to model phenomena such as the passing of

286 messages between threads running on separate processors. Note that the ordering in E allows us to compare programs according to the extent of their definition. By (1.4) there is a corresponding lax transformation Pu: PA===\ PE. The program steps are the values of P on the l-cells of K. Thus given j: K i ---> Kj in K there is a program step jP: KiP ---> K._jPin ( C,~,E ). By (2.3) and (1.9) we write jP = (jPA, jPu) and picture it by KiPA .... jPA .... >KjPA

(2.2)

K i P ~ E~ = j p u = > KjP A program step jP determines action on two levels. There is the abstract level A on which the functor jPA connects the categories ]~_iPA and KjPA of abstract data structures and there is the underlying or concrete value level u on which the natural transformation jPu assigns a function c(jPu) connecting the underlying pointed set of each object c of the abstract data structure category J~PA with the underlying set of its image object c(jPA) in K._jPA. The control category K is used in exactly the same way that sketches are used. That is, certain cones are required to be taken to limit cones by the models (i.e. the programs). See the section 3 for an instance of this (cf. Barr-Wells[2], p228, and Guitart[6]).

3.

Control categories for assembly language programs.

The control category K determines the flow of control in the program, it is an obvious category theoretic translation of the old idea of a flowchart. By examining one of the mathematical idealizations of the computer, namely, the unlimited register machine (URM) of Cutland [4], it is easy to see theoretically what data should be recorded by a control category K. In Cutland's model there are an infinite number of registers R 1..... Rn . . . . each containing a natural number. Contents of registers are altered by instructions and a finite list of instructions is called a program. There are four types of instructions, namely, zero (replace contents of R n by 0), successor(replace contents rn of Rn by rn+l ), transfer T(m,n) (replace contents of R n by rm, the contents of Rm) and jump J(m,n,q) (if rm=r n proceed to qth instruction of P, if not, proceed to next instruction). We remark that URM computable is the same as Turing computable (cf.[4], page 57). On theoretical grounds the control category K and program functor P: K --->

287 (C4,E) should thus deal with arithmetic operations, transfer (i.e. load and store) and jump (i.e. branch and loop) in the appropriate context. Implementation on actual computers appears more complex because of the variety of programming languages, microprocessors and machines available. For this reason we restrict our attention to programs written in assembly language and leave for later an analysis of "higher" imperative languages in the Algol family like C or Pascal. Instructions for assembly language programs are "close to the machine" and as such can be seen as variations on or as closely related to the types of instructions used in the URM. The disadvantage is the large number of instructions used as well as the fact that the specific form of the instructions themselves vary, as for example between those used for the M68000 family of microprocessors and the RISC instructions for the M88000 which we examine next (cf. [12] and [13]). For example, consider the following 14 line program in assembly language. This program computes a function, the Boolean AND of two bit vectors. It has one loop, calling for one label ("loop") and one branch point. The input is a linked list A,B,Tail where A and B hold arbitrary precision unsigned integers. For simplicity we assume A and B to have the same length. Each begins with a pointer to the next object on the list, so that the lengths of A and B are determined by these pointers. The output is a linked list C,Tail where C holds the logical AND of the arbitrary precision integers that were in A and B. ;rl=adr(B)=adr(B,Tail) (3.1) 01 Id r l , sp, r0 02 Id r2, r l , r0 ;r2 = adr(Tail) ;r3 =len(B), will be loop counter 03 subu r3, r2, rl ;sp=r2=adr(Tail) 04 add sp, r2, r0 loop: ;rl=adr(LSW(A)),A shrinks 05 subu r l , r1,#4 ;sp=adrLSW(B);Bshrinks,Cgrows 06 subu sp, sp,#4 07 Id r4, r l , r0 ;load argument from A 08 ld r5, sp, r0 ;load argument from B 09 and r4, r4, r5 ;r4=AAB, boolean operation 10 st r4, sp, r0 ;store AAB in C 11 subu r3, r3,#4 ;decrement loop counter ;branch on cond. loop counter 40 12 bcnd ne0,r3,1oop output: ;make space for pointer to Next 13 subu sp, sp,#4 ;storepointer 14 st r2, sp, r0 Only six different machine instructions are used in this bit vector program, Id, subu, add, and, st, and bend. These are instructions for the MC88100, which is the processor chip in the current version of Motorola's M88000 family

288 of processors(cf [13]). We call this simply the 88000, There are 52 machine instructions available on the 88000 if we count all the distinct mnemonic roots in the user's manual. Some of these instructions are just variations on each other. For example, included among the 52 are four variations on add, namely, add, add.ci, add.co and add.cio in which .ci stands for carry in , .co for carry out and .cio for carry in and out . For a d d . c i , the current value of the (one-bit) carry register (left there by some earlier computation) is added to the sum; for a d d . c o the carry resulting from the sum is stored in the carry register; a d d . c i o does both and the unaltered add does neither. Most of the 52 instructions are sufficiently like the six in the program (3.1) so that our analysis go through unchanged except for details and hence for other programs on the 88000 as well. Other microprocessors, such as the MC68000, use instruction sets written in different form although these can be analysed in the same way by dividing them into those of arithmetic, transfer or branch type. We remark that the above program does contain instructions representing arithmetic (subu, and, add), transfer (id, st) and jump or branch (bend) phenomena, hence representing the types of instructions found in the URM. Here, are the machine instructions used in the program above. Id a, b, c Load register a with data pointed to by register b, offset by the address in register c. So if c above is rO, which always stores zero, then the instruction simply loads register a with the data stored at the location pointed to by register b. add a, b, c Add the numbers stored in registers b and c and put the result in register a. subu a, b, c Subtract the number stored in c from that in b and put the result in register a. and a, b, c Perform the bit-wise Boolean and of the numbers (now regarded as elements of the boolean algebra with 232 elements) in registers b and c, and put result in a. st a, b, c Store the contents of register a at the location pointed to by register b, offset by the address in register c. bcnd neo, b, label Branch, if this condition is satisfied: the number in register b is not equal to zero, to the point in the code indicated by label. We remark that there are restrictions in the 88000 instructions on the

289 values that a, b, and ¢ can take, particularly for the arithmetic operations. For example, for add we must have b and c take values in the processor registers. For this 14 line example the control category K is the free category generated by the following diagram.

K

I

J~-l->

Loop

. . -4-> . . K 4. -5-> .

1 2 ' ! t ~ ~ 1 2 #~lbb s 2B

-11->

11 .('; "" t

--12 -13->

(3.2)

... -14-> _K.t4

The 1-cells of the control category are named with the line numbers of the program. Nodes of the diagram, labelled _K.i, 0 b AND c 10: ~ ---> KlO, D n ---> Nil (D n pointed to by R31), R5 ---> R5 ]-[ D n with b~~~> .

11:K10 ---> K l l , R3 ---> R3, b~~->

b-4 (decrement loop counter in R3). The next program step12 uses two equalizers in ( C ~ E ) . More precisely program step 12 is b c n d , branch on condition. The machine branches if the value in R3 is not zero. The part of the control category K involved is

K

~I Loop

K12#

12'

,~

B

(5.6)

S~1'~3

K 4 -5-> ... -11->

The value of P: K ..... > ( C ~ E ) on B is called the boolean concrete category and may be written as B P A ---BP--> E where the abstract (or addressed) category B P A has two objects B and Nil. The underlying pointed set functor maps B to the pointed set {0,1,*} and maps Nil to {*}. This B corresponds to a register the programmer doesn't see, something used by the processor itself, in its own computations in the background. BPA is a pointed category with B]-[Nil = B. Now we return to program step 12.

The control 2-category K has objects

296 K12#, K12 and B as above. The functor PA maps these to K12#PA, K12PA and BPA where K12PA and K12#PA are copies of the abstract category K 11A. As before we mention only the "interesting" parts of each abstract functor and of each underlying natural transformation. There are three 1-cells b 1, b 2, b 3 : K 11 ""> B in K. The abstract 2-functor PA maps these to abstract 1-functors that are identical , mapping R3 ---> B and everything else that is not a product with R3 to Nil and an object R3 ~ X gets mapped to B ~ Nil = B. biPA

(5.7)

K11 PA ........ > BP A biP:

The concrete functor P__ puts different underlying natural transformations under these 1-cells. That is, blPA = b2PA = b3PA but biPu ~ bjPu for iCj. The interesting components are defined on R3 which has for underlying set under K 11P' a pointed set with, say 232(defined) elements and the 1 undefined element or point. The interesting components map a. every defined element to 0 (blPU). b. all nonzero defined elements to 0 and 0 to 1 (b2Pu). c. every defined element to 1 (b3Pu). The first and third are "constant" pointed functions. We require sP: K12#P ---> K11P to be the equalizer of blP = (blPA, blPu ) and b2P and tP: K__.12P---> K11P to be the equalizer of b2P and b3P. That is, we require them to be equalizers in (C~,E). In this sense we regard K as a sketch with cones based on s, b 1, b2 and t, b2, b 3 and require P (but not PA) to be a model of K. For the specific form taken by the equalizer in this case we first note that biP = (biPA, BiPu) has been described in (5.8) and now descdbe sP = (sPA, sPu) and tP = (tPA, tPu). The abstract 2-functor PA takes the same category as value on control objects K l l , K12 and K12# and the functors sPA: K12#PA ---> K11PA and tPA: K12PA ---> K l l P A are identity functors. However, the concrete 2-functor P does not take the same value on K l l , K12, and K12# since the concrete categories K11P: K11PA ..... > E, K12P and K12#P differ in their values on R3. The value of K11P on R3 has already been described. In the concrete category K12#P the object R3 has an underlying set with 232-1 defined elements(nonzero elements)

297 plus point while in K12P the underlying set of R3 has one defined element plus point. Then sPu and tPu are the inclusion of nonzero elements and inclusion of zero, respectively, when applied to R3. The horizontal composite in (C.JtE) of the path from the beginning, starting with the initial input (that is, with the underlying set of the structured element) can be continued nontrivially through only one of the concrete categories K12#P and K12P. This is the case because the composites of the whole path with the parallel pairs (blPu, b2Pu ) and (b2Pu, b3Pu ) will be equal for exactly one of the parallel pairs. Once started, the program can continue only in one way once it reaches the branch point. Finally program step 13 is like step 3 and step 14 is like step 10.

References

[1] L. Allison, A Practical Introduction to Denotational Semantics, Cambridge Computer Science Texts 23, Cambridge University Press, 1986. [2] M. Barr, C. Wells, Category Theory for Computing Science, Prentice Hall (1990). [3] M. Barr, C. Wells, Toposes, Triples and Theories, Springer(1985). [4] N. Cutland, Computability, Cambridge University Press, 1980. [5] J . W . Gray, Formal Category Theory h Adjointness for 2-Categories. Springer Lecture Notes in Mathematics 391 (1974). [6] R. Guitart, On the geometry of computations, Cahiers Top. et Geom. Diff. XXVII-4(1986), 107-136. [7] C. B. Jay, Extending properties to categories of partial maps. Laboratory for Foundations of Computer Science, Report 90-107, Edinburgh(1990). [8] L.A. Leventhal, D. Hawkins, G. Kane, W.D. Cramer, 68000 Assembly Language Programming, 2nd Edition, Osborne McGraw-Hill (1986). [9] J. MacDonald, A. Stone, Soft adjunctions between 2-categories. J. Pure Applied Algebra 60 (1989), 155-203. [10] J. MacDonald, A. Stone, A class of 2-adjunctions invariant under perturbation. To appear. [11] S. Mac Lane, Categories for the Working Mathematician, Springer(1971). [12] MC 68000 16-Bit Microprocessor user's manual, 3rd Edition, Prentice Hall (1982). [13] MC 88100 RISC Microprocessor user's manual, 2nd Edition, Prentice Hall (1990). This paper is in final form and no similar paper has been or is being submitted elsewhere.

WEAK

PRODUCTS

OVER A LOCALLY HAUSDORFF

LOCALE

Susan B. Niefield Union College, Department of Mathematics Schenectady, NY 12308, USA

Introduction

Let T o p and L o c denote the categories of To topological spaces and locales (in the sense [3]), respectively. For an introduction to locales, we refer the reader to [5] and [9]. It is well-known that if X and Y are topological spaces, then the IocMe product f~(X) x f~(Y) of their open set lattices may differ from f~(X timesY), i.e. the induced locale morphism f: f~(X x Y) --+ ~ ( X ) x f~(Y) need not be an isomorphism. In particular, Isbell [3] showed that f is not an isomorphism, when X = Y = Q, the rationals with the subspace topology of the real line. In a positive direction, it is not difficult to show (and part of the folk-lore of the subject) that if X is locally compact, then f is an isomorphism for all spaces Y. For a partiM converse to this result, the reader is referred to [4]. Instead of further investigating the relationship between the products of spaces and of locales, Peter Johnstone and Sun Shu-Hao [6] introduced a monoidal structure ® on Loc called the weak product and showed that f/(X) ® ~(Y) ~ f~(X x Y), for all spaces X and Y. They also considered the largest coreflective subcategory W S p of L o c for which ® is the categorical product. Now, if X and Y ave spaces over a base space T, then one can consider the induced locale morphism f:f~(X XT Y) ---* f~(X) xfl()r) f~(Y). It is well-known that if X is a locally closed subset of T (i.e. the intersection of an open and a closed), then f is an isomorphism for every space Y over T. Generalizing this result and the Folk Theorem mentioned above, Niefield [8] showed that if X is exponentiable as a space over T, then f is an isomorphism for every space Y over T. Of course, the exponentiable sober spaces are precisely the locally compact ones (see [1] and [2]). In this paper, we consider a generalization of the weak product to Loc/L. After a brief summary of the relevant results from [6], in Section 2 we consider the question of preservation of equalizer by the functor f~: T o p --+ Loc. In Section 3, we investigate locally Hausdorff locales, providing examples of locales satisfyinga condition arising in the previous section. Next, a definition of ®L is introduced in Section 4 and shown to give the product in W S p / L . We conclude by showing that ®L is an associative bifunctor on Loc/L, but that L is not a unit.

299

1. W e a k P r o d u c t s

and Hausdorff Locales

In this section, we recall the properties of the weak product ® which will be used throughout this article. For details, we refer the reader to [6]. Property

1.1 ® is a symmetric monoidal structure on L o c .

Property

1.2 ® is a subfunctor of ×.

T h r o u g h o u t this paper, we will write A ® B ~ A x B to denote this subfunctor. Note that since we wilt not work directly with elements of A ® B, we will omit a description of the nucleus on A x B which defines it. An inclusion A' ~-* A of locales is called locally closed if A' is the intersection of an open and a closed sublocale of A, or equivalently, if it is the composition (in either order) of an open and a closed inclusion. P r o p e r t y 1.3 If A' --* A and B' --* B are locally closed inclusions, then the following diagram is a pullback in Loc. A'®B I

'

,

A®B

l A' x B'

J, ~ A ×B

A locale A is called weakly spatial if the diagonal morphism A: A --* A x A factors through A ® A. The full subcategory of L o c consisting of the weakly spatial locales is denoted by W S p . For any locale A, we can consider the following pullback. A

~ A®A

$

$

A

~ AxA

We will refer to the morplfism A --~ A ® A as the weak diagonal of A. T h o u g h A need not be weakly spatial, repeating this process (infinitely often, if necessary) gives rise to: Property

1.4 W S p is a corefiective subcategory of L o c containing all spatial locales.

Property

1.5 ® is the product in W S p .

P r o p e r t y 1.6 For any To spaces X and Y, the induced map f t ( X x Y ) ---* f ~ ( X ) ® ~ 2 ( Y ) is an isomorphism. A locale A is called Hausdorff if whenever a and b are elements with a ~ 1 and a ;~ b, there exist x and y in A such that x A y = 0, x ~ a, and y ~ b. Property

1.7 If A is Hausdorff, then the weak diagonal of A is locally closed.

Note that although the converse to P r o p e r t y 1.7 does not hold, an additional condition can be added to obtain a characterization of this Hausdorff property [6].

300 P r o p e r t y 1.8 For a To space X, ~ ( X ) is Hausdorff if and only if X is Hausdorff. P r o p e r t y 1.9 If A is Hausdorff, then A is the spatial part of A. Moreover, a Hausdorff locale is spatial if and only if it is weakly spatial. 2. E q u a l i z e r s In this section, we consider which equalizers of T o p (respectively, W S p ) are preserved by f~ (respectively, the inclusion W S p --* Loc). In either case, if the morphisms X -~ Y form the cokernel-pair of E ~ X, then the equalizer E "-* X ~ Y is preserved, since the functor in question preserves pushouts being a left adjoint. Also, every equalizer in T o p (as well as Loc) factors through one of this form. However, not every equalizer if preserved, for consider the following example. E x a m p l e 2.1 Let A ~ ~ ( X ) be any regtflar monomorphism in Loc, where A is not spatial and X is a topological space, for example, ~(l=t).., ~ ~/(R). Let f, g: ~ ( Z ) -~ L be the cokernel-pair of A ~ ~ ( X ) in Loc. Note that L is spatial since it is a quotient of a spatial locale. Write L = Y/(Y), where Y is a sober space and consider the equalizer ]

E

(.)

X- Y

where fl(f) = f and g~(9) = g. Since A ~-* ~ ( X ) is the equalizer of f and g in Loc, it follows that g~ does not preserve the equalizer (*). If A is not weakly spatial, a similar argument gives rise to an equalizer in W S p which is not an equalizer in Loc. Note that g~(R)-,~ is not weakly spatial by 1.8 and 1.9. Next, we will consider condition on Y such that every equalizer diagram with codomain Y is preserved. We begin with a 1emma. L e m m a 2.2 A locally closed sublocale of a (weakly) spatial locale is (weakly) spatial. P r o o f For the spatial case, we refer the reader to [5]. The weakly spatial case follows directly from Property 1.3. P r o p o s i t i o n 2.3 Suppose E ~-4 A ~ L is an equalizer in Loc, where L is a locale whose weald diagonal is locally closed. Then E is a locally closed sublocale of L. Moreover, if A is (weakly) spatial, then so is E. P r o o f Suppose A is (weaidy) spatial and f, g: A --* L. Then (f, g): A -* L x L factors through L ® L, since the diagonal A ~ A x A factors through A ® A. Since the equalizer of f and g can be obtained by pulling back (f, g) along the diagonal of L, we have the following diagram of pullbacks in Loc E

,

L

~

L

A

~

L®L

"

LxL

301

Since the weak diagonal of L is locally closed, it follows that E is a locally closed sublocale of A, and so E is (weakly) spatial by Lemma 2.2. C o r o l l a r y 2.4 If E ~-~ A ~ L is an equalizer in W S p and L has a locally closed weak diagonal, then it is an equalizer in Loc. P r o o f Since W S p is a coreflective subcategory of Loc, equalizers in W S p are obtained by applying the coreflection to the equalizer of the morphisms in Loc. Thus, the desired result follows from the Proposition. In the next section, we will show that every spatial locale with a locally closed weak diagonal is of the form ~(T), where T is locally Hansdorff. Thus, we have the following corollary. C o r o l l a r y 2.5 If T is locally Hausdorff and E ~ X ~ T is an equalizer in Top, then ~ ( E ) ~ f/(X) ~ f2(T) is an equalizer in Loc.

3. L o c a l l y H a u s d o r f f Locales Next, we consider a class of examples of locales with locally closed weak diagonals, including the (weakly) spatial ones. A locale L is called locally Hausdorffif L is a join of Hausdorff open sublocales, i.e. L = V L~, where each L~ is a Hausdorff open sublocale of L. L e m m a 3.1 If L = VL~, where each L~ is an open sublocale of L, then La is open in L and L = V L . . P r o o f Consider the following commutative cube, where the front and back faces axe pullbacks by definition of the weak diagonal and the right face is a pullback by Property 1.3.

L~

, L~ ® L ~

\

\ L

,

L®L

1 ---*

L~

\

L~ x L~

\ L

, LxL

302

Thus, the left face is also a pullback and so L~ is an open sublocale of L (since pullbacks preserve open sublocales) and L = V La (since pullbacks preserve sups of open sublocales). P r o p o s i t i o n 3.2 If L is locally Hausdorff, then the weak diagonal is locally closed. P r o o f Write L = V La, where each La is a Hausdorff open sublocale of L. Since La is Hausdorff, we know that the weak diagonal La ~ La ® La is closed by Property 1.7, mad so La ~ L ® L is locally closed. Thus, each L~ is open in the closure of L in L ® L. Since/~ = V L~ by Lemma 3.1, it follows that L is open in its closure. P r o p o s i t i o n 3.3 The following are equivalent for a space T: (1) Every point of T has a Hansdorff neighborhood (2) T is a union of Hausdorff open subspaces (3) The diagonal T -~ T x T is locally closed (4) The diagonal T ~ T x T is exponentiable in T o p / T x T P r o o f The equivalence of (1)- (3) is left as an exercise and (3) ~ (4) appeared in [7]. A space satisfying the equivalent conditions of Proposition 3.3 is called locally Hausdorff. It is not difficult to show that such spaces are necessarily sober. C o r o l l a r y 3.4 For a To space T, ~(T) is locally Hausdorff if and only if T is locally Hausdorff. P r o o f If T is locally Hausdorff, then (2) of Proposition 3.3 implies that ~(T) is locally Hansdorff by Property 1.8. Conversely, if f~(T) is locally Hausdorff the weak diagonal g/(T) ~ ~ ( T x T) is locally closed, and so the diagonal T ~ T x T is locally closed, as desired. P r o p o s i t i o n 3.5 A locally Hausdorff locale is spatial if and only if it is weakly spatial. P r o o f Suppose L = V L~,, where each La is a Hausdorff open sublocale of L. If L is weakly spatiM, it follows that L~ is weakly spatial (Property 1.3), and so L~ is spatial (Property 1.9). Therefore, L is spatial.

4. P r o d u c t s in W S p / L As in any slice category, products in W S p / L can be obtained via an equalizer E ~--~A ® B - ~ L

in W S p . If L has a locally closed weak diagonal, then we know that this is just the equalizer in Loc (Corollary 2.4).

303

Now, if A and B are any locales over L, then we define their weak product A ®L B by the following equalizer in Loc. A @L B '--* A @ B---~ L

P r o p o s i t i o n 4.1 The canonical diagram A®LB

,

A®B

AXLB

,

AxB

is a pullback. P r o o f Consider the diagram A®LB

,

A®B

AXLB

,

AxB

\ /

Since the bottom row is an equalizer, the desired result follows. T h e o r e m 4.2 If X and Y are spaces over a locally Hausdorff space T, then the canonical morphism f: fZ(X XT Y) -* ~/(X)@n(T)Y/(Y) is an isomorphism. P r o o f Consider the diagram

n ( x ×r Y)

,--,

n ( x × Y)

\ \

a(X)®a(n a(Y)

~

n(T)

a ( x ) ®a(Y)

where the rows are equalizers by Corollary 2.5 and the definition of @n(T). Since g is an isomorphism by Property 1.6, it follows that f is an isomorphism. T h e o r e m 4.3 If L has a locally closed weak diagonal, then ®L is the product in WSp/L. P r o o f Since W S p is a coreflective subcategory of Loc, equalizers in W S p are obtained by applying the coreflection to the equalizer formed in Loc. Thus, Corollary 2.4 gives the desired result.

304

5. A s s o c i a t i v i t y o f ®L We know that if L has a locally closed weak diagonal, then ®L gives a symmetric monoidal structure on W S p / L , being the product. We conclude by considering ®L as a bifunctor on L o c / L . Clearly, ®L is a symmetric bifunctor. T h e o r e m 5.1 If L has a locally closed weak diagonal, then @L is associative. P r o o f Consider the following diagram

A®L(B®LC)

,

A®(B®LC)

,L

,

A®(B®C)

J,

1

AxL(B®LC)

,

Ax(B@LC)

~ Ax(B®C)

AxL(BXLC)

,

Ax(BxLC)

* Ax(BxC)

where the northwest square is the pullback defining A@L (B @L C) using Proposition 4.1, the northeast square is a pullback by Property 1.3 and Proposition 2.3, the southwest square is a pullback since both horizontal arrows are equalizers of morphisms with codomain L, and the southeast square is a pullback since A x - preserves pullbacks. Thus, we obtain a pullback A ®L (B ®L C)

) A @ (B @ C)

AXL(BxLC)

,

Ax(BxC)

Similarly, (A @L B) ®L C is given by a pullback. Using the associativity of @, x, and xL, it follows that @L is associative. Finally, we note that L is not a unit for @L. In fact, A @L L -~ A if and only if the graph A --~ A x L of the structure morphism A ~ L factors through A ® L. In particular, L would have to be weakly spatial. But, even when L is spatial this condition presents a problem, for example, when L = fl(R) and A = f/(l=t)-,-~. Moreover, any definition of @L which is compatible with "change of base" gives rise to a morphism A @L B --* A ® B, and hence the same difficulty.

305

References [1] B.J. Day and G.M. Kelly, On topological quotients preserved by puUbacks or products, P r o c . Camb. Phil. Soc. 67 (1970), 553-558. [2] K.H. Hofmann and J.D. Lawson, The spectral theory of distributive continuous lattices, Trans. A m e r . Math. Soc. 246 (1978), 285-310. [3] J.R. Isbell, Atomless parts of spaces, M a t h . Scand. 31 (1972), 5-32. [4] J.R. Isbell, Product spaces in locales, Proc. A m e r . M a t h . Soc. 81 (1981), 116-8. [5] P.T. Johnstone, Stone Spaces, Cambridge University Press, 1982. [6] P.T. Johnstone and Sun Shu-Hao, Weak products and Hausdorff loc~es, in Categorical A l g e b r a and its Applications, Lecture Notes in Mathematics 1348, SpringerVerlag, 1988. [7] S.B. Niefield, Cartesianness: topological spaces, uniform spaces, and afflne schemes, J. P u r e Appl. Alg. 23 (1982), 147-67. [8] S.B. Niefield, Cartesian spaces over T and locales over f/(T), Cahiers de Top. et G~om. Diff. 23 (1982), 157-67. [9] S. Vickers, Topology via Logic, Cambridge University Press, 1989.

This paper is in final form and will not be publishedelsewhere.

Categorical Interpolation: Descent and the Beck-Chevalley Condition without Direct Images Du~ko Pavlovi6 Zevenwouden223, 3524 CR Utrecht,The Netherlands

Fibred categories have been introduced by Grothendieck (1959, 1971), as the setting for his theory of descent. The present paper contains (in section 4) a characterisation of the effective descent morphisms under an arbitrary fibred category. This essentially geometric result complements a logical analysis of the Beck-Chevalley property (section 1) - which was crucial in the well-known theorem on sufficient conditions for the descent under bifibrations, due to Brnabou-Roubaud (1970) and Beck (unpublished). We describe the notion o f

interpolants

(sections 2 and 3) as the common denominator of the concepts o f descent and the BeckChevalley property. (For the basic notions and facts about fibred categories, the reader can consult Gray 1966, or BEnabou 1985. A survey can also be found in Pavlovi6 1990.)

1. T h e B e c k - C h e v a l l e y c o n d i t i o n / p r o p e r t y 11. Proposition. Let E: ~----> ~B be a bifibration, Q = (f,g,s,t) a square in ~B such that fog = sot, and O = (~p,T,~,O) a square in g such that ¢poT= coO, with Eq0=f, ET=g, Ecr=s and EO=t. The following conditions are equivalent: a) if 1} and q~are cartesian and if o is cocartesian then y must be cocartesian; b) if o and y are cocartesian and if O is cartesian then cp must be cartesian; c) if O is cartesian and if c is cocartesian then q~is cartesian iff y is cocartesian.

307

If some inverse image functors f* and t* and some direct image functors g! and sI are chosen, then every square O over Q satisfies conditions (a-c) iff there is a canonical natural isomorphism d)

f's! " g!t*.

12. Definition. A square Q in the base of a bifibration E:8 --9 ~ satisfies the Beck-Chevalley condition if every square O over Q satisfies conditions (a-c). E is said to have the Beck-

ChevalIey property if all the pullback squares in ~ satisfy the Beck-Chevalley condition. "Beck-Chevalley" will be abbreviated to "BC". 13. Proposition. A bifibration E has the BC-property iff the cocartesian arrows are stable under those pullbacks along cartesian arrows which E preserves. 14. Sources. The Beck-Chevalley condition has arisen in the theory of descent - as developed from Grothendieck 1959. Jon Beck and Claude Chevalley studied it independently from each another. The former expressed it in the form 1 l(d), the latter as in 1 l(a). It is conspicuous that neither of them ever published anything on it. Early references are: Btnabou-Roubaud 1970, Lawvere 1970. The proofs of propositions 11 and 13 are elementary. They can be found in my thesis (1990). 15. Logical meaning of the BC-property. Consider a fibration E:8 ---9 ~ as a "category of predicates": the base category $ is to be thought of as a category of "sets" and "functions", while the objects and arrows of a fibre eI represent "predicates" ct(x I) over the "set" I, and "proofs" between them. In this setting, the logical operation of substitution is interpreted by the inverse images. An inverse image functor over a "function" t: I---) J in :B can be understood as mapping

t*:Sj ---) 8I: ~(yJ) k---)15(t(xI)). Lawvere (1969) noticed that the quantifiers are adjoint to the substitution:

O~(xI) I- ~(t(XI)) ~(t(xI)) t-- (g(XI)

¢:~ ¢=>

3XI(t(XI)=y J ^ ~(xI)) I- ~(yJ), ~(yJ) F- VXI(t(xI)=y J ~ ~(XI)),

so that the logical picture of the direct image functors (t!-q t*---t t,) becomes t! : el ----) 8j: 0~(xI) ~ 3x I (t(xI)=yJ A ~(x I) ), and t,: el-'--)8j: 0t(xI)k---)VxI(t(xI)=yJ ~ Ct(xI)). What does the Beck-Chevalley property mean in this context? The simplest case is when the commutative square Q consists of projection arrows. A direct image functor along a projection ~: KxM----) K just quantifies a variable, while an inverse image functor adds a dummy:

308

X*:~K'-'~ ~KxM: X(XK) I----)x(xK,/xaM) X! :~KxM--') ~K : ~( XK, zM) t'-) 3zMAI/(XK, zM), X*:~KxM--) ~K : ~(X K, zM) t--) VzM.~I/(xK, zM).

The picture of the BC-conditionis:

g(x,~

~ , ~

n ~ ~

3z.xg(x~,z) 3z.~(x, z) E

KxLxM

""

K

/

KxL

"rhe quantifier 3z and the variable y do not interfere"- says the BC-condition here. If we apply 3z on ~(x,/y~,z), we get the same result as when when we apply it on ~(x,z) and then add y. Over a general pullback square Q, this picture becomes 13(t(x)), I .

s

i

~

3 ~ ) = f ( w ) ^ ~(y) 3y(s(y)=z^ ~(y))

I

E

309

A proof 3x(g(x)=w ^ ~(t(x))) F- q y (s (y)=f(w) ^ [3(y)) can be derived from a proof that Q is commutative: V- f(g(x))=s(t(x)). The converse proof 3y(s(y)=f(w) ^ ~3(y)) ~- 3x (g(x)=w ^ 13(t(x))) - follows from s(y)=f(w) v- qx(t(x)=y ^ g(x)=w), which tells that Q is a (weak) pullback. In this way, logic suggests the demand for an isomorphism f'st(13) = g!t*(~) when Q is a pullback.

2. Interpolation condition 21. Motivation. The fact that variables do not interfere with each other can be expressed in a different way, without quantifiers: a(x,y) ~- T(y,z) ¢~ there is an interpolant 13(y), such that a(x,y) ~- [~(y) e- 7(y,z). At the first sight, this seems to be a different idea of the independence of variables. Surprisingly, it is not. We show in the sequel that the BC-property - whenever it can be expressed- is equivalent with the existence of a certain kind of interpolants.

22. Notation. For a given fibration E:~ ---> a3, an arrow te ~B(I,J) and an object Ye ICjI (i.e. EY=J), O~,: t'Y----) Y denotes an arbitrary cartesian lifting of t at Y; if E is a cofibration (i.e. if its dual EO:~°--> ~3o is a fibration), then ok: X---) t!X will be an arbitrary cocartesian lifting of t at an object X over I (i.e. a cartesian lifting with respect to E°). In general, we do not choose the whole (co)cleavages, but we do use these generic symbols for an arbitrarily chosen cartesian or cocartesian arrow. Moreover, the unique vertical arrow by which o k factorizes through Otl~£ will be TI: X----) t*t!X - for the obvious reason that this arrow would be a component of the unit of the adjointness t!--t t*/f functors t! and t* were chosen. Similarly, given a vertical arrow q: tlX--> Y, we denote by q': X---> t*Y the unique vertical arrow such that ~ ~o q' = q o o)~: this is the "right transpose" of q by t!--4 t*. Given vertical p:X---> t ' Y , its "left transpose" 'p: tlX---> Y is the unique vertical arrow such that O~,op = 'poo)~. The unique vertical isomorphisms between various inverse images of an object along an arrow will all be denoted by "¢. E.g., if fog = sot, g*f*B and t*s*B are inverse images of B along the same arrow, and there is a unique vertical iso 'c: t*s*B---) f*g*B. Note, finally, that the thick points o.... enclose the (sketches of) proofs.

310

23. Definition. Let E: ~ --) ~B be a fibration, and Q = (f,g,s,t) a commutative square in B. An (Q-)interpolant of an arrow de ¢ i ( t * A , g ' C ) is a triple (a,B,c), where Be leKI, ae ~j(A, s'B), c~ ~L(f*B, C), such that d = g*(c)oxot*(a).

Ot

A

4s

-t~ A d ' ~ g *

t*s*B "~ g*f*B

s*B

,z B

f*B

E

K A square Q in the base of a fibration E satisfies the interpolation condition if there is a Q-interpolant for every arrow de ~i(t*A, g'C). 24. Proposition. Let E: ~ ~ ~ be a bifibration. A commutative square Q = (f,g,s,t) in B satisfies the interpolation condition iff the vertical arrow P=PA : g!t*A----> f*s!A is a split mono for every A~ I~jI. (The arrow p is defined by the equation: Oslf opoot,~A = ~ o O ~ . ) • If: Suppose eop = id. We claim that an interpolant (a,B,c) is given by: B := slA, a := rl : A----) s*s!A, c := 'doe : f*s!A---) g!t*A---) C.

311

On the diagram t*A

d

-"- g * C

ot A I

0g ~,t*a

g*c I ' ~

•

-

I

/ t*s*B z ' -x- ~ g * f * B, , .

~,_ ~

Id

~" glt*"A

s's! AF

f*sl A

we see that ~dgh = eoOp~hoxot*(a) ~ d = g*(c)oxot*(a). But the antecedens of this implication is a consequence of the fact that eop = id and of lemma 25 below. Then: Let (a~, B~I, c~l)'be an interpolant of the "unit" r 1 : t'A---> g*g!t*A, (defined as in 22) and let 'a~I : s!A---) B be the "left transpose" of arl. By lemma 26, the arrow e := crlof*('aq) : f*s!A---> f ' B - - ) g!t*A is then a left inverse of p.Lemmas. The following statements are true for any bifibration E.

25. poag = asgoxot*(rl). • By the definition of p, the left side is the unique factorisation over g of ¢s~oO]~ through Ostf. But the right side is such a factorisation too, as the next diagram shows. ..

t*A

~ A

t*s*s t A ~ "c~ ~ s * s l A g*f*s • IA " ~

(l s

O O f ~ f*slA

s! A

312

26. Each of two squares below commutes iff the other one does. d

t*A

~- g*C

~1t*a

g*c 'd

gi t*A t*s*B

--- C

x ~ g*f*B P f's! A f*('a)- f*B

• In the following diagram, each of the triangles clearly commutes iff the other one does. t*A .,_

d

~ g*C..~

o:"---o, "?, f*B

Thus we are done if we prove f*('a)op = '(xot*(a)). As for this equality, compare the following two diagrams: t*A

*--g!t*A

(I g

g!t*A ~

t*A

'(xot*(a)) ,~,~ t s B

x, og V g*f*s!A -'~"~f*s!A

L, g~*}*t3 ~g ~f*'B

v

g

I t*(a)

J

t*s*B

f * t ~ ~ g*f*B

The pentangle R commutes by lemma 25, the rest by definitions. It is easy to see that a = s*('a)orl; hence t*(a) = t*s*('a)ot*(ri). The arrows '(xot*(a)) and f*('a)op are thus the ver-

313

tical factorizations of the same arrow Ogoxot*(a) = Ogoxot*s*('a)ot*(rl) through og. By the uniqueness, they must be equal.. 27. Corollary. For fibred preorders, the BC-condition is equivalent with the interpolation condition. 28. Remark. The connection of interpolation and the Beck-ChevaUey condition in the category of Heyting algebras has been noticed by A.M. Pitts (1983a). He also studied the interpolation condition for a special sort of fibred Heyting algebras (1983b), showing how Craig's Interpolation Theorem and Beth's Definability Theorem can be presented in this setting.

3. Uniform

interpolation

31. Definition. For functors S::K ---) ,~ and F::K ---->£, the category of interpolants/S,F/ consists of: - the triples (a,B,c), where Be I:~1, a~ ,~(A, SB), c~ £(FB,C); - a morphism (a,B,c)--) (a',B',c') is a triple (p,q,r), such that the squares on the following diagram commute.

FB c__£.~C

p al:q A'----~SB' a'

Fq{ tr B

FB' ----~C' c'

B' 32. C o m m e n t s . / S , F / ( w i t h obvious projections) is a certain type of lax limit of the diagram J s 3 ( F-F->/Cin the category of categories. It can also be obtained by strict pullbacks, as the next picture shows.

314

/S,F/

/v',, F/f~ / v x. / v ' , , j/J N 11 / om ,]Is

J

Given functors T:~ ---) ) and G:I~ ---) ), every natural transformation cp:TS ---) GF induces a functor from the category of interpolants of S and F to the comma category o f T and G: R : / S , F / - - - ) T / G : (a, B, c)~---) (A, GcocpBoTa, C) (p, q, r)v---) (p, r). In the obvious sense, (a,B,c) is an interpolant of R(a,B,c), relative to q~. 33. Definition. Given a square of functors Q = (F,G,S,T) as above, with a natural transformation cp:TS---) GF, we define initial interpolants to be those objects o f / S , F / w h i c h are initial among the interpolants of the same arrow. In other words, (a, B, c) is an initial interpolant if for any other interpolant (a',B',c'), such that R(a',B',c') = R(a,B,c), there is unique qe ~ ( B , B'), with a'= Sqoa and c ' o F q = c . We consider, thus, the initiality in R-fibres. We say that the interpolation in Q is uniform if an initial interpolant can be recognized by the f'n'st component. In other words, with uniform interpolation, an interpolant (a, B, c) must be initial w h e n e v e r for every (a',B',c'), such that R(a',B',c') = R(a,B,c), there is unique qe ~ (B, B'), with a'=Sqoa. 34. Back to fibrations. For a fibration E: ¢ ---) a3 and a commutative square Q = (f,g,s,t) in a3, the categories/s*,f*/obtained for various choices of the inverse image functors s* and f* are all isomorphic. Moreover, these categories are equivalent with the category of all the interpolants for all the possible inverse images along s and f. Similarly, the comma category t*/g* for some chosen t* and g* is equivalent with the category of triples (A, d, C), where d: t'A----) g*C is a vertical arrow from an arbitrary inverse image of A along t to an arbitrary inverse image of C along g. Note that Q satisfies the interpolation condition i_fthe functor R:/s*,f*/---) t*/g* induced by the canonical isomorphism x: t's*---) g*f'* is a retraction, i.e. if there is a functor

315

M: t*/g* "-'>/s*,f*/ such that RM=id. This functor M gives a choice of initial interpolants if it is left adjoint to R. On the other hand, the interpolation condition does not mention the arrows of t*/g*, so that it does not seem to imply the existence of M. 35. Terminology. We shall say that a commutative square Q in the base of a fibration satisfies the un/form interpolation condition if it satisfies the interpolation condition, and if the interpolation over it is uniform. For the next proposition - which explain s what is uniform about the uniform interpolation - we need the notion of trifibration. The notion of bifibration is standard: A functor E:~ ~ ;B is a bifibration if both E and its dual E°:~°--b ~Bo are fibrations. (The dual categories and functors are, of course, obtained by formally changing the directions of all morphisms.) We say that E is

a trifibration if both E:~ ---> :B and E°P: ~op---) ;B are bifibrations. The category ~op, fibred over :B, is obtained by changing the direction of all the vertical arrows in ~. (The arrows of ~op are the equivalence classes of spans with a vertical arrow pointing at the source, and a cartesian arrow pointing at the target.) A fibration E:¢ ---) B is a bifibration iff every inverse image functor t*: ~J---~ ~I has a left adjoint t!: ¢I---> Cj. It is a trifibration iff there is also a right adjoint t,: ~I'--)

~J of t*.

A cocartesian lifting by Eop of t at X is generically denoted xgt:XXo(--->, I )t,X. (The barred arrows --~ always belong to ~op.) 36. Proposition. Let E: ~ ---->:B be a trifibration, and Q = (f,g,s,t) a commutative square in :B, satisfying the interpolation condition. The interpolation is uniform iff c'o'a = ~"o'~" is true for any two interpolants (a, B, c) and (a', ~, c') of the same arrow.

316

t*A ~t*a

~ A ~

s~B.

g*C g * c / ~

t*s*B.-~ g*f*B .

~

s! A f * C

~

~ 0

- if, B ~/

B I

K • Then: If (a,B,c) is an interpolant of d, the triple (rl, s!A, cof*('a)) is another interpolant of d since a = s*r'aao~ , , .,, so that the next diagram commutes. t*A

d

---- *~g .

t*rl

__ ,.,,,],s*n ....

l/*s*('a)x t*s*s !A ~ g * f * s :

g*f*B /g*f*('a) A

It follows from the uniformity that (rl, s!A,cof*('a)) is an initi~ interpolant. Hence an arrow (idA, 'a', idc): (1], s!A, co f*('a)) --) (~',~,~') for each interpolant (ff,~,~') of d. On the other hand, for any two lnterpolants ' (a, A t~ , CA), (a', ]g, 6-) of d, the existence of an arrow (idA, q, idc): (~, i~, ~)---> (~',~,~') (in/s*,f*/), implies c 'o'a = c o a.

317

~*q

slA

f,C

f.q~/~

B A

Putting ~ := ~1 and c := cof*('a), we get ~"o'~" = (cof*('a))'o'rl

=

(cof*('a))'

--C

'o' a .

If: Let (a,B,c} be an interpolant such that for every interpolant (a,~,c') of the same arrow there is unique ~ ~K(B, l~) with ~'=s*(~)oa. We must prove that (of*(~)=c. Since (T1, s!A, cof*('a)) is an interpolant of the same arrow, there is q~le ~K(B, s!A), such that Tl=s*(q11)oa. It is easy to see that 1) qTio'a = id ond 'aoq~ = id. On the other hand, from s * (~) o a = a": A---) s* ~ follows ~ o' a =' a': s!A ---->~. Using (1), we get 2) q = 'goq n. Finally, the "left transpose" of c'o'a: s!A---> f , C is cof*('a): f*s!A--> C; the hypothesis c'o 'a = ~"o'~" imvlies 3) cof*('a) = ~of*('~'). N o w we can derive:

~'of*(~) ~) ~'of*('~'oqrl) (3)cof,(,aoqrl) ~) c.. 37. Theorem. A commutative square Q = (f,g,s,t) in the base of a bifibration E: ~ ---) ~ satisfies the uniform interpolation condition iff it satisfies the Beck-Chevalley condition. • In section 2 we proved that the interpolation condition is satisfied iff p: g!t*A---) f*s!A is a split mono. It is now sufficient to show that the interpolation is uniform iff p: glt*A---) f*s!A is an epi. If: The if-part of the previous proof can be copied almost completely. The only difference is that equality (3) must be derived in a different way this time: namely, from lemma 26 and the fact that p: g!t*A--)f*s!A is an epi. Then: For arbitrary arrows Cl, c2: f*s!A---) C, the equivalence

318

Then: For arbitrary arrows Cl, c2: f*s!A---) C, the diagram

ag

t*A p'

,~ g~t*A

t's'st A

g*f*s~A

og '~ f*s!A

g*(cl) g*(c2) Cl j C2 g*C

og

~

C

shows that ClOp = c2op ¢~ g*(cl)oXot*(rl) = g*(c2)oxot*(rl). But this means that clop = c2op implies that (rl, s!A, Cl) and (rl, slA, c2) must be interpolants of the same arrow. The uniformity now imDlies that these interpolants must be initial; thus Cl =c2. So we derived that clop = c2op implies Cl =c2."

4. Descent by interpolation 41. Geometric motivation. Let a3 be a site (cf. Artin et al.), and D: ~ ---> a3 a discrete fibration. If f: L----) K is a covering morphism in a3, and 00,31: M----> L its kernel pair - obtained by pulling back f along itself - the sheaf condition (ibid.) on D tells that for every A¢ ;DL, such that 30*A=31*A, there must exist a unique Be 33 K with A = f * B . In other words, every "vertical arrow" 30*A----> 01*A (which is of course identity, since the fibration D is discrete) must have a unique interpolant over the kernel square (f, 30, f, 31) of f. For arbitrary coveting family {fn:Ln---)KI n¢N }, the sheaf condition can be expressed by saying that every family of "vertical arrows" {n30*An---)n31*Aml n,me N} determines a unique common interpolant Be ;D K. The arrows n30: Mnm---)Ln and n31: Mnm--)Lm here are obtained in a pullback of fn and fm. The notion of descent lifts the sheaf condition from the discrete fibrations to fibrations E: ~ ---> ~Bin general. For simplicity, we shall consider covering by one arrow; the passage on covering families only requires some more involved formulations. The question will be: When

319

can one descend along a morphism f: L---) K, and represent ~K in terms of ~L? If this is possible, f is said to be an effective descent morphism. 42. Notation, terminology. To fix the notation, consider the following cube of pullback squares.

M_

L

T

0

P2

_

L

~

M

1 L

K A kernel square of f is a pullback (f):= (f, 30, f, 31). Note that the diagram above contains not only (f), but also (30) and (31). Three different pullback squares are obtained by pulling back (f) along f: (30), (31) and S := (30,Pl,31,P0). By r 1: L-'-') M we shall now denote the unique arrow such that 30oT1= 8~1orl = idL. For cartesian liftings of 30 we shall use a}0: 30*A ---b A (instead of ~3~0). Let ~0: A ----) 30*A be the unique splitting of ~0 over r I (i.e., ~0o~0 = id and Ev0=rl). It is straightforward to show that ~0 must be cartesian. - Idem for the splitting 1.}1 over rl of the cartesian lifting ~I of 31. Given a fibration E:S ----) a3 and an (f)-interpolant (a,B,c) of de SM(30*A, 31"C), the triple f*(a,B,c) := (~o30*(a), f ' B , 31"(c)o'¢), where xo30*(a): 30*A---) 30*f*B ----)31*f'B, 31"(C)O~: 3 0 * f ' B - - ) 31*f*B --) 31"C, is clearly an S-interpolant of dS: = "¢op2*(d) o'c: p0*30*A----) p2*30*A---) p2"31"C-'-) p1"31"C. We say that the interpolant (a,B,c) is f-simple if (id, id, id): f*(a,B,c) ---) f*(a,B,c) is the only arrow in the category of S-interpolants. In other words, the object f*(a,B,c) has just one endomorphism in its R-fibre. (In general, every pullback square Q and arrow q with the same target span a cube of pullbacks as above; and every Q-interpolant induces an interpolant over the square opposite to Q in this cube. So we can speak of q-simple interpolants in this general situation. For Q:=(f) and q:=f, we should actually consider two more interpolants induced by (a,B,c): namely, those over (30)

320

and (31)- and include them in the definition of f-simplicity. This would, however, only add a couple of inessential sentences to the proofs below.) 43. Definition. Let a fibration E: ~ ---) a3 and an arrow fe 3(L,K) be given. An f-descent data (for E) is a pair (A,d}, Ae I~LI, de ~M(30*A, ~I*A), such that

RE)

Olodo~o0 = id; i.e. ~*(d)=id: 0

A - ' ~ O ° 3o*A --,,---~-~ A d A'~-~I

31*A 4 30

L ~

M~

] id ~A

"q --->(f*~,'cI~> in DesE(f) is in

the form f*b for a unique be ~K(B,~). In other words, f is effective iff each f# is essentially surjective; f is a descent morphism iff each f# is full and faithful; f is an effective descent morphism iff each f# is an equivalence of categories.

322

46. T e r m i n o l o g y . An interpolant (a,B,c) of a descent data d: ~0*A----) ~I*A is natural if a: A----) f*B and c: f*B ---) A are morphisms of descent data, i.e. if they satisfy ~ l * ( a ) o d = XBO30*(a) and ~l*(C)OZ B = do~0*(c). An arrow cp from a fibred category ~ is an E-coequalizer of a pair ~0,~51 of parallel arrows if - 9 o 5 0 = 9 o ~ 1 and - for every Z in ~, ZoiS0= ZO~l and E z = E q) imply that there is a unique vertical arrow b with Z =bocp. 47. T h e o r e m . Let E: ~ ----) :13 be a fibration and f an arrow in its base. (Notation as abovel) i) f is an effective morphism iff each f-descent data has a natural, f-simple interpolant.

ii) f is a descent morphism iff every cartesian lifting "Of:f*B ---) B is an E-coequalizer of its kernel pair. • i) If: Let (a,B,c) be a natural, f-simple interpolant of f-descent data (A,d). So we have d" 30*A----) ~I*A, a: A---) f*B and c: f*B ---) A such that 1) d = ~l*(C)OXO~0*(a), 2)

~l*(a)od = "co~0*(a), and

3)

~l*(C)O'~ = do~0*(c ).

We shall prove coa = idA onO aoc = idf*B. eoa = id : As before, choose the cartesian liftings x)0 and a) 1 ofrl to be the splittings ofa~0 and ~1 respectively. ,0.o A ~ [a I

ajo ,90~.A~

o

[00*a l

f*B is a kernel pair of 9f: f'B---> B

(for every B). Chasing the next diagram - where h is vertical, and X =~foh - one easily proves that 31*(h)ox = ~'o~o*(h ) ¢=~ XoO0 = xoOlox. ~0,f* B

---~l,f, B

f, B ~ 0 * I f * B ~.---~ D1:

B

I B

%

f*B

324

Thus, if h is a morphism of descent data, the assumption that "0f is an E-coequalizer of its kernel pair gives a unique arrow be CK(B,~), such that X = boOf. In other words, h = f*b. Conversely, if an arrow Z: f*B ----) ~ over f satisfies Xo'o0--zo~lox, its vertical part h must be a morphism of descent data (f*B,XB)--) (f*~,'q~). The hypothesis that f is a descent morphism means that there is a unique arrow be ~K(B,t]), such that h = f*b. Thus, %= boOf. 48. L e m m a . Let Q = (f,g,s,t) be a pullback square in ~ and O = ((p,7,~,O) a commutative square in ~ over Q (i.e., (poT = ~o'O, E(p=f, Ey=g, E~=s, E'O=t). If (p and "O are cartesian, then O is a pullback square. 49. C o m m e n t . This descent theorem is a far descendant of the method which Joyal and Tiemey (1984) used to prove that open surjections of toposes are effective descent morphisms - in absence of the Beck-Chevalley property. More recently, Moerdijk (1989) observed that an

appropriately saturated class O of arrows in :B must consist of effective descent morphisms with respect to the fibration Cod: ~ / ~ ----) ~Bif it satisfies the following axioms: i) A coequalizer of every parallel pair of arrows from O exists, and it is stable under pullbacks; ii) Each arrow belonging to O is a coequalizer of its kernel pair. The two parts of theorem 47 clearly correspond to these axioms. On the other hand, it is perhaps interesting to put theorems 37 and 47 together, aligning the Beck-Chevalley property and descent - on the common ground of interpolation. This way, one can analyze how B6nabou-Roubaud-Beck's theorem (cf. Hyland-Moerdijk 1990) provides some sufficient conditions for descent in presence of the Beck-Chevalley property.

References

Art-in, M. Grothendieck, A., Verdier, J.L. (1972)

Thdorie des Topos (Expos6s I-IV from SGA 4, 1963/64), Lecture Notes in

Mathematics 269 (Springer, Berlin) B6nahou, J. (1985)

Fibered categories and the foundations of naive category theory, J. Symbolic Logic 50(1), 10-37

B6nabou, J., Roubaud, J. (1970)

Monades et descente, C2L Acad. Sc. Paris (S&ie A), t. 270, 96-98

325

Giraud, J. (1964) Gray, J.W. (1966)

Th6orie de la descente, Bull. Soc. math. France, M~moire 2

Fibred and Cofibred Categories, Proceedings of the Conference on Categorical Algebra, La Yolla 1965, (Springer, Berlin) 21-84

Grothendieck, A. (1959) Technique de descente et th6or~mes d'existence en g6om6trie alg6brique, I. G6n6ralitfs. Descente par morphismes fid~lement plats, Sdminaire Bourbaki 190 (1971) Expos6s VI, VIII, IX (from SGA 1, 1960/61) in: Rev~tements Etales et Groupe Fondamental, Lecture Notes in Mathematics 224 (Springer, Berlin), 145-260 Hyland, J.M.E., Moerdijk, I. (1990) An application of Beck's Theorem, Manuscript Joyal, A., Tierney, M. An extension of the Galois theory of Grothendieck, lvlemoirs of the A.M.S., (1984) No. 309 (American Mathematical Society, Providence, RI) Lawvere, F.W. (1970) Equality in hyperdoctrines and comprehension schema as an adjoint functor, Applications of Category Theory, Proceedings of A.M.S. Symposia on Pure Mathematics XVII (American Mathematical Society, Providence RI), 1-14 Moerdijk, I. (1989)

Descent Theory for Toposes, Bull. Soc. Math. de Belgique (S~rie A), XLI(2), 373-391

Pavlovi6, D. (I990)

Pitts, A.M. (1983a) (1983b)

Predicates and Fibrations: From Type Theoretical to Category Theoretical Presentation of Constructive Logic, Thesis, State University Utrecht

Amalgation and Interpolation in the Category of Heyting Algebras, J. Pure Appl. Algebra 29, 155-165 An Application of Open Maps to Categorical Logic, J Pure Appl. Algebra 29, 313-326

This paper is in final form and will not be published elsewhere.

AN n-CATEGORICAL PASTING THEOREM A.J.Power* Department of Computer Science University of Edinburgh King's Buildings Edinburgh EH9 3JZ Scotland. d e d i c a t e d t o M a x K e l l y on t h e o c c a s i o n of his 6 0 t h b i r t h d a y

Abstract In order to facilitate the study of 2-categories with structure, we state and prove an n-categorical pasting theorem. This is based upon a new definition of n-pasting scheme that generalises Johnson's definition of a well-formed loop-free pasting scheme by weakening his no direct loops condition. We define n-pasting, prove the theorem, and show that for n=3, it incorporates all possible composites of n-cells. We show that that is not true for higher n. We define the horizontal n-category of an (n+l)-category to gcneralise that of a 2-category, we define horizontal and vertical composition for an (n+l)-category and we state and prove an interchange law. We also study further conditions on a pasting diagram and their impact upon how one may evaluate a composite, and we express Street's free n-categories in terms of left adjoints.

1

Introduction

"Pasting" has become a well known and valuable tool in the study of 2-categories. In [8], a "2-categorical pasting theorem" was proved in order to make precise the notion of pasting as outlined in [7]. Now, in various articles such as [2], [9], [10] and implicitly in "This research was supported by the Australian Research Council and by ESPRIT Basic Research Action 3245: Logical Frameworks Design, Implementation and Experiment.

327

[11], 3-categories or mild generalisations of them are used to study 2-categories with extra structure. So we seek a 3-categorical pasting theorem in order to facilitate that. In fact, an n-categorical pasting theorem for each positive natural number' n is only a little more difficult, and it seems a more natural level of generality; so that is the aim of this paper. The idea of pasting is to unify the n different sorts of composite in the definition of n-category into one general notion of "pasting composite". For instance, for n=3, given 3-cells 7 : a ~ a ' and 7' :/9 "-* fi', we should like to know directly from the definition of pasting composite that a diagram such as "

>.

>.

>

>.

>

commutes, where the 3-cells are the evident ones generated by 3' and 7'There has been and there continues to be much work on this problem. Michael Johnson produced an n-categorical pasting theorem in his thesis [3] and :published the result in [4]. Johnson investigated two classes of schemes. The first class, loop-free schemes, are of interest because the free n-category on an n-dimensional such admits a particularly easy construction. We do not handle these: a counterexample appears in Section 2. The second class, well-formed loop-free schemes, are the pasting schemes of interest to us. These parametrize composable diagrams in n-categories. Johnson realised that his definition was not broad enough to include all possible composites of n-cells. In particular, it was not broad enough to allow all of the constructions he sought, specifically that of a certain tensor product of pasting schemes, as outlined in his lecture [5]. The difficulty was that his "no direct loops" condition was too strong. Here, we generalise Johnson's definition. Specifically, we drop most of the no direct loops condition, and what remains of it is true of all composites of n-cells. This allows us to capture all possible composites for n=3; we also capture a large class of composites for higher n. Alas, we do not capture all: an example appears in Section 3. That example is interesting as it is not merely a technical example but acts as a counterexample to a large class of conjectures. For instance, a variant of that example exhibits a composite of 4-cells that fails to satisfy

328

the fourth (and most substantial other than no direct loops) requirement of Johnson's "loop-freeness" condition. It is not immediately evident from the two definitions that the definition here generalises that of Johnson, but it does follow from an analysis of the early results of Johnson's paper [4] as he has indicated in a private communication for general n and shown in detail for n=2 in a Sydney Category seminar in 1988. It should be noted that greater generality for pasting schemes does not necessarily imply a more general pasting theorem. For instance, our Example 3.12 is not one of Johnson's well-formed loop-free pasting schemes, but diagrams of that form can be handled by Johnson's pasting theorem by parametrizing them by loop-free schemes. Our approach is to give a topological foundation and to proceed by induction. Typically, one does not try to use 3-pasting schemes until one already has 2-pasting schemes to act as domains and codomains of the 3-cells; similarly for higher dimensions. So it seems reasonable to use the definition and results of n-pasting schemes in the definition of an (n+l)-pasting scheme. This is a convenient way to order the information so that it is less overwhelming in its detail. Just as a 2-cell can be seen as two parallel directed paths in a graph, together with a homotopy in 2-space from one to the other [8], we wish to regard a 3-cell as a homotopy in 3-space, and similarly for highcr n. Using this paradigm, one may regard Ross Street's pictures [13] as n-pasting schemes together with a choice of the order in which some n-cells are composed. Central to our approach is our "interchange law". Just as one speaks of horizontal and vertical composition for a 2-category, the various n + l sorts of composite in the definition of an (n+l)-category may also be viewed naturally in terms of two sorts of composition: vertical composition, which is the top level composition of the (n+l)-category; and horizontal composition, which is the n-pasting composition of the (n+l)-category. These two sorts of composition satisfy an appropriately generalised interchange law given in Section 6. It is from this result that we deduce the unicity of a pasting composite. There is other current work on giving a topological analysis of n-categories. Specifically, Kapranov and Voevodsky [6] are also working on pasting results that are topological in flavour, and A1-Agl and Steiner [1] have characterised oo-categories in terms of simplexes. However, in both cases, the work is somewhat different in direction to ours. Of course, much of the work here was ultimately inspired by Street's fundamental paper [13], as explained in Section 8. It is possible to exhibit in terms of left adjoints the free n-categories described internally by Street [13]. A consequence of our result is that one can do so for a large class of them by means of pasting diagrams, hence in topological terms. In Section 2, we commence the induction with the definitions for ordinary categories, i.e. in the case n = l , and we recall the definitions for 2-categories as in [8]. The definitions for 2-categories are not needed for the induction, but they illustrate the first nontrivial case. In Section 3, we give the inductive definitions needed to define pasting diagrams. In Section 4, we give the simple topological results that we require. In Section 5, we define horizontal pasting. In Section 6, we define horizontal and vertical composition in an (n+l)-category and give our "interchange law" and our "n-categorical pasting theorem". In Section 7, we show that we have captured all possible composites in a 3-category, we discuss our failure for higher n, and we illustrate a strengthening of the conditions that

329

allow us greater flexibility in describing a composite. Finally, in Section 8, we explain limiting behaviour as n approazhes infinity, and show how the work relates to that of Street [13], specifically to his free w-categories. Throughout, we shall assume that n-space is oriented for every n, and all embeddings into n-space are piecewise linear. This avoids irrelevant topological detail. I should like to offer warm thanks to Michael Johnson and to lq:oss Street for valuable discussions on this work. This paper is in final form and no similar paper has been or is being submitted elsewhere.

2

Commencing the induction

A O-pasting scheme is the point of O-space. A 1-computad is a finite directed graph; a 1-computad morphism is a map of directed graphs.

A 1-pasting scheme is a finite non-empty set G together with an embedding g of G into the oriented line. We identify G with its image, and we call the elements of G O-cells. G partitions the rest of the line into a finite number of disconnected open intervals and two rays. We call the open intervals 1-cells and we define the domain and codomain of a 1-cell f to be its endpoints, written domf and cod f, with domf < codf in the order determined by the orientation of the line. The domain of the pe~sting scheme, denoted doinG, is the least element of G in the order determined by the orientation of the line; dually, the codomain of (G,g), denoted codG, is the greatest element. The 0-cells and 1-cells of (G, g) together with the domain and codomain functions form a directed graph, which we call the underlying 1-computad of (G,g), and which we denote by C(G). A 1-pasting diagram in a 1-computad H consists of a 1-pasting scheme (G, g) together with a 1-computad morphism h : C(G) ~ H. The domain of a 1-pasting diagram ((G,g), h) is doing together with the restriction of h to C(domG); dually for codomain. A 1-pasting diagram ((G,g), h) is called simply a 1-pasting scheme in H if h is a monomorphism in the category of directed graphs. Every 1-pasting scheme with at le.ast two 0-cells is called spherical. A 1-computad is locally spherical if each 1-cell together with its domain and codomain is a spherical 1-pasting scheme, i.e. if every 1-cell has its domain and codomain different 0-cells. A 1-subpasting scheme of a 1-pasting scheme (G,g) is a subset J of G for which given j < g < j' with j and j' in J and g in G, it follows that g lies in J. A labelling of a 1-pasting scheme (G,g) in a category A is a 1-pasting diagram ((G,g),a) in A. A 1-pasting composite, or equally a strong composite, of a labelled 1pasting scheme in a category A is the composite in A of the sequence of arrows determined by the diagram. We call any composite in a category A a vertical composite. We now recall the definitions and a few results for n=2, as appear in [8].

330

Definition 2.1 A 2-computadconsists of a directed graph G, together with a set G~ and two functions dora, cod : G: --* diags(G), where diags(G) is the set of directed paths in G, such that dora dora = dora cod and cod dorn = codcod. A 2-computad morphism from (G2, G, dom, cod) to (H2, H, dom, cod) consists of a morphism of directed graphs f : G --* H and a function f2 : G2 --~//2 such that domf2 = fdom and cod f2 = fcod. The category 2-Cornputad is the category of 2-computads and 2-computad morphisms, with the evident composition. D e f i n i t i o n 2.2 A 2-pasting scheme consists of a (non-empty) connected finite directed graph G together with an embedding g of G into oriented 2-space subject to the following conditions. The image of G divides the rest of 2-space into finitely many disconnected regions, called faces. The boundary of each interior face F must be of the form a(F)(r(F)) -1 for directed paths a(F) and r(F) in G of non-zero length, if moving clockwise around the boundary; and there must exist vertices s(G) and t(G) in the exterior face for which given any vertex v, there exist directed paths from s(G) to v and from v to t(G). R e m a r k 2.3 In [8], it was assumed in the definition of 2-pasting scheme that s(G) and t(G) are distinct, but that assumption was inessential to the argument. If they are the same, then the 2-pasting scheme must be trivial. With the above notation, the interior faces F of (G, g) are called 2-cells, and domF and codF are defined to be or(F) and r(F) respectively. So, (G,g) has an underlying 2-computad given by the graph G and the 2-celts of (G,g); we denote this 2-computad by C(G). It also follows that the exterior face E also has boundary of the form a(E)(r(E)) -~ for directed paths a(E) and r(E) of non-zero length, if moving around the boundary with E always on one's right side; these are called codG and doinG respectively. P r o p o s i t i o n 2.4 If (G,g) is a 2-pasting scheme, then G has no directed cycle. P r o p o s i t i o n 2.5 If a 2-pasting scheme (G,g) has at least one interior face, then there exists an interior face F such that domF lies entirely on doinG. D e f i n i t i o n 2.6 A 2-pasting diagram in a 2-computad H consists of a 2-pasting scheme (G, g) together with a 2-computad morphism h : C(G) --+ H. A labelling of a 2-pasting scheme (G,g) in a 2-category A is a 2-pasting diagram ((G,g),a) in A. T h e o r e m 2.7 (A 2-categorical pasting theorem) Every labelling of a 2-pasting scheme has a unique composite. P r o o f i The existence of a composite follows from Proposition 2.5 and induction on the number of interior faces; unlcity also follows directly by induction. D R e m a r k 2.8 For n=2, pasting composition describes all possible composites of a set of 2-cells. This is evident because a horizontal composite of labelled 2-pasting schemes is a labelled 2-pasting scheme, and a vertical composite of labelled 2-pasting schemes is a labelled 2-pasting scheme.

331

In the introduction, it was mentioned that our pasting schemes generalise the wellformed loop-free pasting schemes of Johnson, but not his loop-free schemes in general• For n=2, our pasting schemes are precisely his well-formed loop-free pasting schemes as he showed in a lecture in Sydney in 1988, but even then, we de not include all of his loop-free schemes, for instance the following. E x a m p l e 2.9 Consider the following diagram:

(2.1)

>.

).

Observe that this is not one of our 2-pasting schemes since there is no candidate for t(G). It is however one of Johnson's loop-free schemes. It is not well-formed since its domain is not even connected. This generality allows Johnson to give an explicit description of the free 2-category on some data, but it evidently does not parametrize a composable d~agram in general since the two 2-cells are not composable. In order to study higher dimensions, we add the following definitions. The domain of a 2-pasting diagram ((G,g), h) in a 2-computad H is domG together with the restriction of h to C(domG); dually for codomain. A 2-pasting diagram ((G,g), h) is called simply a 2-pasting scheme in H if h is a monomorphism in 2-Computad, i.e. if h is injective on kcells for k=0 to 2. A 2-pasting scheme is called spherical if domGU codG is homeomorphic to the 1-sphere. A 2-computad G is called locally spherical if each 2-cell in it is a spherical 2-pasting scheme containing just one 2-cell. A 2-pasting inclusion iinto a 2-pasting scheme (G, g) is a directed graph inclusion j into G such that the composite gj is a 2-pasting scheme for which every 2-cell is a 2-cell of (G, g). A 2-subpasting scheme is an equivalence class of 2-pasting inclusions under the evident equivalence.

3

C o n t i n u i n g the i n d u c t i o n

This section is devoted to defining the terms n-computad and n-pasting scheme. In order to simplify the exposition, it is convenient to define several subsidiary terms. We have

332

already commenced with n = l and exhibited the ideas to follow by recalling the definitions for n=2. Generalising the situation for n=2, an (n+l)-computad will be an (n+l)-dimensional cellular complex, together with an orientation assigned to each k-cell for k=0 to n + l , subject to some axioms. Each (n+l)-cell will have a domain and a codomain, each of which will be an n-pasting scheme, hence the induction. An (n+l)-pasting scheme will consist of an n-computad embedded in (n+l)-space: this is why we have a mutual induction on pasting schemes and computads. An (n+l)-pasting scheme (G, g) will be called spherical if its domain and codomain, which will be subcomputads of G, are spherical n-pasting schemes and if doing U codG is homeomorphic to the n-sphere. An (n+l)-computad will be called locally spherical if for each (n+l)-cell, the subcomputad generated by taking its domain and codomain is a spherical (n+l)-pasting scheme (with only one (n+l)-ceU in it). Given an (n+l)-pasting scheme (G,g), it will generate an (n+l)-computad by taking the embedding and the region enclosed by it; this (n+l)-computad will be denoted by C(G). An (n+l)-pasting diagram in an (n+l)-computad H will be an (n+l)-pasting scheme (G,g) together with an (n+l)-computad morphism h : C(G) ---, H. Definition 3.1 An (n+l)-computad consists of an n-computaxt G., a set G.+I, and two functions dora, cod : G,,+I ~ diags(G.), where diags(G,,) is the set of n-pasting diagrams (See Definition 3,13) in G., such that domdom = domcod and coddom = codcod. An (n÷ l )-computad morphism from ( Gn+ I , G.,, dorn, cod) to (H~+a, Hn, dora, cod) consists of an n-computad morphism f . : G. ~ H. and a function f.+1 : G.+I --~ H~+I such that dora f,,+l = f . dora and cod f,,+l = f,, cod. The category (n + 1)-Computad is the category of (n+l)-computads and (n+l)-computad morphisms, with the evident composition. P r o p o s i t i o n 3.2 Let (G,g) be a locally spherical (See Definition 3.15) n-computad G together with an embedding g of G into oriented (n+l)-space. Then the orientation of (n+l)-space determines an orientation for each n-cell of G, giving it a "left hand side" and a "right hand side". Proof: Since G is locally spherical, every n-cell in G is given by an inclusion of an n-pasting scheme with just one n-cell into G, and so composing this inclusion with g gives a piecewise linear embedding of an n-ball into oriented (n+l)-space. So the orientation of n-space specifies an orientation for the embedded n-cell. O N o t a t i o n 3.3 Given a finite locally spherical n-computad G together with an embedding g of G into (n+l)-space, the image of G divides the rest of (n+l)-space into finitely many disconnected regions. For each such region F, we write i n F for the set of n-cells of G that are incident to F and with orientation into F, as given by Proposition 3.2. Dually, we write outF for the set of n-cells of G that are incident to F and with orientation out of F. We shall call the interior regions (n+l)-cells of (G, g). Let the exterior region determined by such an embedding be denoted by E.

333

Definition 3.4 An (n+l)-embedding is a (non-empty) connected, finite, locally spherical n-computad G together with an embedding g of G into (n+l)-space subject to the following conditions: 1. if F is an interior region, then F is homeomorphic to the (n+l)-ball, and inF and outF, together with their domains and codomains, each forms a spherical npasting scheme, denoted domF and codF respectively, such that domF U codF is the boundary of F; dom(domF) = dom(codF); and cod(domF) = cod(codF). 2. there exist n-pasting schemes doinG and codG in G such that the n-cells of doing are precisely the elements of outE; the n-cells of codG are precisely the elements of inE; dora(doinG) = dom(codG); cod(doinG) = cod(codG); and doing U codG is the boundary of E. Moreover, doing extends to an n-sphere S(domG) such that G - doing lies strictly inside S(domG). N o t a t i o n 3.5 Given an (n+l)-embedding (G,g), we shall denote dora(doinG) by S and cod(doinG) by T, and we shall denote the region enclosed by S(domG) by B(domG). Observe, more generally, that if X is any n-pastir,.g scheme in G with domain S and codomain T, then X together with S(domG) - doing forms an n-sphere. We shall denote this n-sphere by S(X) and we shall denote the region it encloses by B(X). Note that every n-celt of X is an element of in(B(X)). If F is an (n+l)-cell of (G,g), we shall denote by d(F) the union over all k.

(3.2)

X'

We shall denote this 3-pasting scheme by F 1. Moreover, we shall denote the somewhat trivial 3-pasting scheme given by removing 7 from (3.2) by F0. Now consider the following diagram, which is similar to the first. It is to be understood to contain two 3-cells H and H'. The idea is that H sits to the left of and below

336

H', and that 6 is directed vertically upwards. The domain of H is given by fl and 6, and the codomain is given by 3, which is not depicted explicitly. The domain of H' is fl', and the codomain is given by 6 and fl', which is not depicted explicitly.

X

Y

/k'

!

(3.3)

k

We shall denote this 3-pasting scheme by Hi; and we shall denote the 3-pasting scheme given by removing 6 from (3.3) by Ho. Now identify (3.2) and (3.3) at X, Y and X' by gradually curving (in a pieeewise linear way) the two diagrams. This gives the following picture.

337

f

>.

~ot

t ! I

I

7/,,

g X

X'

Y

(3.4)

gl !

To the left side of the picture, we add two new 2-cells. The domain of one of them is given by g and codS, and its codomain is k. The domain of the other is given by f and codT, and its codomain is h. Then, these two new 2-cells, together with ~ and ~ enclose a region K with domain given by ~ and one of the new 2-cells, and with codomain given by fl and the other new 2-cell. Dually, on the right side of the picture, we can form another 3-cell K ' whose domain includes/3-7 and whose codomain includes a'. Having made these constructions, we have a 3-pasting scheme, which we shall call G. The domain of G is given by a, fl', and two of the new 2-cells. T h a t it is a 3-embedding is easy; to check the 2-cell extension condition, observe that it is true for any 2-cell in the domain, then by removing a topmost 3-cell, say F, one can see that it is true for all 2-cells in codF, etcetera. For instance, the 2-pasting scheme given by starting with doing and removing F and H ' is depicted by

338

X

> Y

>X'

(3.5)

where the unlabelled 2-cells are two of the new 2-cells. The domain replacement condition can be most easily seen as a result of Theorem 4.14. The 3-pasting scheme G has several 3-sequences, i.e. sequences of all the 3-cells in G such that the first has domain on doinG, then removing it from G, one continues inductively. A formal definition appears in Corollary 4.6. Two important 3-sequences in G are H', K', F, F', K, H and F, K, H', H, K', F'. Observe that every 3-sequence must have both F and H ~ occurring before F ' and H. If one removes 3' from G, then one still has a pasting scheme which we shall call GI. A 3-sequence in G1 is given by H', K', Fo, K, H, i.e. the replacement of the subsequence F, F ' in our first 3-sequence above by the singleton Fo. Dually, if one removes 8 from G, one also has a pasting scheme G~, and G2 has a 3-sequence given by F, K, Ho, K', F'. However, if one removes both 7 and ~ simultaneously from G, then one no longer has either a composite or a pasting scheme: any 3-sequence must have both F o before Ho and Ho before Fo, and the 2-ceil extension condition fails. The latter is easiest seen as a consequence of Proposition 4.3. Given all of these definitions, one may have a 4-embedding with domain G1, and with two 4-cells Y and Z. The domain of Y is F 0 and its codomain is F1. The domain of Z is //1 and its codomain is Ho. So the codomain of the 4-embedding is G2. The 4-composite given by composing first tile evident extension of Y followed by that of Z is certainly possible in any 4-category, but the 4-embedding does not satisfy condition 1 of domain

339

replacement since Z is a topmost 4-cell. Note that the 4-embedding does satisfy the 3-cell extension condition. So this shows both that condition 1 of domain replacement is not redundant, and that there exist composites of 4-cells that do not form 4-pasting schemes. Also, one may have a 4-embedding with domain G, and with 4-cells whose domains are F 1 and H1, and with codomains given by new copies of the interiors of F 1 and H 1. This would satisfy the 3-cell extension condition and condition 1 of domain replacement, but it would not satisfy condition 2. Finally, supposing one reverses the direction of "7 in G, one would still have a 3-pasting scheme, and it may appear as part of the domain of a 4-cell in a 4-embedding; but then, the 4-embedding would fail to satisfy the fourth (and most substantial other than no direct loops) requirement of Johnson's "loop-freeness" condition, as one may take H 1 to be the subpasting scheme to act as a counterexample. Again, the 4-embedding could still satisfy the 3-celt extension condition and could in fact form a 4-pasting scheme with just one 4-cell and hence even satisfy the no direct loops condition• E x a m p l e 3.12 We now give an example of a 3-pasting scheme with just one 3-cell that does not satisfy Johnson's "no direct loops" condition since its underlying graph has a cycle.

(3.6) ~.

>

Such a scheme could arise in practice as follows. In [13], Street defines a "2-adjunction" in a 3-category• Some of the data is depicted by •

1

>.

1

~.

(3.7) >. 1

:>. 1

where the unmarked interior region represents an identity 2-cell. Then, if one appends an identity 3-cell, one obtains

340

1

>.

1

>-

1

>. (3.8)

). 1

>. 1

)1

where the unmarked interior regions represent identity 2-cells. In fact, Vaughan Pratt observed some time ago that (3.7) itself fails the no direct loops condition, but his argument is somewhat more complex than ours. Finally, to end this section, we need four more definitions, which have been used above for n, in order to allow us to continue the induction. D e f i n i t i o n 3.13 An (n+l)-pasting diagram ((G,g),h) in an (n+l)-computad H is an (n+l)-pasting scheme (G, g) together with an (n+l)-computad morphism h : C(G) ---*H. The domain of an (n+l)-pasting diagram ((G, g), h) is given by doing and the restriction of h to doinG; dually for the codomain. We shall call an (n+l):pasting diagram ((G,g), h) simply an (n+l)-pasting scheme in H if h is a monomorphism in the category (n + 1)Computad, i.e. if h is injective on k-cells for all k=0 to n + l . D e f i n i t i o n 3.14 An (n+l)-embedding (G,g) is spherical if doing and codG are spherical n-pasting schemes and doing U codG is homeomorphic to the n-sphere. D e f i n i t i o n 3.15 An (n+l)-computad G is locally spherical if each (n+l)-cell together with its domain and codomain is a spherical (n+l)-pasting scheme in G containing one (n+l)-cell. D e f i n i t i o n 3.16 An (n+l)-pasting inclusion into an (n+l)-pasting scheme (G,g) is an n-computad inclusion j into G such that the composite gj is an (n+l)-pasting scheme for which every (n+l)-cell is an (n+l)-cell of (G,g). An (n+1)-subpasting scheme is an equivalence class of (n+l)-pasting inclusions under the evident equivalence.

4

Simple topological results

The main result of this section is Theorem 4.5, which states that given any (n+l)-pasting scheme (G, g) with at least one interior region, there exists an interior region F such that domF is a subpasting scheme of domG, and such that if one removes d(F) from G, then one has a new (n+l)-pasting scheme containing one fewer interior regions. We now give a series of Propositions leading to that result.

341

D e f i n i t i o n 4.1 An (n+l)-embedding (G,g) is acyclic if there exists no alternating sequence Fo, xo, F1,..., F,~ with m>0, with each Fi an (n+1)-cell and each xi an n-cell of (G,g), with F0 = Fro, and zl an element of outFi f3 inFi+~. Acyclicity is the highest dimensional part of Johnson's "no direct loops" condition, and it is the only part of the condition that we keep, as we shall show that it is a consequence of the n-cell extension condition. Example 3.12 shows that we do not require lower dimensional parts of the no direct loops condition. P r o p o s i t i o n 4.2 If an (n+l)-embedding (G,g) is acyclic and has at least one interior region, then there exists an interior region F for which domF is a subpasting scheme of

doinG. P r o o f : Let F 0 be any interior region. If every n-cell of inF lies on doinG, then since domF is spherical, it follows that domF is a subpasting scheme of doinG. If not, then let x o be an n-cell of inF that is not on doinG. It must therefore be an element of outF1 for some interior region F 1. Continue inductively. Since (G, g) is finite and acyclic, there must exist some m for which domFm is a subpasting scheme of doinG, o

We shall call any interior region F of (G, g) such that domF is a subpasting scheme of doing a topmost region of (G,g). P r o p o s i t i o n 4.3 If an (n+1)-embedding (G,g) satisfies the n-cell extension condition, then it is acyclic. Proof." Suppose there is a cycle F 0 , x 0 , . . . , F , n . Then, x 0 extends to an n-pasting scheme X from S to T in G, which in turn extends to S(X) as in 3.5, enclosing the region B(X). Recall that every n-cell of X is an element of in(B(X)), and S(X) extends X through the exterior region of (G,g). So Fo is outside B(X) but F 1 is inside it. Since Fo = F,~, it follows that there exists an i such that Fi is inside B(X) but Fi+l is outside it, but this is a contradiction since every n-cell on X is oriented into B ( X ) , so cannot be an element of outFi. [3 In the presence of condition 1 of the domain replacement condition, Proposition 4.3 has a converse given by Proposition 4.9, but before proving that, we shall develop some of the consequences of Proposition 4.3. C o r o l l a r y 4.4 If an (n+l)-embedding (G,g) with at least one interior region satisfies the n-cell extension condition, then there exists an interior region F for which domF is a subpasting scheme of doinG. Proof." Propositions 4.2 and 4.3.

[]

This leads us directly to the main theorem of this section. It allows us to use induction to show t h a t any labelled (n+l)-pasting scheme has a composite.

342

T h e o r e m 4.5 For any (n+l)-pasting scheme (G,g) with at least one interior region, there exists an interior region F such that domF is a subpasting scheme of doinG, and such that removing d(F) from G gives a new (n+l)-pasting scheme containing one fewer interior regions. P r o o f i This follows immediately from the definitions and Corollary 4.4 given the observation that any n-pasting scheme X from S to T in G that includes any n-cell in the domain of a topmost (n+l)-cell F necessarily includes every element of inF. []

C o r o l l a r y 4.6 For any (n+l)-pasting scheme (G,g), the (n+l)-cells of (G,9) can be ordered F o < F1... < Fm such that there exists a sequence G0,G1,.-.,G,~ of (n+l)pasting schemes with (G,g) = Go and such that Gi+x is the result of removing d(Fi) from

Gi. We shall call any such sequence an (n+l)-sequence. Note that we have not yet used condition 2 of the domain replacement condition. It will be used later to show the unicity of the possible composites of a labelled pasting scheme. Several other results follow from Proposition 4.3. One is that the apparent asymmetry in the definition of (n+l)-pasting scheme is illusory. P r o p o s i t i o n 4.7 Let (G,9) be an acyclic (n+l)-embedding. Then (G, 9) satisfies domain replacement if and only if it satisfies codomain replacement. P r o o f i Suppose (G,g) satisfies domain replacement. Let X be an n-pasting scheme from S to T in G and let F be an (n+l)-cell of G with codF a subpasting scheme of X. We shall show that if one replaces codF by domF in X, one has an n-pasting scheme embedded in (n+l)-space. We proceed by induction on the number of (n+l)-cells of (G,g) that lie outside B(X). If it is 1, then the (n+l)-cell must be F, so the replacement of codF by domF in X is doinG, hence an n-pasting scheme embedded in (n+l)-space. If it is m + l , then there exists an (n+l)-celt F x different to F and outside B(X). If F I i s not a topmost (n+l)-cell, then there exists F 2 outside B(X) such that outF2 tq inF~ is nonempty. Note that F2 is not F since codF is a subpasting scheme of X but F 2 lies outside B(X). So, by induction, since (G,g) is acyclic, there exists a topmost (n+l)-cell F ' of (G, g) that lies outside B(X) and is not F. Then, if we remove d(F) from G, we are done by induction. Thus we have proved part of codomain replacement. The rest follows directly from this part together with domain replacement. The converse is dual. []

C o r o l l a r y 4.8 Let (G,g) be an (n+l)-embedding that satisfies the n-cell extension condition. Then (G,g) satisfies domain replacement if and only if it satisfies codomain replacement.

343

[]

P r o o f i Propositions 4.3 and 4.7. Now we give the promised converse to Proposition 4.3.

P r o p o s i t i o n 4.9 If an (n+l)-embedding (G,g) satisfies condition 1 of the domain replacement condition and is acyclic, then it satisfies the n-cell extension condition. Proof." This follows by induction on the number of (n+l)-cells in (G,g). If (G,g) has no (n+l)-cells, the result is trivial since every n-cell lies on domG. If (G,g) has m + l (n+l)-cells, let F be a topmost (n+l)-cell: one such exists by Proposition 4.2. Then, given an n-cell z, either z lies on domF, therefore on doinG, or z is in Gx, the acyclic (n+l)-embedding given by removing d(F) from G. Trivially, G1 satisfies condition 1 of domain replacement, so by induction we are done. D C o r o l l a r y 4.10 Let (G,g) be an (n+l)-embedding that satisfies condition 1 of domain replacement. Then (G, g) satisfies the n-cell extension condition if and only if it is acydic. []

Proof'. Propositions 4.3 and 4.9.

C o r o l l a r y 4.11 Let (G,g) be an (n+l)-pasting scheme and let X be an n-pasting scheme in G from S to T. Then the collection of (n+l)-cells of (G,g) that lie in B(X) form an (n+l)-subpasting scheme of (G, g) with domain given by X and codomain given by codG. P r o o f i Domain replacement and the fact of being an (n+l)-embedding are immediate. By Corollary 4.10, (G, g) is acyclic, and so our collection of (n+l)-cells is also acyclic, and hence by Corollary 4.10 again, satisfies the n-cell extension condition. [] Now, as indicated in Example 3.11, we shall exhibit, for the case of 3-embeddings, a simpler condition than domain replacement, but which in the presence of the 2-cell extension condition, is equivalent to it. Consider the following condition on an (n+l)embedding (G, g) (*) if z is a k-cell of G with k n - Cat . . .

1 - Cat

> 0 - Cat

(8.2) (n + I) -- C m p t d

> n - Cmptd...1 - Cmptd

> 0 - Cmptd

So, assuming the existence of at least three strongly inaccessible cardinals, we may now define Catoo to be the object of the large 2-category [w°p, CAT] given by the top line of diagram (8.2). Similarly, we may define Crnptd¢~ to that object of [w°p, CAT] determined by the bottom line of diagram (8.2). So we may strengthen Proposition 8.1 to C o r o l l a r y 8.3 The collection (F, -I U, ln < w) forms an adjuntion F -I U in [w°r, CAT]. Now we may define oo-Cat to be the limit in C A T of the diagram Cat.o. Similarly, define oo-Cmptd to be the limit in C A T of the diagram Cmptdoo. Then, taking the limits induces an adjunction Foo -t Uo~ : oo-Cat ~ oo-Cmptd. Our category oo-Cat does not quite agree with Street's w-Cat. Ours is the full subcategory of his determined by those of his w-categories for which every element is an n-cell for some n. This agrees with the notation of AI-Agl and Steiner. Nevertheless, we can still deduce

357

C o r o l l a r y 8.4 An oo-category is free in Street's sense if and only if it is in the image of

Foo. We now seek to compare our definition of n-computad with the above. First note that at several points we have implicitly used the notion of the underlying (n+l)-computa~ of an (n+l)-category, for instance in defining a labelling of an (n+l)-pasting scheme in Definition 6.1. The underlying (n+l)-computad is subtly different to the underlying weak (n+l)-computad: where the latter uses F,U,(A,) as in diagram (8.1), the former uses diags of the underlying computad of A,. So although for each n, there is an evident fully faithful functor J~ : n-Computad ~ n-Cmptd expressing n-Computad as a full subcategory of n-Cmpld (and this lifts to a map in [w°p, CAT]), U,, does not factor through J , in general. Our n-computads amount to those weak n-computads whose domains and codomains ~tre given by pasting diagrams, and so the underlying n-computad and weak n-computad of an n-category differ accordingly. Since we do not capture all possible composites in general, J~ generally expresses n-Compu~ad as a proper subcategory of n-Cmptd. However, since we do capture all composites for n = l to 3, it follows that we have P r o p o s i t i o n 8.5 For n = l to 4, the functor d, is an equivalence of categories. There is a strong case that what we have called a weak computad would be better called simply a computad, and what we have called a computad would be better called a strong computad. Our choice in this paper has been just for brevity. However, it should be noted that our computads do incorporate a large class of the more general weak computads. For instance, it follows from Johnson and Street's work that all of the simplexes are computads. Moreover, Examples 3.11 and 3.12 illustrate several pasting schemes that may form domains and codomains for higher dimensional computads such as those describing at least some of the higher level adjuncions. Finally, Johnson's "coherent situations" [4] may be discussed more directly in the geometrical terms of our computads.

9

Bibliography

[1] F.A.A1-Agt and R.Steiner, Nerves of multiple categories, preprint (1990) [2] J.W.Gray, Formal Category Theory: Adjointness for 2-Categories, in Lecture Notes in Math 391 (Springer, Berlin, 1971) [3] M.Johnson, Pasting diagrams in n-categories with applications to coherence theorems and categories of paths, Doctoral thesis, University of Sydney (1987) [4] Michael Johnson, The combinatorics of n-categorical pasting, J. Pure Appl Algebra 62 (1989) 211-225 [5] Michael Johnson, Finding a representable family and the free construction on 1, International Category Theory Meeting 1990 (lecture)

358

[6] M.M.Kapranov and V.A.Voevodsky, Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders, preprint, Steklov Mathematical Institute (1990) [7] G.M.KeUy and R.H.Street, Review of the dements of 2-categories, in Lecture Notes in Math 420 (Springer, Berlin, 1974) 75-103 [8] A.J.Power, A 2-categorical pasting theorem, J. Algebra 129 (1990) 439-445 [9] A.J.Power, An algebraic formulation for data refinement, in Lecture Notes in Comp. Science 442 (Springer, Berlin, 1990) 390-401 [10] A.J.Power, Coherence for bicategories with finite bilimits II (in preparation) [11] M.B.Smyth and G.D.Plotkin, The category-theoretic solution to recursive domain equations, SIAM J. Computing 11 (1982) 761-783 [12] Ross Street, Limits indexed by category-valued 2-functors, J. Pure Appl Algebra 8 (1976) 149-181 [13] Ross Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-336

TOPOS-THEORETIC

APPROACHES by

MODALITY

TO

Gonzalo E. REYES and Houman ZOLFAGHARI Ddpartement de matl~matiques et de statistique Universit~ de Montrdal Dedicated to Max Kelly on the occasion of his 602 birthday.

Introduction In the last few years, two topos-theoretic approaches to modal logic have been developed simultaneously but independently: one started by Reyes [9] and further developed in Lavendhomme, Lucas and Reyes [7], and the other due to Ghilardi and Meloni [2]. At first sight, these approaches appear quite unrelated and the formal systems to which they lead are different. It may come, therefore, as a surprise that both are particular cases of a more general one which we shall describe in detail in this paper. There is a fundamental difference between modal operators in a topos such as D (necessity) and other logical operators such as --1 (negation): while -a is functorial in the sense that it commutes with pull-backs (and hence it defines a map -7 : f2 --~ f2), this is not so for 13, Indeed, the only operator [] : f2 --->f2 such that • p ~ p and DT = T is the identity. In Reyes [9], this difficulty is circumvented by considering a topos over a base topos and restricting the domain of application of modal operators to predicates of "constant" objects only, but keeping functoriality with respect to "constant" maps. Ghilardi and Meloni [2] on the other hand, restrict the domain of application as in Reyes [9] but relax functoriality to lax functoriality for constant maps. This feature of the approach of Ghilardi and Meloni [2] will be kept in our work. Indeed, it has to be kept in any context which aims to generalize their approach. In a sequel to this paper, we shall show that by imposing a restriction on the topos itself, we may define lax modal operators on predicates of all objects of the topos.

§1.

T h e general context

We consider a topos ~ (thought of as a universe of "variable sets") defined over a topos (thought of as a universe of "constant sets"). In other words, we have a geometric morphism (A,I"), i.e., a diagram A F

such that A-~ F and A preserves finite limits. Since A is uniquely determined by F , we shall write ~ morphism, leaving A implicit, as is customary in topos theory. From now on we make the following assumption on (A,F): (*)

r > ,~ for the above geometric

A preserves infima of subobjects of ,~

It is convenient to reformulate (*) in terms of the existence of adjoint functors. To do so, consider the diagrams, for each S ~ : g ,

360 As

(**)

Sub,(S) ~

Sub~(AS).

Fs

where Sub,~(S) is the ordered category of subobjects of S and AS, FS are defined as follows: given p c_~ S apply A to obtain AP ~-~ AS. Define As(P~-+S) = AP ~ AS. On the other hand, given K ~ AS, apply 1" and take the pull-back F(K)

,

) FAS

l~

T~s

Fs(K) '

~ S

where TI : Id ---* FA is the unit of the adjunction A -~ F . It is easy to check that AS -~ FS. Therefore our assumption (*) may be reformulated as

(*)'

for each S e /3 , we have a diagram

(

Sub,8(S)

~s As

(

) Subs(AS)

F, with I'ls -~ AS -~ r s . From these adjunctions, we obtain at once explicit descriptions for H s and FS. 1.1

Proposition. Let ~

F

) /8 be a geometric morphism satisfying (*)'. Then Us(K) = A{P ~ Fs(K) = V {P ~

SIK~--~ AP ~ SIAP~--~ K ~

AS} AS}.

This proposition tells us that 1-Is(K) is the smallest subobject P of S such that K is contained in AP, whereas Fs(K ) is the largest subobject P of S such that K contains AP. We notice the following difference between I-Is, AS and FS : whereas AS and FS are easily seen to be natural in S , this is not so for HS (unless we impose further conditions on the morphism (A,F)). The naturality of AS and FS implies the existence of a diagram (in /3):

n, 7

8

r(ns)

where 8-~ ~ and 8 ( T ) = T , 8 ( p A q ) = 8 ( p ) A 8(q). Although this is well-known, let us elaborate a little: the domain category of AS is Sub,8(S) = /3(S,O,~) , whereas the target category of AS is Subs(AS) = 8(AS,f~8) -~/3(S,F(f~8)). By identifying the domain and the codomain of AS with /3(S,f~A~) and /3(S,F(~8)) respectively, we may consider AS as a map which sends S-elements of f~,~ into S-elements of F(f~8) in a way that is natural in S , i.e., (As),6 defines a map f~.~ ~

F(f28)

such that 8 o p = AS(p) for all S-elements

361

./ p : S ~ f~,g of fl~5 • Similarly, .(Fs)~ defines a map F ( f ~ ) ~ fL8 such that T o K = F s ( K ) for all S-elements K : S ----> F ( f l S ) of F(f2~). On the other hand, (xs).8 is only lax natural in S in the sense that for all S

f

> T a/8

Hsf* < f ' F I T . This means that we have a cell as indicated in the diagram S

~(S,f~)

~

eg(S,F(t2~))

,1 ,.T T

T,*

eS(T,~,8) < . ~(T,F(f2~))

In fact, K < ATI-ITK for all K ~ ,'8 ( T , F ( f l S ) ) , since I-IT --t A T . Hence f*(K) < f*ATFITK = A s f * H T K , by naturality of (As).~ • Since I-Is-q A s , this implies that Hsf*(K) < f*HT(K). To express this lax functoriality of (1-1s).6 we shall write F ( t 2 ~ ) - - -~ t2,~. With this convention we may rewrite our main assumption on 8 (*)" There is a lax ~ such that

F

> /8 as follows

(... -- --

8

~,8

~ F(f~)

(

with g-q 8v- T . Although several examples of our general context will be given later on, we mention 2 cases where our main assumption (*) (equivalently, (*)") is satisfied. E x a m p l e 1. Let u • C ----> ID be a functor. Then the induced geometric morphism Set ID° Set ~:* is essential, i.e., A = u* has a left adjoint I1 = u l . This example can be generalized: any essential geometric morphism obviously satisfies (*)" E x a m p l e 2. An open geometric morphism 8 ~ there is a diagram (in /8)

eg,

i,e., a geometric morphism such that

(

f~,8

) Ffl~ (

"t where ~, is an actual map such that ~,-q 8--~ ~/. Notice that ~. satisfies the following Frobenius condition: ~(kA 8(p)) = ~,(k) A p .

362 §2.

Modal operators

In this section we define modal operators [] ("necessity") and 0 Cpossibility") starting from a F geometric morphism ~ > /g satistying (*). Recall that we can reformulate (*) as the existence of a lax n in the diagram

(

7

with n -~ 5 -~ Y. 2.1 Definition. We define the necessity operator D as follows [] = 5~, : F ( ~ C ) ~

F(~).

2.2 Proposition. The map D has the following properties [] < I d F ( ~ ) l-q2 = D DT=T D(KAL) = DK A DL Proof. Direct check. Alternatively, observe that [] = 5y is a l e x cotriple. Notice that we have not used our assumption (*) to define D , since it is definable from 5 and 7 alone. From [] we can further define D s : ~(S,F(f2~))---~ A(S,F(f2~)) by composition with D , i.e. Ds(K) = [] o K = As o Fs o K . On the other hand, to define we need our assumption. 2.3 Definition. We define the possibility operator ~) to be the lax (0s)A where, for each S e , ~ , 0s : ~ ( S , F ( f ~ ) ) ---~ 2~(S,F(f~)) is given by 0s = AS ° U s . 2.4 Proposition. The couple (0,D) has the following properties for all S e A~ DS < Idr(f~) -< Os • os2 = os

, O~ = Os

Os -~ Ds Furthermore, for all S

f

> T e eft, Dsf* = f * D T

363

Osf* < f*OT Proof. Straight forward computation using the adjunctions Us -~ AS -~ F s . Using our convention once again, we may write 0 : F ( ~ ) family (0s)z which is lax natural in S . With this convention

- - ~ F(f~g) to indicate that we have a

8~ Since we are interested in modal operators and n : F(f~a) - - - ~ F ( ~ ) always exists (D = By), r it is natural to ask for a characterization of those geometric morphisms ~ > /g for which there is a lax 0 : F ( ~ 6 ) - - ~ F ( ~ ) such that the couple (0,O) has the properties listed in the previous proposition. Let us call these MAO couples for short. The answer to this question justifies, to some extent, the choice of our context: 2.5

F Proposition. Let C ) /g be a geometric morphism. Then the following are equivalent: (1) There is a lax 0 : F ( f ~ ) - - - ~ F ( ~ a ) such that (0,El) is a MAO couple. (2) If F

ro~

/"~r ~

is the canonical sur]ection/inclusion factorization of F , then ro satisfies (*)", i.e., there is a diagram Eo (~-----

8o (

%

with a lax ~o such that ~o -~ 8o -~ ~o • Furthermore, by identifying r o ( ~ )

and 0=8o~o.

Proof. As we saw before, the geometric morphism ~

~* 7

8

with r ( ~ 6 ) ,

[] = 8oYo

r ) ~ gives rise to a diagram

r(~)

such that 5-~ y , 8 T = T and 8 ( p A q ) = 8 ( p ) A 8(q). W e d e f i n e [] =&~'. We now describe the surjectionhnclusion factorization of F in terms of rn by using some elementary notions of the theory of locales (cf. [6]). The fixpoints of O constitute a sub locale f~o of the locale F(ff28), i.e., we have a diagram in

28

~o HAHF ~env

rI(F) In conclusion, the transpose of ~ is the morphism (rlvOrq,rc2) = rlF x AX : F x AX

> AHF x A X .

( ~ ) : We start by remarking these natural correspondances: H(FxAX) F x AX

) HF×X

~IF×~X

>Y ACt

~ AHFxAX , > d~Y x(ac0 AHF ; AYaX

a x(eOxNxa

~

o

a(Y x)

where 'r(Aa) and x(cc) are the exponential transposes of Acx and cx and the last factorization follows from the commutative diagram. HFxX

>Y

x(c0×xN'xa 7 e v yXxx given by the adjunction (-) x X-~ ( ) x , on which we apply A and then take the transpose with respect to ( . ) x A X -~ ( ) a x

Now suppose that 0 is mono, and let o~,j3 be morpbisms such that tx~ = 13lP. We have: IX

H(FxAX) F

1IF

> HFxX

, a',I, = ~ q '

> ArIF

>>dlY ax

\ \ Ax@) 7 o ~(yX) But 0 is mono and

') y )

369

))

AHF

)

ax(l~) x(a) .........~ H F

HF

Id

a(yX) )yX ...>

x(l~)

therefore x(ot) = x(13) and a = 13 . (~)

Suppose ~t is epi and let a,13 be morphisms such that Oa = 013, F~

a

0

A(Y x)

) AY aX

Remark that transposing t(a)

HF Id%

> yX

~t(a)

where t(cO is the transpose of ~ , we have a factorization F

) A(Yx)

nrNN //za~(~) AliF and we have the same thing for [3. We get the following correspondances. F

a

O

)) A ( y x )

) A y ax

AI-IF FxAX

~ A(yX)xAX

r t , × a X % at(a)×aXs~tO)×,aX AliF×,t~ II(FxAX) )) y X x A X

Aev

ev

) AY

)Y

I~×X W being epi, ev equalizes t(a) x X and t(~) x X . But the exponential transpose of this part of the diagram is t(a)

lIft)--~ t(l~)

therefore we have t(o0 = t(13) and a = t3 .

yX

Id

) yX,

370

The equivalence (3) ¢:~ (5) follows from (4) ¢~ (6) by using slice toposes, and the following result of Johnstone [5]: For an object Y of "6, let rl/Y

)

A/Y ~/AY < ,6/Y )

FlY

be the induced morphism. F is open iff for all Y , IVY is pre open. (Just notice that A/Y(X Y) = AX ~ AY and YI/Y(F ~ AY) = I-IF ~ Y which is the transpose of F ~ AF). F/Y being pre open means that all the morphism I'I/Y((F--~ AY)×(AX--~ AY)) 'e/Y) (FIF--->Y) × (X----~Y) . are epi. We can verify that ~/Y is just the morphism FI(FXayAX )

~F,x,v

) FIFxX

%/ Y in ,6/Y. And ~g]Y is epi iff 0F,x,Y is epi. So F is open iff all the maps ~F,x,Y are epi, which is precisely condition (3). We shall give counter examples to reverse arrows in the next section.

§4.

P a r t i c u l a r cases and examples

In this section we show that the approaches to modal logic mentioned in the introduction may be seen as particular cases of the one developed in this paper. Furthermore we show, by means of examples that some of the implications of 3.2 Theorem cannot be reversed. The context of Reyes [9] is a locally connected geometric morphism ~

F

) ,6, F

whereas the

context of Lavendhomme, Lucas and Reyes [7] is an arbitrary geometric morphism ~ ) ,6 with ,6 a Boolean topos. In the last case, F is automatically open (cf. [1] ) and 3.2 Theorem show that (*) is satisfied and, furthermore, 0 is actually functorial. On the other hand, the context of Ghilardi and Meloni [2] is the essential geometric morphism Setl•l Set •° derived from the obvious functor ICI---~ C . (Here ~2 is a small category and I•I the discrete category of its objects). Since A has a left adjoint, Ylo, condition (*) is satisfied, but 0 is not functorial, only lax functorial. We shall investigate functoriality of 0 in this case in 4.2 Proposition. In some sense, the approach of Lavendhomme, Lucas and Reyes [7] is at the opposite extreme from the one of Ghilardi and Meloni [2]. Whereas in the first, the base topos ,6 is Boolean, in the second it is the topos ~ which is Boolean. The motivation for the first was to view ~ as a universe of variable sets and ,6 as a universe of constant sets, providing a "standard of constancy" for change and modalities in . Modalities were defined for predicates of constant sets only and functoriality of modal operators resulted as a consequence. This is clearly not the motivation for the other approach. We shall study these extremes in some detail. Case A: /5 is Boolean. In this case we have the following

371

F 4.1 Proposition. Let ~ ) ,6 be a geometric morphism with ,6 Boolean. Then the resulting modal operators are endomaps of F(f2~) and have the following properties [] < Idr(n~ )_ Set, where IB is a Beth tree, gives rise to the "Graft semantics". See Lavendhomme, Lucas and Reyes [7]. The interest of this example is the existence of a completeness theorem for IBM with respect to this "graft" semantics (see the above mentioned paper). Case B: ~ is Boolean. As we mentioned already, the basic example of this context is the one studied by Ghilardi and Meloni [2]: the essential geometric morphism u!

SetI£1 (

)

U*

u.

SetC° )

where u is the obvious functor u : I£1--~ C . We have a MAO couple (0,D) on SetlCI. Objects of SetC° are presheaves on C and those of Setlq are just families indexed by objects of C . The functor u* applied to a presheaf X just forgets all the transition morphisms and "sees" X as a family. In the following discussion we will identify, by abuse of language, X and u*X. We have now the following nice characterization of the resulting modal operators (see Proposition 1.1): if X is a presheafand F a subfamily of X , then DF is the largest subpresheaf of X contained in F. OF is the largest subpresheaf of X containing F. So DF and OF are the two best "presheaf approximations" of the family F . We can go further and give explicit descriptions of 0 and [] (cf. Ghilardi & Meloni [2]), by recalling that a subpresheaf is simply a subfamily closed under the action of the transition morphisms. So if F is a subfamily of the presheaf X , OF is the closure under this action and DF is the closed part:

372

{

xE 0F(C) iff for some C f..~ C ' i n C and y e F ( C ' ) , x = X ( f ) ( y )

CM)

f

x e OF(C) iff for all C'

> C in ~ , X(f)(x) e F(C').

R e m a r k . We can think of OF as the orbits of the elements of F under the action of all the transition morphisms of X . One interesting presheaf is hA, the representable presheaf associated with an object A of E . Then {1A} is a subfamily containing only 1A, the identity of A , at stage A and nothing at other stages. (We will write families simply as sets, where each element belongs to its implicit stage). The transition morphisms hA(f) correspond to right composition by f , and we have by the above caracterizadon

0({IA} ) = hA • Here is one nice and natural application of this idea. We know from general topos theory that a f morphism X ~Y in Set @° induces a diagram 3f

~(X)
hB

\ 7Im hf Im hf is not empty (we have at least f as an element) so we must have Im hf : hB • hf is then surjeetive and we have g : B---~ A such that g f = 1B. Each f e Mor(C) has a left inverse, so it has an inverse by a standard argument, therefore 112 is a groupo'fd. In the case Setl(:l ~ Set (:° , we even have a finer result, implicit in Ghilardi and Meloni [2]. 4.4 P r o p o s i t i o n . Let 0 be the possibility operator associated with following conditons are equivalent: (i) C is a groupo'/d

Setl(:l--~ Set (:° . Then the

(ii)

0 is functorial i.e. Of* = f*O for all X

(iii)

07~* = r~*O for all projections X x y

f ~ Y in Set(:*

(iv)

Od* = d*O for all the diagonals X

(v) (vi)

All the transition morphisms of all objects X in Set (:° are surjective. All the transition morphisms of all objects X in Set ~° are injective.

n > X in Set (:* . d ~ X × X in Set (:°

374

Proof. We already know (i) ¢~ (ii). The implications (ii) ~ (iii), (ii) ~ (iv) and (i) ~ (v) are immediate. (v) ~ (i) follows from the analysis of the representables. Let hA be the one associated with A in hA(f) f C , hA(B) is the set hom~:[B,A] and the transition morphism hA(B) > hA(C) for C > B in C is just fight composition by f . Putting C = A taking an inverse image of 1Ae hA(A) (hA(f) being surjective), we obtain a left inverse for f . Each f in C has a left inverse, therefore it is a groupo'fd. Next we show (iii) ~ (i) and (iv) ~ (vi) ~ (i) and we wilt have all the equivalences. (iii) ~ (i): Suppose that 0 commutes with pulling back along projections. We show that every f morphism B > A in C has a left inverse, and so C is a groupoi'd. Let f be such a morphism. Take the first projection hA × hB > hA . { 1A} is a subfamily of hA, and as we argued earlier {){ 1A} = ha thus r~*0{ 1A} = hA × hB • On the other hand: 0~*({ 1A}) = 0({ 1A} ×hB), therefore we must have hA × hB = 0([ 1A}×hB). Now take (f, lB) e (hA×hB)(B). By the caractefization (M) there are morphisms g and k in C such that (f, lB) = (hA×hB)(k)((1A,g)) = (hA(k)(1 A),hB(k)(g)) = (k,gok) So k = f and g o f = IB wich gives the desired left inverse for f . ( i v ) ~ (vi) Let X b e a p r e s h e a f a n d f : B --~ A a m o r p h i s m i n C . We show that X(f) is injective. Take the diagonal d : X---> X × X . We have 0d* = d*0 . Suppose that X(f)(a) = X(f)(b) = c for some a,b E X ( A ) . We have (XxX)(f)(a,b) = (X(f)(a),X(f)(b)) = (c,c) = d ( c ) , therefore d(c) = (c,c) e 0{(a,b)} because it is in the orbit of (a,b). So we have c e d*0{(a,b)} = 0d* {(a,b)} , and by (M) it must come from some element in d*{(a,b)} under the action of a transition morphism. But d*{(a,b)} = {(x,x)lxeX} n {(a,b)} and d*{a,b} beeing non empty, we must have a = b . (vi) ~ (i): Suppose all the transition morphisms are injective for all preshseaves. We show that f every arrow B ) A in C has a right inverse, therefore C is a groupo'fd. Let X be the colimit of the following diagram in Set c° hB

h~

> hA

hA X(C) for C ~ C consists of two copies of hA(C) quotiented by the congruence - generated by hB(C) and the two versions of hf(C). More precisely, the elements are of the form (x,i) where x hA(C) and i = 0 or i = 1. The generators of the congruence are the basic relations (hf(y),i) R (hf(y),j) where y ~ hB(C) and i,j e {0,1}, or equivalently (foy,i) R (foy,j). Now take C = A . We have X(f)(1A,O) = (f,O) = (f,1) = X(f)(1A,1)

375

and X(f) being injective, we must have (1A,0) - (1A,1). But each non trivial congruence comes from a chain of basic relations. Therefore we must match (1A,0) with some (foy,i). So there is an y e hB(A) such that f o y = 1A giving the desired right inverse.

Remark on connectivity properties Another major difference between the approaches A and B is given by connectivity properties. A geometricmophism ~

r > .6 is connected if F A = I d (or equivalently I ] ( 1 ) = 1

if it is essential).

In this case Set c° r ) Set, F is connected iff C is connected (the connected components of 1 are precisely the connected components of C ) , whereas for SetlCI ~ the trivial case:

Set c° , it is possible only for

Proposition. Let u : Set Ict r ~ Set~:o be the geometric morphism induced by tCI ~

C. Then the

following are equivalent: (1) F is connected

(2)

C =

(3)

For all presheaf X , and A,B subfamilies of X , D(AuB) n OA n OB c_ O(ANB).

IC I

Proof. (2) ~ (3) and (2) ~ (1) are immediate. (1) ~ (2): since (1) is equivalent to A being full and f

faithful. (3) ~ (2): suppose we have C' > C in C . We show that C = C', and f = Id. Take X = h c , A={1c} and let B the complement of A . We have 0 ( A n B ) = ~ , O ( A U B ) = h c , 0 A = 0 { 1 c } = hc therefore the inequality forces 0 B = ~ , which implies B = O . Now f e h c , therefore f ~ A and so f = l c •

Examples and counter examples. (a) For the following essential geometric morphism, we don't have Frobenius property, although it is open as we show by using our results on modality operators. The functor u : (o ~ o) --~ (o --~ o), sending the two parallel arrows to the unique non-identity arrow of the right category, induces an essential morphism

u! ) --4

Set o ~

u*

o
Xo) where X1 and Xo are sets and ~o,51

functions. We interpret X as a graph, X1 and Xo being the sets of arrows and points, respectively, and ~o,~1 the source and target functions. Set --~ ° thus becomes the topos of (irreflexive) graphs. (cf. Lawvere [8]). In Set ° --~ ° , an object Y consists of two sets Yo,Y1 and a function 8 : Y1 -'-) Yo • In the same fashion, we will think of Y as a graph whose only arrows are loops, and ~ is at the same time the source and target functions. Then the functor A = u* is simply the "translation":

376

8

A(YI

8 > Yo) = (YI

> Yo) a S e t ° ~ ° 8

The functor F = u. just extracts the loops out of all arrows of a graph, keeping all the vertices. The functor II = u! collapses every arrow into a loop, identifying the source and target vertices of every arrow. The Frobenius property does not hold here: take X = ( 0 --~ I ) , the "generic vertex" and io

X = (I ~

2 ) , the "generic arrow". Then Y x AX = ( 0 __~ 2) which is a graph with two vertices ii

and no arrows. W e have H(Y×AX) = (O---) 2 ) . On the other hand H ( Y ) = (1 --> 1), the "generic loop", which is also the point in this topos. Therefore H ( Y ) x X = ( 0 ---) I) . For openness, we observe that u. is surjective and by 3.1 theorem, it is open iff the induced possibility operator is functorial. But here 0 is the identity operator: let X be an object of Set ° --~ ° and E ~ AX in Set ° ~ ° . AX is a graph consisting only of loops and vertices, so E which must be the same, is already of the form AS' where S' ~---~ X . 0 E being the smallest AS' where S 'c--> X and E AS' must be E itself. W e have 0 = Id (and [] = Id) which is functorial. (b) Here we give an example where we have functoriality of 0 but not openness. In fact the "Frobenius morphism" ~P here is mono but not epi in general. Let C be a small category, and F be the global sections functor in the geometric morphism: A

Set C° (

)

Set

F

The functor 1-" has a left adjoint B (the "codiscrete" or "chaotic" functor) if some representable ot Set C° has a point (see Lawvere [8]). Suppose we have such a left adjoint, we get an essential morphism A

Set
AS x X

which is not iso in general. In fact we have hu = AS x ex and W is iso iff the counity AFX-----* X is iso, which happens only if X is a sum of points. Let us look at the particular example where ~ = A1, the monoid with three elements: 1, 80 and 51 , and identities ~ l ~ j = ~i for i,j e {0,1 }. This example is studied in Lawvere [8]. An object X of Set A: corresponds to a reflexive graph: a set X of arrows and two endofunctions X(8o) and X ( S t ) , the source and target functions, respectively. Here the vertices are represented by reflexive arrows: i.e. arrows which are their own source and target. The set of vertices of X is precisely the set ot points F ( X ) and A F ( X ) is the discrete reflexive subgraph of the points of X . The counity ex : AI-'X ~ X is just the inclusion as a subgraph. It is mono and therefore W is mono. It is not epi except if X is discrete. So (F,B) is not open. On the other hand, 0 is functorial as we see either by checking that the image factorization of (1-',B) is

377

Set

>Set A~

2",,/ Set

where the surjective part is trivially open, or by analyzing AS for S E Set and remarking that is subgraphs come from subsets of S and therefore t) = Id which is functorial. (c) ttere is an example where the modality operators are nontrival. Let C be the category oO

So

) ol

o2 8o and ID = ( o

> o)

)

)

and let u : C - - ~

ID be the functor which sends ao to 8o and oq to 81. As

81 before we interpret an object X = (Xt - - ~ Xo) of Set D°

as a graph. In Set C° an object Y is of the

type

Y(O)

(

Y(o.~)

Y(1)

Y(al) l

Y(2) In the following discussion, we think of Y(0) as the set of vertices of Y , and Y(1) and Y(2) as sets of loops, where we have two kind of them Xl ~ Y(1) is a continuous loop and x2 e Y(2) is a dotted one. The Y will be like this: fh'2.

Y Now if X i s a g r a p h A X ( 0 ) = X ( 0 )

. ,,.

and A X ( 1 ) = A X ( 2 ) = X ( 1 )

x(0)


Ex

,~........

,

F x -t U x

U x

be the corresponding adjunction exhibiting E x as a category of algebras. T h e crucial point now is to observe that, for a fixed X , the condition (M) of Proposition 1.4 is the condition (1) of Proposition 2.3 applied to the above adjunction. With this observation and the propositions of the last section, we have : P r o p o s i t i o n 3.1. A category E with pullbacks and finite coproducts is modular iff, given any c o m m u t a t i v e diagram of the f o r m

X

X

f

......

,~

Z

e

÷

X+Y

'

ix

k

iy

W

Y

then the top row is a coproduct iff the right-hand square is a pullback.

P r o o f . This follows from Proposition 1.4 and the equivalence (1) ¢~ (2) of Proposition 2.3.

[]

Assuming the existence of a terminal object we have : P r o p o s i t i o n 3 . 2 . A left exact category E with finite coproducts is modular (or a ~ n e ) iff, given any c o m m u t a t i v e diagram o f the f o r m

X

f

I I X

"

Z

,

• .........~ X + 1

ix

k

W

l, ~

i 11

then the top row is a coproduct iff the right.hand square is a pullback.

398

P r o o f . It follows from Proposition 1.4, the equivalence (1) ¢~ (3) of Proposition 2.8 and Proposition 2.3. [] R e m a r k 3.3. The last two propositions should be compared with the characterization of a distributive category, either as a category with pullbacks and finite coproducts such that, given any commutative diagram j

V

....~ ..

Z

k

W

q

X

ix

....~. X + Y

iy

Y

then the top row is a coproduct iff b o t h squares are puUbacks or, assuming the existence of a terminal object, as a left exact category satisfying the same condition, where it suffices now to take those diagrams with the coproduct 1 --* 1 + 1 ~-- 1 as the b o t t o m rOW.

P r o p o s i t i o n 3.4. A category E with pullbaeks and finite coproducts is modular iff~ for

all objects X and Y of E, the adjunction

E/Y

,

u#

.....'

E X / ( X 25-+ X +1)

(induced, for any Y, by the adjunetion F x q U x ) is an equivalence. P r o o f . It follows from Proposition 1.4 and the equivalence (1) ~ (3) of Proposition 2.3. [] Assuming the existence of a terminal object we have : P r o p o s i t i o n 3.5. A left ezact category E with finite eoproducts is modular (or affine)

iff, for every object X of E, the adjunetion

E

,

"

EX/(x-~x+I)

tr( (where E is identified with E / l ) is an equivalence.

399

P r o o f . This follows from Proposition 1.4, the equivalence (2) ~, (3) of Proposition 2.8 and Proposition 2.3. [] Remark that condition (2) of the characterization theorem of [1] is a particular case of the condition above, where X = I .

4.

The fibration of pointed objects

Let E be a category with puilbacks, and let p : P t E -+ E be the fibration of pointed objects in E. (An object in P t E is a pair of maps 8

X~

~Y f in E with f s = l x , a map in P t E is a morphism of such diagrams, and p ( f , s ) = X . ) Let us denote by P t E [X] the fiber over X of P t E. We shall now prove that, for a modular category E, P t E is equivalent to a constant fibration, and then that P t E is additive. P r o p o s i t i o n 4.1. I f E is a modular category, then the universal map K : PtE--~E×Pt

El0]

from the fibration P t E to a constant fibratlon is an equivalence.

P r o o f . Identifying P t E [X] with E X / ( x = X) and P t E [0] with E / 0 , one checks that, over an object X of E, the functor K is the functor U0x : E X / ( x = x ) ~ E / 0 which, by proposition 3.4, is an equivalence. [] P r o p o s i t i o n 4.2. I f E is a modular category, then P t E is an additive fibration. P r o o f . In view of Proposition 4.1, it suffices to show that E / 0 is additive. First, the terminal object of E / 0 , the identity on 0, is also an initial object. Let now x : X --* 0 and z : Z ---* 0 be two objects in E / 0 . Applying condition (1) of Definition 1.1 to the map X ~ ~ 0 ---+ Z (denoted by 0) in E / 0 and identifying 0 × Z with Z and X + 0 with X in E / 0 , shows (using a matrix notation) that the map

is an isomorphism.

400

Applying the same condition to the map l x in E / 0 shows that Lhe map

is an isomorphism and this, by a standard argument (see [2] for example), complete the proof that E / 0 is additive. [] Assuming the existence of a terminal object we have, as in [1] P r o p o s i t i o n 4.3. A modular category E with a terminal object is a ~ n e , i.e. E iJ equivalent to a slice of a left exact additive category. P r o o f . It follows from Proposition 3.5, by taking X = I , that E is equivalent to the category EX/(1 ---} 1 + 1). E 1, deaoted Pt(E) in [1], is left exact and, being the fiber over 1 of P t E, is additive by Proposition 4.2. 13

Aknowledgements The author was in 1989/1990 on a study leave from the Coll~ge de Stall in Geneva and spent that year in Sydney as a Visiting Fellow at Macquarie University, where the work reported on here was done. The author wishes to thank the D~partement de l'instruction publique du canton de Gen~ve for this opportunity, the Sydney Category Theory Research Group for inviting him to Sydney and its members for their hospitality.

l~eferences [1] A. Carboni, Categories of affine spaces, J. Pure Appl. Algebra 61 (1989), 243-250. [2] B. Mitchell, Theory of categories, Academic Press, 1965. [3] P. Taylor, The trace factorization of stable functors (paper circulated at the International Category Theory Meeting '89, in Bangor).

This paper is in final form and will not be published elsewhere.

SOME CONSTRUCTIVE RESULTS RELATED TO COMPACTNESS AND THE (STRONG) HAUSDORFF PROPERTY FOR LOCALES J.J.C. Vermeulen Department of Mathematics, University of Cape Town Private Bag, Rondebosch 7700, South Africa 0. I n t r o d u c t i o n The categorical characterization of the Hausdorff property for topological spaces, viz. closedness of the diagonal, was for locales first considered by J.R. Isbetl [1]. Despite being a natural and well-behaved property, it was, however, not immediately accepted as the %orrect" notion of localic Hausdorffness, since an ordinary Hausdorff topological space need not have a closed 1ocalic diagonal: the standard and localic products of topological spaces are in general different. On the other hand, having a closed Iocalic diagonal is sufficient for a topological space to be Hausdorff, and therefore the term "strongly Hausdorff" was used to refer to the former property. In the constructive context of a topos, where locales are the "true" spaces, it is however evident that strong Hausdorffness is more fundamental than the various other definitions (e.g. [5], [7] and [9]) that have been proposed. In particular, with the strong definition, and the standard localic usage of the terms closed, compact, locally compact and regular (cf. [4]), the following familiar relationships hold in an arbitrary topos: (I) A regular locale is Hausdorff; (II) A compact Hausdorff locale is locally compact; (III) A locally compact Hausdorff locale is regular; (IV) A compact sublocale of a Hausdorff locale is closed. Now, proofs of statements (I) and (implicitly) (II) may already be found in [1], that of (I) being essentially constructive, whereas (II) needed a choice principle: it was shown [1, Thm. 2.6] that all compact, (strongly) Hausdorff locales are spatial, and for such spaces the relevant localic and topological notions coincide. The latter also applies to statement (III); cf. [2] or [4, II 2.13], [4, VII 4.3, 4.5]. We are not aware of a published proof of statement (IV). Our purpose here is to establish the constructive validity of (II) (Prop. 4.2), (III) (Prop. 4.5) and (IV) (Prop. 3.3). In accordance with the spirit of [1] and particularly [6], our language and style of proof have been chosen to exhibit locales as geometric objects rather than lattices. The results presented here have a close connection with the stability under pullback of perfect maps of locales [3], [10]; this is briefly discussed in the final section. NOTATION. Sets are objects of an unspecified topos S. Spaces are locales in $; a space X is therefore completely determined by its frame OX of open subspaces, and a continuous map X {, Y by the inverse-image frame homomorphism f - [-]: OY --+ OX. If S is a subspace of X, -~S stands for the largest open subspace of X disjoint from S. If it exists, the (true) subspace-complement of S is denoted by - S ; in particular, if S is

402

closed, - S = -~S. More generally, - S = -~S where S is the closure of S. T h e interior of S is denoted by ;~. We say 0.1. D e f i n i t i o n . A space X is Hausdorffif the diagonal A = (id, i d ) : X '--+ X x X is closed. Thus defined, the Hausdorff property is evidently inherited by subspaces and products. The interested reader might wish to consult standard references like [4] and [6] for the necessary background on locales; cf. also [3]. 1. S o m e M a c h i n e r y We need a few results of a technical nature, involving binary products of spaces. Let X and Y be spaces. T h e n a base for the product space X × Y is provided by open "squares", that is, open subspaces of X × Y of the form U × V -~ r x U A ~r~.V, where U E 0 X and V E OY and r x , ~rr are the projections. We say a binary relation riU, V) between opens U of X and V of Y is left-downstable, left-stable under suprema, etc. if for each Y E OY, the corresponding closure property holds for the set {U E O X I flU, V ) } in 0 X . By (finitely) left-stable we mean left-downstable and left-stable under (finite) suprema. The corresponding notions of right-stability are defined similarly. With the relation r we assosiate the mapping jr: O X -+ OY given by

Jr (P) = V{Q I riP, Q/}One checks that 1.1. L e m m a .

The following are equivalent: i) r is right-stable; ii) For all P E O Z and Q E OY, Q < jr(P) ~

riP, Q).

•

L e m m a 1 says the assignment r H Jr restricts to a bijection between right-stable relations, and functions O X ~ OY. For convenience, we say the relation r is admissible if i) r is left-downstable, ii) r(0, X) holds and iii) r i p V P', Q A Q') holds whenever both r i P , Q) and r i P ' , Q') do. An admissible relation r is finitely left-stable; the converse is true provided r is right-downstable. Expressed in terms of Jr, 1.2. L e m m a . If r is admissible, then jr transforms finite joins into meets. converse holds if r is right-stable.

The

Thus, the mapping r ~ Jr restricts to a bijection between finitely left-stable, right-stable relations, and functions O X --* OY which transform finite joins into meets. Relations r which are b o t h left- and right-stable, or stable for short, are the familiar "product ideals" of O X × OY, and correspond to mappings jr: O X ~ O Y which transform all joins into meets (Galois-connections). These may be used to represent the opens of X × Y, since: 1.3. P r o p o s i t i o n [4, II 2.12]. Let r be a relation between opens X and Y which is both left- and right-stable. Then, for all U E O X and V E OY, UxV X ~ x X is closed in Shy(X), which implies t h a t it is a closed map in the ordinaray sense; this provides a conceptual proof of Proposition 3.2. Generalizing openness from subspaces to arbitrary maps gives the notion of open map; perfect maps may be viewed as the corresponding extension to maps of the idea of closedness of a subspace (perfect embeddings into X are exactly closed embeddings, by interpreting in Shy(X) the fact that the compact subspaces of the terminal space are its closed subspaces). In this sense, the following is a generalization of statement (IV): 5.2. P r o p o s i t i o n . A n y continuous map f : K ~ Hausdorff space is perfect.

X from a compact space to a

PROOF. T h e graph (id, f ) : K ~ K × X o f f is the inverse image along f x i d : K x X --~ X × X of the diagonal of X; it is therefore closed, hence a perfect embedding into K x X. But this means f = r x o (id, f ) is perfect, being the composite of perfect maps. • Now, as an immediate corollary of the (constructive) regularity of compact Hausdorff spaces, we recover the description of a proper map (cf. [4 III 3.8]) in its familiar form: 5.3. P r o p o s i t i o n . A perfect map E A : E ~ E × x E is closed.

P ~ X is proper if and only if the diagonal

PROOF. A : E ¢--* E > x E is the (ordinary) diagonal of E P~ X in S p / X . statement therefore follows by interpreting Proposition 2.3 in Shv(X). • 5.4. C o r o l l a r y . PROOF.

A n y perfect map E

The

P, X with Hausdorff domain is proper.

If E has a closed diagonal, it has a closed diagonal over X.

•

We are grateful to Bernhard Banaschewski for drawing our attention to an error in a preliminary draft of this paper.

409

REFERENCES

[1] J.R. Isbell. Atomless parts of spaces. Math. Scand. 31 (1972), 5-32. [2] J.R. Isbell. Function spaces and adjoints. Math. Seand. 36 (1975), 317-39. [3] P.T. Johnstone. Factorization and pullback theorems for Iocalic geometric morphisms. Univ. Cath. de Louvain, S~m. de Math. Pure, Rapport no. 79. [4] P.T. Johnstone. Stone Spaces. Cambridge studies in advanced Math. no. 3 (Cambridge University Press, 1982). [5] P.T. Johnstone and Shu-Hao Sun. Weak products and Hausdorff locales. Springer Lecture Notes in Math. vol. 1348 (Springer-Verlag, 1988). [6] A. Joyal and M. Tierney. An extension of the Galois theory of Grothendieck. Memoirs Amer. Math Soe. 309 (1984). [7] J. Paseka and S. ~marda. T2-frames and almost compact frames. Preprint (?). [8] J. Pazeka and B. ~marda. On some notions related to compactness for locales. Acta Univ. Carolinae (2) 29 (1988), 51-65. [9] H. Simmons. The lattice-theoretic part of topological seperation properties. Proc. Edin. Math. Soc. (2) 21 (1978), 41-48 [10] J.J.C. Vermeulen. Pullback-stability properties of perfect maps of locales. Preprint.

This paper is in final form and will not be published elsewhere.

AN INTRODUCTION TO TANNAKA DUALITY AND QUANTUM GROUPS

by

Andr6 JOYAL and Ross STREET Universit6 du Qu6beca Montr6al, Canada and Macquarie Universi~, Australia Introduction The goal of this paper is to give an account of classical Tannaka duality [C ] in such a way as to be accessible to the general mathen~atical reader, and to provide a key for entry to more recent developments [SR, DM] and quantum groups [D1]. Expertise in neither representation theory nor category theory is assumed. Naively speaking, Tannaka duality theory is the study of the interplay which exists between a group and the category of its representations. The early duality theorems of Tannaka-Krein [Ta, Kr] concentrate on the problem of reconstructing a compact group from the collection of its representations. In the abelian case, this problem amounts to reconstructing the group from its character group, and is the content of the Pontrjagin duality theorem. A good exposition of this theory can be found in the book by Chevalley [C]. In these early developments, there was little or no use of categorical concepts, partly because they did not exist at the time, and, partly because categorically concepts were not yet familiar to the mathematical community and it was possible to do without them [BtD]. To Grothendieck we owe the understanding that the process of Tannaka duality can be reversed. In his work to solve the Weil conjectures, he constructed the category of motives as the universal recipient of a Weil cohomology [KI]. By using a fiber functor from his category of motives to vector spa.ces, he could construct a pro-algebraic group G. He also conjectured that the category of motives could be recaptured as the category of representations of G. This group is called the Grothendieck Galois group, since it is an extension of the Galois group of F~/Q. The work spreading from these ideas can be found in [SR, DM]. An entirely different development came from mathematical physicists working on superselection principles in quantum field theory [DHR] where it was discovered that the superselection structure could be described in terms of a category whose objects are certain endomorphisms of the C*-algebra of local observables, and whose arrows are intertwining operators. Reversing the duality process, they succeeded in constructing a compact group whose representations can be identified with their superselection category [DR]. Another impulse to the development of Tannaka duality comes from the theory of q u a n t u m groups. These new mathematical objects were discovered by Jimbo [J] and Drinfel'd [D1] in connection with the work of L.D. Faddeev and his collaborators on the quantum inverse scattering method. V.V. Lyubashenko [Ly] initiated the use of Tannaka duality in the construction of quantum groups. We should also mention S.L. Woronowicz [W] in the case of compact quantum groups. Recently, S. Majid [M3] has shown that one can use Tannaka-Krein duality for constructing the quasi-Hopf algebras introduced by Drinfel'd [D2] in connection with the solution of the Knizhnik-Zamolodchikov equation. Also see K.-H. Ulbrich [U]. Another motivation for studying Tannaka duality might come from the theory of angular m o m e n t u m in Quantum Physics [BL1]. The Racah-Wigner algebra, the 9 - j and 3 - j symbols, and, the Racah and Wigner coefficients, all seem to be about the explicit description of the structures which exist on the category of representations of some compact groups like SU(2) or SU(3) [BL2]. Here the theory of orthogonal polynomials and special functions comes into play [AK]. Also, the q-analogues of the classical orthogonal polynomials show up as spherical functions on quantum groups [Ko].

414

Another essential aspect of the picture that should be mentioned is the connection between knot theory, Feynman diagrams, category theory, and quantum groups. It was discovered by Turaev [T] that the new invariants found by Jones [Jnl] could be constructed from Yang-Baxter operators. The method was formatised by showing that certain categories constructed geometrically from tangles of strings or ribbons could be given a simple algebraic presentation [FY, T, JS1]. It is remarkable that notation, introduced by Penrose [Pn] for calculating with ordinary tensors, has the right degree of generality to express correctly the calculations in any tensor category [JS2]. What is emerging here is a new symbiosis between algebra, geometry and physics, the consequences of which are not yet fully understood. A good review of the recent research on the whole subject is given by P. Cartier [Car]. After a brief review of Pontrjagin duality and Fourier transforms for locally compact abelian groups, we give a treatment in Section 1 of the classical Tannaka theory for compact (non-abelian) groups. In the compact case, representations must be considered, because characters are no longer sufficient to recapture the compact group. There is also a notion of Fourier transform applying, and we examine this in detail. Where possible we work with a general topological monoid M. A central object of our analysis is the algebra R(M) of representative complex-valued functions on M. In Section 2, we show that R(M) is a bialgebra and compute it in the case where M is the unitary group U(n). Section 3 and Section 4 begin the modern treatment of Tannaka reconstruction, motivated at each stage by the example of a topological monoid. Instrumental here is the Fourier cotransforra which can be seen as the continuous predual of the Fourier transform. In fact, the Fourier cotransform provides an isomorphism between the reconstructed object and R(M). Section 5 is an introduction to Tannaka duality for homogeneous spaces [IS]. The proof of the duality theorem of Section 5 is independent of the proof of the one appearing in Section 1. Sections 6 and 7 are devoted to the characterization of the category of comodules over a coalgebra. Sections 8 and 9 study extra structure which is possessed by the coalgebra Endv(X) of Section 4. Section 10 introduces the concept of braided tensor categories and Yang-Baxter operators at the appropriate level of generality. Section 11 is a brief description of the categorical axiomatization of the geometry of knots and tangles. Section 12 is a too brief introduction to quantum groups. There are many important aspects of the theory of quantum groups which could be mentioned, such as the work of Lusztig [Lu], Rosso [R], and Deligne. Our goal here is a modest one (already longer than our editors expected), and we apologize to those whose work we have not mentioned. Section 1 : Classical Tannaka duality. Section 2 : The bialgebra of representative functions. Section 3 : The Fourier cotransform. Section 4 : The coalgebra Endv(X). Section 5 : Tannaka duality for homogeneous spaces. Section 6 : Minimal models. Section 7 : The representation theorem. Section 8 : The bialgebra EndV(X) and tensor categories. Section 9 : Duality and Hopf algebras. Section 10 : Braidings and Yang-Baxter operators. Section 11 : Knot invariants. Section 12 : Quantum groups. This paper is in final form and will not be published elsewhere.

415

§1. Classical Tannaka duality. Before describing Tannaka duality, we briefly recall Pontrjagin duality. Let G be a commutative locally-compact group. A character ~ of G is a continuous h o m o m o r p h i s m x:G *T where T is the multiplicative group of complex numbers of m o d u l u s 1. The characters form a group G v which is given the topology of uniform convergence on compact subsets of G. It turns out that G v is locally compact. There is a canonical pairing (,) : GVxG ~T, and we obtain a canonical h o m o m o r p h i s m i : G ~ (GV)v.

Theorem 1 (Pontrjagin). The topological groups,

canonical

homomorphism

i

is an

isomorphism

of

The group G is compact if and only if the dual group G v is discrete. We have Tv=z, ZV=T, RV=R, (Z/n)V = Z / n . Many groups are self dual, such as the additive groups of the local fields. Pontrjagin duality goes hand-in-hand with the theory of Fourier transforms which we briefly describe. There is a positive measure dx on G called the Haar measure. It is the unique (up to scalar multiple) Borel measure which is invariant under translations. Using it, we can define the spaces Lt(G) and L2(G) of integrabte and square integrable functions. The Fourier transform F : U(G)c~L2(G) , L2(Gv) is defined as follows (Ff)(s) = SGf(x) (S, x)dx

p

The set U(G)c~L2(G) is a dense subspace of L2(G) and we have: Theorem 2 (Plancherel). With correct normatisation of the Haar measure ds on G v, the mapping f ~ F f extends uniquely to an isometry Y : L2(G)

~

~ L2(GV).

The inverse transformation F -1 is given by f ~ ( g ) ( x ) = fGv g(s) ~ - / ~ ds. The measure ds on G v which produces the isometry is unique, and is called the measure associated to dx. When G is compact, we often choose dx so that the total mass of G is 1. In this case, the corresponding associated measure ds on the discrete group G v assigns mass 1 to singletons. It is an open problem to formulate and prove a general duality theorem for noncommutative locally compact groups such as Lie groups. Even the case of simple algebraic groups is not well understood despite the enormous accumulating knowledge on their irreducible representations. However, when the group is compact, there is a good duality theory due to H. Peter, H. Weyl and T. Tannaka. In this case, the dual object G v is discrete and so belongs to the realm of algebra. In order to describe this theory, it will be convenient to introduce some of the needed concepts in a more general setting.

416

Let M be a topological monoid. A (finite) representation of M consists of a finited i m e n s i o n a l complex vector space V together with a continuous h o m o m o r p h i s m ~v : M ~ End(V) into the m o n o i d End(V) of linear e n d o m o r p h i s m s of V. Real representations are d e f i n e d u s i n g real vector spaces. W e denote b y ~J~ep(M,C) [respectively, ~¢p(M,R)] the category of all complex [respectively, real] finite representations of M. There is a forgetful functor U : Rep(M, C) where

q~ctc

~ q&ctc

d e n o t e s the category of all complex vector spaces. Recall that a natural

transformation u : U---~ U is a family of m a p s u v : V - - * V i n d e x e d b y V~ Rep(M, C) such that the square UV

V

~V

W

>W UW

c o m m u t e s for all m o r p h i s m s h : V

~ W of representations ( " i n t e r t w i n i n g operators").

Clearly, each element x e M produces such a natural transformation n(x) : U

* U whose

V c o m p o n e n t is the element ~v(X) : V----* V. There is a t o p o l o g y on the set End(U) of natural transformations from U to U. It is the coarsest t o p o l o g y r e n d e r i n g all the projections

u ~

C o m p o s i t i o n a n d a d d i t i o n of n a t u r a l t r a n s f o r m a t i o n s t u r n s algebra. There is a conjugation operation End(U) * End(U), given b y ~v(X)

=

u v e End(V) End(U)

continuous.

into a topological

ub---~ ~ ,

uv(~)

w h e r e V denotes the conjugate representation of V (the elements of the vector space ~¢ are the s a m e as those of V b u t the identity m a p V .....~ V, d e n o t e d x ~ ,~, is an antilinear i s o m o r p h i s m ) . W e w o u l d like to characterize the natural transformations of the form ~(x). Recall that a natural transformation u is tensor preserving (or " m o n o i d a l " ) w h e n w e have Uv®W = u V®u w

and

u I = 11

w h e r e I denotes the trivial 1-dimensional representation of M. W e say that u is selfconjugate w h e n u -- •. The set of tensor p r e s e r v i n g self-conjugate transformations is a closed subset of End(U)

which is also closed u n d e r composition. We call this subset the

Tannaka monoid of M a n d denote it b y 'T(M). For a n y x ~ M, the n a t u r a l transf o r m a t i o n ~(x)~ End(U) belongs to '/'(M). We h a v e a c o n t i n u o u s h o m o m o r p h i s m : M . . . . * OF(M).

417

Proposition 3. T(M) is a topological group if M is. Proof. S u p p o s e that M is a topological group. Then any representation ~v : M } GL(V) has a dual, or "contragredient", representation :¢vv : M * GL(V v) where W

is the d u a l vector space of V and

~ w (x) = tXv(X)-'. W e have a m o r p h i s m of representations ¢ : V v ®V

.~ I = C

defined b y e(s ® x) =

(s, x). F r o m the n a t u r a l i t y of u~ T(M), w e have the equality £ UW® v

=

U I E,

Using the equalities Uvv®v = u w ® u v

and

UI =

11 ,

w e see that This shows that

< u w ( s ) , Uv(X) ) = <s, x>. Uvv is the contragredient transformation of U W o tu v =

Iw

Uv,

t

a n d implies that u v is invertible.qe d When

M = G is a compact group, the algebra End(U)

has a p a r t i c u l a r l y simple

structure. Let G v be the set of i s o m o r p h i s m classes of irreducible representations of G. Let us choose a representation ( V~, x~) in each such class ~. For any u~ End(1,L), let us write u~. for the V~. c o m p o n e n t of u, a n d p u t q(u) = ( u~ I k~ G v ).

Proposition 4. The map q : End(U)

)

1-I End(V~)

is an isomorphism of topological algebras. Proof. U s i n g naturality, w e have the c o m m u t a t i v e d i a g r a m UV

~V

V

1

V~W

U

• V~W VOW

W U

W

418

for any V, W e R e p ( G , C ) , which shows that UV@w

= Uv • Uw .

This i m p l i e s that u is entirely d e t e r m i n e d b y q(u) since any r e p r e s e n t a t i o n of G d e c o m p o s e s as a direct s u m of irreducibles. To p r o v e that q is surjective, let t = ( t~. I k~ G v) be a family of elements t~. End(V~). W e w a n t to construct an element u ~ End(U) so that u~ = t;~ for every ;t~ G v . Let S ~ e p ( G , C ) . components

There is a u n i q u e d e c o m p o s i t i o n of S as a direct s u m of isotypical S=

Y,S~. ~.~G

v

M o r e o v e r , for each ~ G v, the canonical m a p xg~. : V~ ® H o m G ( V ~ , S;~) is an i s o m o r p h i s m of G-modules. We p u t

> S~

Us = ~g;~o (t~ ® 1) o~g~-I and US =

~ vUsx • ;~ G

It is not difficult to verify that the family u = (Us I S~ Kep(G,C)) defines a n a t u r a l t r a n s f o r m a t i o n

u : U ----* U

h a v i n g the r e q u i r e d property. The

continuity of qq is a consequence of the fact that the topology on E n d ( U ) is the coarsest r e n d e r i n g continuous all the projections u ~-* u~. , ~,e GV.qed R e m a r k . The proof of Proposition 4 is based on the fact that the category ~ p ( G , C ) closure u n d e r direct sums of its subcategory of irreducible representations.

is the

Proposition 5. '/'(G) is compact if G is a compact group. Proof. A n y r e p r e s e n t a t i o n V of a c o m p a c t g r o u p a d m i t s a p o s i t i v e definite i n v a r i a n t hermitian form g : V®V---~ C. We can view g as a C-linear p a i r i n g h : V¢®V

~ C,

h(~, y) = g(x, y).

For any u~ End(U), w e have hour® v =u I oh. But, if u is tensor preserving a n d self conjugate, w e have U ~ ® V = U-~

® UV,

U I= 1,

uv(~)= Uv(X).

This m e a n s that h(uv(X),Uv(Y)) = h(~, y) ; that is, g(uv(x), uv(y)) = g(x, y), which m e a n s that u v belongs to the unitary group U(V, g). It follows from this that '/'(G)

419

is a d o s e d subgroup of a product of compact groups.qe a To better understand the structure of involution

'iV(G) we need to take account of the * -

( )* : End(U) ,~ End(U). Its definition can be given in terms of two other involutions: u*

=

(~)v

=

( u v)

where, by definition, ~ is the conjugation operation considered earlier and (uV) v

=

(u

V9 ~

.

A better description, valid only for compact groups, is to say that it transports across the isomorphism q of Proposition 4 to the canonical C*-algebra structure on each End(V x) for %eG v : the adjoint h* of an element heEnd(Vx) is defined by the equation g(h*(x), y) = g(x, h(y)) and does not depend on the choice of g since g is unique up to a scalar multiple (by S c h u r ' s Lemma applied to the irreducible representation Vx). Remark. The algebra End(Vk) does not depend on the choice of VX in the class k. For, it follows from S c h u r ' s Lemma that the algebras End(V 1) and End(V 2) are canonically isomorphic for any two isomorphic irreducible representations V 1, V 2 . An element u e E n d ( U ) is unitary if u*u = uu* = 1. The group of unitary elements is isomorphic to the product I-I U(d

KE Gv

X)

where U(dx) c End(V x) is a unitary group of dimension d~. = dim V k . We have proved that 'T(G) is a d o s e d subgroup of this product. The theory of Fourier transforms on compact groups can n o w be described. The Fourier transform of a continuous map f : G ---* C is an element 5rf e End(U). It is defined by the integral (Ff) V = fGf(X) Xv(X) dx

It is easy to see that 5r(f* g) = (yf) (5rg), where f * g is the convolution of f and g given by (f* g)(x) = Scf(x y-l) g(y) dy Also, we have Yf* = (Ff)* where f* is defined by f*(x) = f-(x -I). It follows from Proposition 4 that irreducible representations

yf

is entirely d e t e r m i n e d by its effect on

(yf)(~.) = S f(x) x k ( x ) d x -G

420

so that 9- defines a m a p 9-: C(G, C)

) H

End(V;~)

k~ G ~

on the algebra C(G, C ) of complex-valued continuous functions on G. We would like to describe an inverse to 9-. It should be noted that 9- cannot be surjective because we have }l 9-f(x)l[ -< t l f L where the norm on the C*-algebra End(V x) is used on the left hand side. This shows that y lands in the set of elements of bounded norm

, u II-- sup

k~ G ~

llu

tt < +-

This set forms a genuine C*-algebra

1-I~"End(V~)

c

l"ifind(V~) .

k~ G ~

KEG

W e begin by describing a partial inverse 9-4 to S: 9--1 : y , End(V~)

) C(G,C)

keG

(9--lg)(x)=

~

T r ( g ( k ) ~ ( x ) -1) d ~ .

k~ G ~

But, to prove anything about this inverse, we need the orthogonality relations. Lemma 6.

If

~, 8 ~ G v and

A e H o m ( V ~ , V;~) then

~G ~ x ( x ) A ~ ( x ) - l d x

=

I~h)id if ~.=~ 0 otherwise.

ProoL This is a classical lemma, but we give the proof for completeness. The left hand side is an operator obtained by averaging the function x ~ n~(x) A ~8(x) q over the group. It is therefore an invariant m a p from V8 to V~. By Schur's lemma, it must be equal to 0 if k # 6 and a scalar multiple of the identity if k = & To check the equality, we just need to take the trace on both sides.qe d L e m m a 7 . If k, 8 ~ G v and v ~ V ~ , w ~ V 8 then [ w®__~v S G ~ k ( x ) ( v ) ® x s ( x ) - l ( w ) dx

= I O~

if

o the r w i s e .

Proof. It is enough to prove that, for any linear form (~ : V~ [we(v) SG ~ x ( x ) ( v ) * ( ~ ( x ) - l ( w ) ) d x

=

l

0~

k=8

if

} C , we have ~,=5

otherwise .

This equality is a special case of L e m m a 6 applied to the operator A given by A(y) =

¢(y) v .q,d

421

Proposition 8 (Orthogonality Relations). If k, 8 E G v and AE End(V~), Be End(Vs) then

[ Tr(AB---'--'~) if ) , = 8 (i) ~GTr(A n)(x)) Tr(B ha(x-l)) dx

=

I

O~

otherwise .

°

Tr(AB )

(ii) JcTr(A n~(x))Tr(B ns(x)) dx

0

if

K=8

otherwise .

Proof. It suffices to verify relation (i) when both A and B are of rank 1, since the identity is bilinear in A, B and the rank 1 operators span all the operators. Put A(x) = ~(x) v and B(x) = ~/(x) v. The result follows from application of the linear form ~ ® ~ : V~. ® V s ~ C to the equality in Lemma 7. The second relation follows from the first and Tr(B~8(x))

=

Tr(B*ns(x)*)

=

Tr(B*n~ (x)-l) "qed

CoroUaryg. If )`, B e G v and A~End(V)) then the following identity holds: dk

~G {A Tr(An;~(x)) ns(x -1) d x = 0

fork=8 otherwise

.

Proof. Since the pairing (A,B) = Tr (AB) is exact, it suffices to check the equation after applying Tr (B - ) on both sides. But then the orthogonality relations give the result.qe a On C(G, C) we introduce the inner product ( f, g ) = Sc:.~(x)g(x)dx • Note that we have (f,g) = eft**g) where e : C(G, C)

~ C is the functional "evaluate at e~ G" : eft) = f(e).

On ~ ~.EGVEnd(V). ) we introduce the inner product (g,h)=

~ Tr(g*()`)h()`))d~. k~G v

Proposition 10. The following equalities hold: (a) YYq g = g, Co) y-1 (g h) = ( y-1 g ) , ( 9:4 h ) ,

(c) 9:-1 (g*) = ( y 4 g ) * , = (g,h). (d) ( y - l g , y - l h ) Proof. We have (F(y'-lg))(8) =

~ ).eG v

SC d x Tr(g()`)nx(x) -1) ns(x) d x .

422

But, by Corollary 9, we have d~,fcTr(g(k)n;~(x-1))n~(x)dx=

{g(0 k)

for k = 6 otherwise

.

The equality 9" y q g = g follows. Similarly for (b). The equality (c) is trivial. To prove the last, we note that ~(y--lg)=

~ Tr(g(k))dx ' ~G ~

and therefore

(F-lg,F-lh)

=

¢((F'lg)**(F-lh))

= =

¢ F -1 ( g* h) ~ T r ( g * (k) h(k)) d~. ~EG v

=

( g , h >.qed

The domain of 9"-1 can be completed by using the norm HUH 2

=

(u,u).

Its completion is a Hilbert space isomorphic to the Hilbert s u m ~h~'b'"End (V~) keG

where End(V~ ) is given the Hilbert space metric defined by HA112 =

d~Tr(A*A).

The continuous extension of y--1 defines an isometric embedding y:-i

Definition. Let

M

:

Eh~lb,-End (Vx) K~G

) L2(G).

be a topological monoid. A function

representative of (V, ~v)e ~ p ( M , C )

f : M

, C

is said to be

when there is a linear form : End(V)

,C

such that f = ¢ o n v. Equivalently, f is representative of (V, n v) w h e n there is some

A ~ End(V)

such

that f(x) = Tr(A ~v(X)) for every x ~ M. This means that f is a linear combination of the coefficients nij(x) of n v in some basis of V. W e denote by R(V, nv), or R(V), the set of functions that are representative of (V, nv). Let R(M) be the set of functions on M

which

are representative of some representation (V, Kv).

Proposition 11. R(M) is a subalgebra of C(M, C) which is closed under conjugation f~--* -f. Proof. The result follows from the easy observations below. R ( V I ( ~ V 2) = R(V1)+R(V2), R ( V I ® V 2 ) c R(VI)R(V2) R(V) = R(V) R( I ) = { constant functions } qed

423

Theorem 12 (Peter-Weyl). For any compact group G and any x~ G, x ~ e, there exists a

finite dimensional representation (V,~v) such that

~v(x) ~ i d .

Corollary 13. On a compact group any continuous function approximated by representative functions.

can be uniformly

Proof. Let x, y ~ G, x ~ y. Theorem 12 says that there exists a representation gv such that gv(X y ' D ~ id; that is, gv(X) ~ r~v(y). But there exists a linear form ~ : End(V) that

~ C such

(nv(X)) * ~ (~v(Y)). This shows that the s u b a l g e b r a R(G) separates points; a n d since R(G) is closed u n d e r conjugation, the result follows from the Stone-Weierstrass theorem.3e a T h e o r e m 14 (Plancherel t h e o r e m for compact groups). The Fourier transform F can be

extended continuously to an isometry F : L2(G)

)

~h~'~'"End(V~) KeG

Proof. W e have a l r e a d y defined an isometric e m b e d d i n g F -1

:

Y~h~lb'"End (V~) keG

) L2(G).

The subspace Ira( F -1 ) is closed since it is complete. But it contains the dense subspace R(G); so Ira( F -1 ) = L2(G), a n d the theorem is proved.qe d The g r o u p G acts on the left a n d right of L2(G) via the equalities: (x-f)(y)

= f(x-ly),

(f-x)(y)

= f(yx-1).

Also, G acts on both sides of each of the Hilbert spaces End(V~. ): x • A = ~;~(x) A,

A • x = A ~(x).

The Fourier transform respects these actions: F(x.f)

= x.(Ff),

F(f-x)

= (Ff) °x.

A function f ~ C(G, C) is a class function if it is constant on conjugacy classes of G; or equivalently, if it is invariant u n d e r conjugation: f = x. f.x-1. The class functions are exactly the m e m b e r s of the centre of the algebra C(G, C) with respect to the convolution product. P r o p o s i t i o n 15. An element u e E n d ( U ) is central if and only if, for every the linear map Uv: V ~ V is a G-homomorphism.

VeRep(G,C),

Proof. If u is central then, for every x e G, w e have u ~(x) = ~(x) u, a n d so

UvXv(x) = ~v(x) Uv for every V. Conversely, if u v is a G - h o m o m o r p h i s m then, for e v e r y h a v e the c o m m u t a t i v e s q u a r e

w e End(U), w e

424

uV

V w

~V

v V

lwv

~---V uV

showing that u w = wu; so u is central.qe a W e w o u l d like to restrict the Fourier transform to its central part. If f e C(G, C) is a class function then x • f = f ° x , and therefore x(x) ( F f ) = ( F f ) K(x), which shows that F f is central. For any K e G v, the m a p ( F f ) ( ; t ) :Vx .... , V x is a G - h o m o m o r p h i s m . A c c o r d i n g to S c h u r ' s lemma, it m u s t be a scalar multiple of the identity. Denoting this scalar b y (Tf)(K), w e have

,

lie

(Tf)(K) = - ~ Tr(Tf)0~) = ~

f(x) Z~(×)dx

where Z~(x) = Tr(~ ~(x)) is the character of the irreducible representation V~. F r o m the relation ( F f ) ( K ) = (Tf)(~) id ~, w e d e d u c e the relations T(f * g) = (Tf) ( T g ) , T(f*) = T f . The centre of each algebra End(V~) is C • id }. A c c o r d i n g to Propositions 4 and 15, the centre of the algebra End(U) is equal to the p r o d u c t rI

c-id

x

~

c Gv.

~, E G v

W e define the inverse Fourier transform of any function g : G v (T-lg)(x) =

,~ C of finite s u p p o r t by

~ g ( k ) Z ~ ( x ) d~.. ~,~ GV

It is a consequence of Proposition 10 that: (a) (b)

T T -1 g = g , T -1 (g h) = ( T -1 g ) * (T -1 h ) ,

(c)

T -1 ( g * ) = (T -1 g ) * ,

(d) ( T q g , T - l h ) = (g,h). In this last equality, the inner p r o d u c t on the right h a n d side is defined b y (g,h)=

~ ~.~G

The spectral measure

~(~,)h(K)d 2 . v

d on G v is the one assigning w e i g h t d 2

The Hilbert space L2(G v) is the space of spectral measure. We write C(G) for the space of G acting on itself b y conjugation. b y taking the i m a g e of the H a a r m e a s u r e

to the singletons { ~, }.

square s u m m a b l e functions with respect to the space of conjugacy classes of G. It is the orbit There is a canonical m e a s u r e o n C(G) obtained along the projection G ~ C(G). Obviously,

425

L2(C(G)) is isomorphic to the subspace of L2(G) consisting of square integrable class functions. 16. The Fourier transforms mutually inverse isometrics

Theorem

T

and

L2(C(G)) ,

~

Td

have continuous extensions to

, L2(G v ) .

ProoL This is just a description of the restriction of the Fourier transforms the central parts.qe d

F and F q to

A collection X of representations of G is closed when it contains (i) xl if Xl is isomorphic to some x2 a X , (ii) n 1 if x I is a subrepresentation of n 2 a X , (iii) Xlf~X 2 if Xl,n2E X , (iv) x l ® n 2

if n 1 , n 2 e x ,

(v) nl if n # X , (vi) I. The set of representative functions of the members of X is a subalgebra R(X) of the algebra R(G). L e m m a 17. Let X be a closed collection of representations of G. Suppose that, for every xeG, x ~ e , there exists a representation n v e X such that Xv(x)~e. Then X = P~p(G,C).

Proof.

If X ~ ~ F ( G , C ) ,

there will be an element

Xe G v

such that

x)~ X.

The

orthogonality relations then imply that the representative functions of x x are orthogonal to the elements of R(X). In particular, X~. is orthogonal to R(X). On the other hand, the hypothesis implies that the subalgebra R(X) separates the points of G and is d o s e d under conjugation. According to the Stone-Weierstrass theorem, R(X) is dense in C(G, C). This is a contradiction.qe d W e would like to prove that, for any compact group G, the map n : G

~ T(G)

induces, via restriction, an isomorphism of algebras n : R(T(G)) W e shall obtain this from the next result. L e m m a 18.

~ R(G).

The restriction functor : Rep(T(G), C)

~ Rep(G, C)

is an equivalence of categories. Proof. W e define an extension functor

e : Rep(G, C)

~ Rep(T(G), C)

as follows. For any V~ ~ p ( G , C), the m a p u ~-~ u v is a representation of T(G) on V. The commutative triangle

426

G

~V

\

/

~ GL(V)

T(G) showsthat

~¢*(pv)=~v. W e p u t e ( V , ~ v ) = ( V , pv). For a n y m o r p h i s m

h : ( V , n v)

* (W,

~¢w) in Rep(G, C) a n d any u~ T(G), w e have ho u v = u w o h . This shows that h is also a m o r p h i s m h : (V, Pv) " ~ (W, Pw) in Rep(T(G), C). It is easy to see that e preserves direct sums, tensor p r o d u c t s a n d conjugate representations. Moreover, f r o m the s a m e c o m m u t a t i v e triangle (above), if (V, ~v) is irreducible then e(V, nv) is irreducible. F r o m this it follows that the image of the functor e is a full subcategory w h o s e objects constitute a closed collection X of representations of T(G). We prove that X separates the elements of T(G). Let u ~ T(G), u # e. There exists a representation V of G such that u v # id v . This m e a n s t h a t pv(u) ¢ id v a n d thus Pv separates u from e. By L e m m a 17, the collection X is all of Rep(T(G),C), and so e is an equivalence of categories. Finally, n* is an equivalence since ~ o e = id "qed L e m m a 19. The restriction map : R(T(G))

~ R(G)

is an isomorphism of algebras. Proof. W e define an inverse e : R(G) has a u n i q u e representation in the form f(x)= ~

~ R(T(G)) in the following manner. A n y f~ R(G) Tr(g(k):~(x))

dx

k~ Gv

where g e ~zaGv End(V~). ( I n fact, g = F f .) We define e(f)(u) =

~

Tr(g(K) p ~ ( u ) ) d;~ .

A c c o r d i n g to L e m m a 18, w e have G v = T(G) v from which the bijectivity of e follows. Finally, ~* is bijective since ~*o e = id "qed Theorem 20 (Tannaka-Krein). For any compact group G, the canonical map : G ~ T(G)

is an isomorphism. Proof. W e first p r o v e the injectivity of ~ . According to Peter-Weyl (Theorem 12), for any x~ G, x # e , there is a representation (V, ~v) such that ~v(x) ~eid. But then ~ (x) # e since pv(~(x)) = 7¢v. To p r o v e the surjectivity of ~ , w e first p r o v e (*)

rE(G) f(u) d u

=

IGf(~(x))dx

for a n y f ~ c(T(G), C ). It is e n o u g h to p r o v e this e q u a l i t y w h e n

f ~ R(T(G)) since

427

R(T(G)) is dense in c(T(G), C) by Corollary 13. Using Lemma 19, we see that a function f R(T(G)) has a unique representation f(u) = ~ Tr(g(K) p;~(u)) dK, KEGv so that f(n(x))=

~ Tr(g(X) nx(x))d;~ EGv

But the orthogonality relations give IGTr(g(K)n;~(u)) du = {g(0I)

for K = I otherwise .

Similarly, f

T r ( g ( k ) p x ( u ) ) d u = {g(0I) for k = I T(G) otherwise .

Equality (*) follows. To prove the surjectivity, let us suppose that Im(~) ~ T(G). Let f be a positive function whose support is contained in the complement of the closed subset Im(~). Then we have ST(G) f ( u ) d u > 0

and

Scf(/~(x))dx=0

which is a contradiction.qea §2. The bialgebra of representative functions. This Section provides a more complete description of the algebra of representative functions on a compact group and, more generally, on a topological monoid. We shall see that R(M) is a biatgebra, and we shall compute it in some special cases. We begin by giving a simple characterization of representative functions on locally compact monoids. Let M be a locally compact monoid. For each y~ M, consider the translation operators Ky(x)= yx a n d py(X) = xy. We enrich C(M, C) with the topology of uniform convergence on compact subsets of M. The maps (y,f) ~ f o p y and (f,y) ~ fO~y define continuous left and right actions of M on C(M, C). Proposition 1. A continuous complex-valued function f on a locally compact monoid M is representative if and only if the linear subspace of C(M, C) spanned by any one of the

three subsets {foky l y~M },

{fopy l y~M },

{ f O~y Opzl y,z~M }

is finite dimensional. ProoL Let ~v : M ...... * End(V) be a representation of M. The set R(V,~v) of representative functions for ~v is closed under left and right translation; for, if f(x) = Tr(A ~v(X)), we have f o ky o pz(X) = Tr(A xv(yxz)) = Tr(xv(z) A rCv(y) nv(X)). Conversely, let us assume, for example, that { f o py Iye M } generates a finite dimensional vector subspace V. The continuous left action of M on C(M, C) restricts to a continuous

428 left action of

M

on

V.

W e o b t a i n in this w a y a c o n t i n u o u s r e p r e s e n t a t i o n

Xv : M

---4 End(V) of M. Let e: V----~ C be the linear form e ( h ) = h(e) w h e r e e~ M is the unit. W e have fix) = ¢(fopx) = e(Xv(x)(f)) = Tr(A Xv(X)) w h e r e A~ End(V) is the rank one linear function h ~

e(h) f .qed

For any topological m o n o i d M, there is a bialgebra structure on R(M) which w e n o w describe. Let A : C ( M , C ) - - - ~ C ( M x M , C) be the algebra h o m o m o r p h i s m defined b y (Af)(x, y) = f(xy). W e h a v e canonical inclusions R(M) ® R(M) c C(M, C) ® C(M, C) c C ( M x M, C). L e m m a 2. For all f~ R(M), Af e R(M) ® R(M). Proof. It is e n o u g h to p r o v e the result w h e n

f(x)

=

/rij(X) w h e r e u

= 0rij)

is some

representation of M. In this case w e have ij (xy) = Y'k ~ ik (x) ~ kj (Y) ; that is, ,,k~ ij = Y'k 7r ik ®/~ kj "qed Hence

A gives a comultiplication for R(M). The counit

8 : R(M)

*C

of the

coalgebra is given b y ~(0 = f(e). W h e n the m o n o i d M is a topological group, the b i a l g e b r a R(M) becomes a H o p f algebra. The antipode v : R(M) ~ R(M) is defined b y v(f)(x) = f(x-1). TO see that v(f)~ R(M), s u p p o s e that f(x) = Tr(~ (x)A). Then w e have f(x-1) = Tr(~ (x-1)A) = Tr(tA t~ (x-1)) = Tr(rcv (x) tA) w h e r e nv (x) = tn (x-l) is the contragredient representation of n . The a l g e b r a R(G) is easy to describe w h e n G is a c o m p a c t abelian group. In this case, R(G) is the linear subspace of C(G, C) s p a n n e d b y the set G v of characters of G. H o w e v e r , the o r t h o g o n a l i t y relations i m p l y that G v is a linearly i n d e p e n d e n t subset of C(G, C). This s h o w s that R(G) is i s o m o r p h i c to the e n v e l o p i n g a l g e b r a C[G v] (usually called the g r o u p algebra) of the dual g r o u p G v. For any X~ G v, the relation

X(xY) = X(x) x(y) implies the relation aZ = Z®X. T h i s shows that the coalgebra structure

extension of the d i a g o n a l m a p

A :

extension of t h e i n v e r s e o p e r a t i o n

A : C [ G V ] - - ~ C[G v] @ C[G v] is s i m p l y the linear G v - - ~ G v x G v. The a n t i p o d e

v

is the linear

Gv ~

G v . It is also w o r t h w h i l e to identify the

operation c o r r e s p o n d i n g to conjugation f ~

f- in R(G): it is the antilinear extension of

429

the inverse operation. If G is the circle group T then R(G) is the ring of finite Fourier series: R(T) = C [ Z ] = C [ z , z - ~ ] where z = ed0 and ~ = z-1. When G is not abelian, the following result is useful in identifying R(G). For any representation ~v of G, let det(~ v) be the one-dimensional representation obtained as the composite of nv : G ~

GL(V) and det : GL(V)

~ GL(1).

Proposition 3. Let G be a compact group, and let nv be a faithful representation of The algebra R(G) is generated by the coefficients of nv together with det(nv)q.

G.

Proof. Let A be the subalgebra of R(G) generated by the coefficients of ~v and det(~v)-L Let us verify first that A is closed under conjugation. But we have det(Uvi-1 = d e t ( ~ v ) , and also R(~ v) = R ( ~ )

= R(~W)-

It is therefore sufficient to verify that the entries of the contragredient representation are contained in A. This is true because of the familiar formula expressing the entries of an inverse matrix as cofactors divided by the determinant. To finish the proof, let X be the collection of representations whose coefficients belong to A. Then X is a d o s e d collection, and, since gv is faithful, w e can apply Lemma 16. This gives X = T~cp(G,C) s o t h a t A = R(G).qe d W e would like to apply Proposition 3 to compact Lie groups, but to do this we need the following result.

Proposition 4. Each compact Lie group G admits a faithful representation. Proof. W e shall use the fact that the poset of closed submanifolds of a given compact manifold is artinian. This means that any non-empty collection of submanifolds contains a minimal element; or, equivalently, that any decreasing sequence of submanifolds must stop. For any Ve ~,p(G, C), the kernel of ~v is a closed submanifold of G. Let ker(~ w) be a minimal element in this collection. According to the Peter-Weyl Theorem, for any x ¢ e, there exists Ve Rep(G, C) such that x ~ ker(~v). But we have ker(:Cv @ ~w) = ker(nw) n ker(Tcv), so that ker(;~ w) c_ ker(nv), and so x~ker(nw). This proves that ker(Kw) = {e}.qea The last two Propositions have the following immediate consequence.

Corollary 5. The algebra of representative functions on a compact Lie group is finitely generated. W e shall n o w give a complete description of R(G) for the compact Lie group G = U(n) of unitary n x n complex matrices. Let C [(Z i j)] be the ring of polynomials in the n 2 indeterminates

z i j ( 1 < i, j < n ) and let d = det(z i j). If we map z i j into the coefficient Pi j

of the standard representation

430

p : U(n) ¢-

* End(cnxn),

w e obtain a h o m o m o r p h i s m t : C[(zij),d "1] sending

, R(U(n))

d -1 to det(p) -1.

Proposition 6. The homomorphism t is an algebra isomorphism, so that R(U(n)) -= C[(zij),d-1]. Proof. P r o p o s i t i o n 3 implies that t is surjective. To s h o w that t is injective, let us first r e m a r k that, for any P(z)e C[(zij) , d -1 ], the function t(P(z)) on U(n) is exactly the function u t---* P(u). Therefore, if t(P(z)) = 0 then P(u) = 0 for every u e U(n). We have to s h o w that this implies that P = 0. This p r o p e r t y is sometimes formulated b y saying that U(n) is Zariski dense in G L ( n , C ) ( it is the famous "unitarian trick" of H e r m a n n Weyl which he u s e d to r e d u c e the theory of rational r e p r e s e n t a t i o n s of GL(n, C) to the easier case of continuous representations of U(n) ). A l t h o u g h this is v e r y classical, w e give a brief proof. The Cauchy transformation Z = C(A) of a matrix A~ End(C n) is defined b y the formula Z = (I+A)(I-A) q. Its inverse is given b y A = ( Z - I ) ( Z + I ) -1. The Cauchy transformation is a birational correspondence which is well defined between a n e i g h b o u r h o o d of A = 0 a n d a n e i g h b o u r h o o d of Z = I in C nxn W h e n A is antiHermitian, Z is unitary, a n d conversely. As well as the C a u c h y transformation, w e shall use the invertible linear transformation A = L(B) defined b y A

=

(B-tB)+i(B+tB),

B

-

A - tA A + tA + 2 2i

W h e n B is real, A is anti-Hermitian, and conversely. By hypothesis, the function Q(B) = P(C(L(B))) is a r a t i o n a l f u n c t i o n of B w h i c h v a n i s h e s i d e n t i c a l l y w h e n B is real in s o m e n e i g h b o u r h o o d of 0. This d e a r l y implies that Q = 0 identically; a n d therefore P = 0 identically since P(Z) = QCLq ( C ' l ( Z ) ) ) . q e d The H o p f algebra structure on C [(zi j), d -1] which corresponds to the one on R(U(n)) is easy to identify. The comultiplication is given b y •zij

= ~kZik@Zkj

a n d the counit b y Ezij = ~ij (the Kronecker d e l t a ) . The a n t i p o d e is given b y VZ = Z"1 where z = ( z i j ) a n d v z = ( v z i j ) . With this notation, w e h a v e Az

= z@z

where Az = (Azij) and z® z (•kZik®Zkj)W e s h o u l d also identify the conjugation operation (-) on C[(zij), d -1 ] which =

431 corresponds to the conjugation f ~ f- on R(U(n)). It is easy to see that unique antilinear ring h o m o m o r p h i s m such that

(-)

is the

Z = tz'l

where 2 = ( ~ij ). For any commutative C-algebra A , we define its spectrum Spec(A) as the set of all algebra h o m o m o r p h i s m s

X:A

~ C.

We give Spec(A)

the topology of pointwise

convergence. When A is finitely generated, say A = C [aI . . . . . an], the m a p p i n g

X

(X(al) . . . . , Z(an) ) is a h o m o m o r p h i s m between Spec(A) and a closed algebraic subset of C n . W h e n A is enriched with an antilinear involution a ~ ~ , we can define a conjugation operation on Spec(A) by putting, for all a~ A, x(a) = X(~) The real spectrum SpecR(A) of A is the set of x~Spec(A) such that X =X • The spectrum of a finitely generated commutative H o p f algebra is a complex algebraic group. If the H o p f algebra is enriched with an antilinear involution (respecting the H o p f algebra structure) then its real spectrum is a real algebraic group. In the example above we see that the spectrum of R(U(n)) is the algebraic group GL(n,C) and its real spectrum is U(n). This is a special case of a more general result. For any compact group G, consider the canonical m a p G ~ SpecR(R(G)) sending x ~ G into the h o m o m o r p h i s m Xx given by Xx(f) = f(x). T h e o r e m 7. For any compact group G , there is a canonical homeomorphism

G

~

~ SpecR(R(G)) .

The proof of the above Theorem will be delayed until the next Section. It follows from Tannaka reconstruction. We note immediately this striking consequence: Corollary 8. Every compact Lie group is (real) algebraic. Proof. If G is a compact Lie group then, according to Corollary 5, the Hopf algebra R(G) is finitely generated. So Spec(R(G)) is therefore a complex algebraic group whose real part is equal to G.qe d

To end this Section, we prove the following result needed in Section 8. Let M and M' be topological monoids. We have an obvious canonical map i : R(M) ® R(M') ~ R(M x M'). P r o p o s i t i o n 9. The canonical map

i (above) is an isomorphism.

Proof. Clearly i is injective. We prove it surjective. It suffices to prove that the matrix entries ~rs of any representation ~v of M x M' belongs to the image of i. We have :Ors(x,y) = Z t ~rt (x,e)~ts (e,y).

This shows that ~rs = i (Z t ~lrt® :c2s) where ~lv(X) = ~v (x,e) and K2v (y) = ~v(e,y).qed

432

§3. The Fourier cotransform. In this Section w e shall describe a transformation which has the Fourier transform as its dual. It is m o r e f u n d a m e n t a l than the Fourier transform in the sense that it behaves better algebraically. Starting with a category C a n d a functor X : C ~ q/ectc w i t h values in finite d i m e n s i o n a l vector spaces, w e shall construct a certain vector s p a c e EndV(X) whose d u a l is the algebra End(X) = Hom(X, X) of natural transformations from X to X. W h e n C = R e p ( M , C ) a n d X = 1~ is the forgetful functor, the Fourier cotransform defines an i s o m o r p h i s m ~-v : End" (U) ~ ~ R(M) w h o s e t r a n s p o s e is the (generalised) Fourier transform. In Section 4 w e shall s h o w that End"(U) s u p p o r t s a n a t u r a l coalgebra structure a n d that coalgebras.

yv

is an i s o m o r p h i s m of

Let C be a small category. For any pair of functors X, Y : C ~ q~ctc, let Horn(X, Y) be the set of n a t u r a l transformations between X a n d Y . N o w Hom(X, Y) is a vector space which can be described as follows. For any arrow f : A ~ B in C, let pf , q f :

1-[Hom(X(C),Y(C))

) Hom(X(A),Y(B))

CEC

be the m a p s o u t of the p r o d u c t whose values at u -- ( u c I C~ C ) are given b y pf(u) = Y(f) u A , qf(u) = u B X(f) . Then clearly, Horn(X, Y) is the ~ommon equalizer of the pairs (pf, qf) : u ~ Horn(X, Y)

~

for all f ~ C, pf (u) = qf (u).

This construction can b e d u a l i s e d w h e n X(C) a n d Y(C) are finite d i m e n s i o n a l for every C ~ C (actually w e just n e e d Y(C) to be finite dimensional for e v e r y C ~ C , b u t here w e shall not consider this m o r e general situation). M o r e precisely, w e w a n t to construct a vector space Horn v (X, Y) w h o s e d u a l will be t h i n k of H o r n " (X,Y) as a predual of c o m m o n coequalizer of the m a p s

Horn(X, Y) (it is therefore suggestive to

Horn (X, Y) ). By definition, Horn" (X, Y) is the

tpf , tqf : H o m ( X ( A ) , Y(B))*

)

~[ H o m ( X ( C ) , Y(C))* CGC

into the direct sum. But, for a n y p a i r of finite d i m e n s i o n a l vector spaces V, W, w e have canonical i s o m o r p h i s m s Hom(V, W)* = (V*®W)* _= W * @ V -= H o m ( W , V ) . The p a i r i n g b e t w e e n Hom(V, W) a n d Horn(W, V) is explicitly given b y (S, T) = Tr(ST) = Tr(TS). Taking this d u a l i t y into consideration, w e see that c o m m o n coequalizer of the m a p s if , jf : H o m ( Y ( B ) , X(A))

)

Horn v (X, Y) can be d e f i n e d as the

~ Hom(Y(C),X(C)) CGC

433

where, for a n y SE Hom(Y(B), X(A)), if(S) = (SoY(f),A),

jr(S) = ( X ( f ) o S , B ) .

In this d e s c r i p t i o n , the s e c o n d c o m p o n e n t of a p a i r

(SoY(f),A)

indicates to which

c o m p o n e n t of the direct s u m this element belongs. A h a n d i e r d e s c r i p t i o n of Horn v (X, Y) is the following. For a n y C ~ C and any S ~ Hom(Y(C), X(C)), let us write IS] for the i m a g e of S u n d e r the canonical m a p Hom(Y(C), X(C))

~ H o r n " (X, Y).

W e have that Horn v (X, Y) is generated as a vector space b y the symbols IS] subject to the relations (i) [czS+f~T] = ~[S]+[3[T] for S, T e H o m ( Y ( C ) , X ( C ) ) , (ii)

[SoY(f)] = [X(f)oS] for f : A - - ~

B and S~Hom(Y(B), X(A)).

It is clear by construction that Horn(X, Y) is the linear d u a l of H o m v (X, Y). The explicit p a i r i n g H o m (X, Y) ® H o m " (X, Y) ..... ,~ C ,

(u, [S]) = Tr(u C o S )

w h e r e S ~ Hom(Y(C), X(C)). Before continuing the general s t u d y of Horn v (X, Y), let us c o m p u t e Horn v (X, X) = Endv(X) in the special case w h e r e C = ~ p ( M , C) and X = 1,[ is the forgetful functor. We define the Fourier cotransform y-v : End"(U)

~ R(M),

as follows: S v (z) (x) --- (~ (x), z ) .

More explicitly, if z = [A] where A ~ End(V) and ~v : M

~ End(V) then

F v ([A]) (x) = (~ (x), [A]) = Tr(KV(X) A ) . Theorem 1. For any topological monoid M, the Fourier cotransform is an isomorphism y-v : End"(U)

~ ~ R(M).

To p r o v e Theorem 1, w e shall need two Lemmas. Lemma 2. If V 1 , V2~ Rep(M, C), and if A ~ End(V 1 • V 2) is written as A

=

( All ~A21

A12~ A22),

then [A] = [All] + [A22]. Proof. Let Ji : Vi ) Vl (~ V2 and Pi : V1 ~ V2"--+ Vi inclusions and projections. We have [A] = [ A ( j l P l + J2P2)] = [AjlPl] + [Aj2P2] = [Pl A Jl] + [P2 A J2] = [Alt1 + [A221.qe d Lemma 3. Any

( i = 1, 2 ) be the canonical

z ~ End"(U) can be represented as z = [ ~ ® v ] for some v ~ V and q~

434

V* w h e r e ~v : M ~ End(V). generates V as an M - m o d u l e .

Moreover,

this can be done in such a way that

Proof. Each e l e m e n t z ~ End v (11) can be r e p r e s e n t e d as z = [ A ]

v

for s o m e A ~ E n d ( W )

a n d W ~ Rep(M, C ) . But w e can express A as a s u m of m a t r i c e s of r a n k one: A =

~

(~i®vi ;

iEI

so that, z =

~

[(~i ® v i ] -

i~I

Let V b e the r e p r e s e n t a t i o n o b t a i n e d as the direct s u m of I copies of W, a n d let v =(viIi~

w h e r e Pi : V ~

I),

(~ = 2 ( ~ i P i ieI W is the i-th projection. U s i n g L e m m a 2, w e see that z : [¢®v].

To p r o v e the last s t a t e m e n t , let V ' be the s u b m o d u l e of V g e n e r a t e d b y v. T h e i n c l u s i o n j : V'----~ V is an a r r o w in the c a t e g o r y R e p ( M , C). Let B be t h e c o m p o s i t e V

¢ )C

v )V'.

Then we have [¢®v]

Proof of T h e o r e m 1.

= [jB] = [Bj] = [(¢lV')®vl.qe

T h e s u r j e c t i v i t y of

r e p r e s e n t a t i v e function. S u p p o s e that

Fv

d

is i m m e d i a t e

f r o m t h e d e f i n i t i o n of

F v(z) = 0. A c c o r d i n g to L e m m a 3, w e can s u p p o s e

that z = [ ¢ ® v ] a n d that v g e n e r a t e s the M - m o d u l e V. T h e n w e h a v e ¢(~v(X)(V))

=

F v(z)(x)

=

0

for e v e r y x ~ M. This s h o w s that the s u b m o d u l e g e n e r a t e d b y v is c o n t a i n e d in ker(~)). T h e r e f o r e (~ = 0 a n d z = 0.qe d A t this p o i n t it m i g h t b e w o r t h w h i l e to e x p l a i n w h y w e call F v t h e " F o u r i e r c o t r a n s f o r m ' . S u p p o s e G is a c o m p a c t g r o u p . The c o n t i n u o u s d u a l of t h e Banach s p a c e C(G, C ) is the space M ( G , C) of b o u n d e d m e a s u r e s o n G. The F o u r i e r t r a n s f o r m of ti M ( G , C) is the e l e m e n t FI~ ~ E n d ( U ) g i v e n by (Fp,) v

=

f ~ v ( X ) d~t(x) . G

T h e F o u r i e r t r a n s f o r m of a f u n c t i o n f ~ C(G, C) w h e r e d x d e n o t e s the H a a r m e a s u r e o n G.

is s i m p l y the F o u r i e r t r a n s f o r m of f dx

Proposition 4. If z ~ E n d v (U) a n d ~t ~ M ( G , C) then

Proof. If z = [ A ]

= < ~,sVz).

w h e r e A ~ End(V) a n d ~ v : G ~ E n d ( V ) (Fg,[A])

= Tr ( ( F ~ t ) v A )

then we have

435

=

Tr ( S K v (x) A d~t(x)) G

= f Tr (:¢v(x) A ) d p . ( x ) G = S FV([A])(x)d~(x) G =

( ~, FV([A]) ).q ed

The above Proposition 4 shows that the Fourier transform F : M(G,C) ~ End(U) is the continuous dual of the linear map F v : End v (12) * R(G) c C(G, C). At this point, it becomes possible to extend the d o m a i n of the Fourier transform. W e have a canonical inclusion M(G,C) c R(G)* since R(G) is dense in the Banach space C(G, C) (Section 1 Corollary 13). We shall extend F to the full linear dual of R(G): F : R(G)* ..... ~ End(U). Let us write ( h , f) for the evaluation pairing between

R(G)* and

R(G). The Fourier

transform of h e R(G)* can be defined by the formula (Fh) v = ( h , ~v) by which we mean that, if (Kij) is the matrix of r~v for some basis of V then ((h, ~ij)) is the matrix of (Fh) v for the same basis. One might think of the elements of R(G)* as generalised distributions on G. More precisely, when G is a compact Lie group, it can be proved that R(G) is a dense subspace of the space C'(G, C) of smooth functions with the smooth topology [Sc]. The continuous dual of C-(G, C) is the space D(G, C) of distributions on G. We thus have a canonical inclusion ~XG, C) c R(G)*, showing that the Fourier transform is defined on distributions. Recall that there is an algebra structure on D(G, C) given by the convolution product of distributions. There is also an algebra structure on R(G)* (and more generally on each R(M)* ) which is the dual of the coalgebra structure on R(G). It is easy to verify that this product on R(G)* extends the convolution p r o d u c t of distributions. In particular, the universal enveloping algebra U(g) of the Lie algebra g of the group G is a subalgebra of R(G)*, since it is equal to the subalgebra of D(G, C) consisting of distributions whose supports are concentrated at the unit element ee G. The Lie algebra g c U(g) c R(G)" corresponds to the E-derivations D : R(G) ~ C; that is, the linear maps such that D(fg) = D(f)¢(g) + ¢(f)D(g) where

e: R(G)

~ C is evaluation at ee G.

In defining the generalised Fourier transform there was no need to restrict to

436

compact Lie groups. Quite generally, we have: P r o p o s i t i o n 5. For any topological monoid M, the (generalised) Fourier transform is an

isomorphism of topological algebras F : R(M)*

N

~ End(U).

Proof. As in the p r o o f of Proposition 4, it can be s h o w n that, for any zeEndV(U)

and

h e R(M)*, w e have (Fh,z)

= ( h , FVz).

This p r o v e s that F is an i s o m o r p h i s m since it is the t r a n s p o s e of yv a n d w e have Theorem 1. Also this shows that F is bicontinuous since the topology on both sides is the usual t o p o l o g y for linear duals (namely, pointwise convergence for linear functionals). To finish the proof, w e m u s t verify that F is an algebra h o m o m o r p h i s m . By definition of the convolution product, w e have, for all h, ke R(M)* a n d fe R(M), (h*k,f)

= ~(h,

fr

(k, gr)

r=t

where n

Af = ~_~ f r ® g r r=l

If w e a p p l y this f o r m u l a to the entries (relative to some basis of V), w e obtain (h*k,

gij ) =

Eij of the matrix of a r e p r e s e n t a t i o n Kv of M (h,~ir)(k,

r=l

grj),

which means exactly that F ( h * k) v = ( F h ) v ( F k ) v .qea W h e n G is a compact group, w e can combine the last Proposition 5 with Section 1 Proposition 4 to obtain: Corollary 6. For any compact group G, the Fourier transform defines an isomorphism of

topological algebras F : R(G)*

)

rI ~eG

End(Vz) . v

W e n o w r e t u r n to the m o r e general situation of a topological m o n o i d that w e h a v e a canonical m a p j : M ~ R(M)* defined b y

M. Recall

( j ( x ) , f ) = f(x). W e m i g h t say that j(x) is the Dirac m e a s u r e concentrated at

xe M. We have a c o m m u t a t i v e triangle M

J

~- R(M)

End(U)

437

since, for every V~ Rep(M,C), F(j(x)) v = (j(x), ~ v ) = g v (x) = :~(X)v Let End®(U) denote the submonoid of End(U) consisting of the tensor-preserving natural transformations (the product is composition). The Tannaka m o n o i d T(M) of M is the submonoid of End®(U) consisting of the self-conjugate natural transformations.

Proposition 7.

The Fourier transform F induces the following isomorphisms topological monoids: Spec(R(M))

~ * End®(U),

SpecR(R(M))

of

- , T(M).

Moreover, the following triangle commutes.

J

M

~

Proof. For any representations

'

~ Spec(R(M))

~

T(M)

V, W e Rep(M,C), let (~ij) and (Prs) be the matrices of

~v and ~w in some basis of V and W, respectively. If we express the equality (Yk)v® w = (Fk) v® (Yk)w in matrix form, we obtain the equality {k, gijPrs) = (k,~ij)(k, pr s) for all i, j, r, s. This proves that k is an algebra h o m o m o r p h i s m if and only if F ke End®(U) (we are using here the fact that R(M) is the linear span of the matrix entries of the representations of M ). This establishes the first bijection. The second bijection is a consequence of the formula Fk= Fk where the conjugate k of an element ke R(M)* is defined by (k,f)

= (k,i).

Bicontinuity of the bijections follows from that of the Fourier transform.qe a Corollary 8. For any compact group G, there is a commutative triangle of compact group

isomorphisms: G

J

> Spec(R(G)) /~

~T(G)/

Proof. According to Tannaka-Krein (Theorem 20), x is an isomorphism. Hence, j = F "1orc is also an isomorphism.qe d ,~4. The coalgebra End v ( X ). In this Section, we shall introduce a coalgebra structure on End"(X) for any functor

438

X : C----* q,~ctc whose values are finite dimensional vector spaces. More generally, u n d e r the same hypotheses o n functors Y, Z, we shall describe a m a p A : H o m v (X, Z) ~ HomV(y, Z) ® Homv(X, Y) which dualises the usual composition map. We shall also show that there is a coaction 7 : X dualising the usual evaluation action

, Y®Homv(X,Y)

Horn(X, Y) ® X

i Y.

W e begin by giving another description of H o m v (X, Y) useful for m a n y purposes. It is based o n the concept of tensor product [F] of a contravariant functor S : C°P ~

q~ctc

with a covariant functor T : C .~ q~ctc . The behaviour of this tensor p r o d u c t is similar to that of the tensor product of right and left modules. To stress this analogy, we write o n

the right the action of f : A ..... ~ B on an element x~ S(B): S(f)(x) = x . f and similarly, we shall write on the left the action of f on ye T(A): T(f)(y) = f oy. The required tensor product is based on the notion of bilinear pairing q:SxT ,V where V is a vector space. This q is a family ( qc I Ce C) of bilinear pairings qc : S(C) x T(C) ~V which respect the actions of S and T. More precisely, it is required that we have qA(x ° f , y ) = q B ( x , f . y ) for all f : A

~ B in C, x • S(B), y • T(A). The universal recipient of such a pairing is a

vector space called the tensor product over C of S with T, a n d is denoted by S ® c T . We n o w formulate one of its f u n d a m e n t a l properties. Note that a family bilinear pairings qc : S(C) x T(C)

( qc I C • C )

of

~V

corresponds to a family ( q'c [ C• C) of linear maps q'c : T(C) , Hom(S(C),V). Let us denote by H o m ( S , V ) space Hom(S(C),V).

the covariant functor whose value at C • C is the vector

Proposition 1. The correspondence described above determines a bijection between linear .~ V and natural transformations T - ~ H o m ( S , V ) . maps S ® c T Suppose n o w that the functors

X, Y : C

, r/&Ctc take their v a l u e s in finite

dimensional vector spaces. For any object C • C, the map [ , ] : Y(C)*®X(C)

; HomV(X,Y),

[ ~ , x ] = [q®x]

is a bilinear pairing b e t w e e n the contravariant functor Y* a n d the covariant functor X since we have, for all f : A k 4

B in C, x• X(A), ¢ • Y(B)*,

[¢Y(f)®x] = [~®Xff)x]. Using this pairing, we obtain a canonical m a p Y'®c X ~ Horny (X,Y). The verification of the next result is left to the reader.

439

Proposition 2. The canonical map described above is an isomorphism Y*®c X

~

~ H o m " ( X , Y).

W e can n o w apply Proposition 1 to this situation u s i n g the canonical isomorphism H o r n c(Y*, V)

~

~ Y®V.

W e see that there is a bijection between the linear maps Horn v (X, Y) ~ maps X ~

V and the linear

Y®V. To describe this bijection explicitly, we need the n a t u r a l transformation 7 : X ~

Y®HomV(X,Y)

defined as follows. For any C e C and any basis e I . . . . . e n of Y(C), we put FL

~/c(X)

=

,#

E e i ® [e i®x] . i=l

W e leave to the reader the verification that ~ is natural.

Proposition 3. For any natural transformation n : X - - - * Y®V, there exists a unique linear inap fi Horn`` (X, Y ) ~ V such that the following triangle commutes. X

Y

-~- Y ® HomV(x, Y)

Y®V W e can n o w describe the map A : H o m v (X, Z) ~ H o m v (Y, Z) ® H o m v (X, Y) whose dual is the usual composition map Horn(Y, Z) ® Hom(X, Y) ~ Horn(X, Z ) . W e have A = fi where n is the composite X.

7 > Y®HomV(X,y)

-/@1 ) Z @ H o m V ( y , Z ) @ H o m ` ` ( X , y ) .

A short calculation gives n

&[¢®x] = ~ [¢®yi]®[y~®x] i=l

where xeX(C), ~eZ(C)* a n d {}'1. . . . . Yn}, {Yl', . . . . Yn" } are dual bases of Y(C), Y(C)*. In particular, d~ defines a coalgebra structure o n End"(X) = Horn" (X, X) : A : End" (X)

} End" (X) ® E n d v (X).

To c o m p u t e A explicitly, recall that, for a n y vector space V, there is a coalgebra structure on End(V) since End(V) is self dual a n d is itself an algebra. If e l , • • - • en is a basis of V then the coalgebra structure 8 : End(V) 8(eij)

~ End(V) ® End(V) is given by =

~

eik®ekj

k=I

where e i j = ei*® ej ; the counit is the trace map Tr : End(V) ~ End (X(C)) CEC

> End V(x)

C. The canonical m a p

440

expresses the coalgebra End ~ (X) as a quotient of the direct s u m of coalgebras. This implies that w e h a v e A[eij]

=

~

[eik]®[ekj]

,

k=l

and that the counit e : End v (X)

~ C is given b y e [A] = Tr A .

The following result s u p p l e m e n t s Section 3 Theorem 1. P r o p o s i t i o n 4. For any topological monoid M, the Fourier cotransform 9rv : End ~(U)

~ R(M)

is an isomorphism of coalgebras. Proof. Let Ve Rep(M, C) and take A e End(V). The calculation ¢ FV[A] = T r ( n v ( e ) A ) = T r ( A ) = a[A] shows that y " preserves counits. If e 1 , . . . , e n is a basis of V then w e have y v ( [eij])(x ) = Tr(:~v(X ) eij ) = e~.(r[v(x)ej ) = gij(x)" This shows that F v preserves A since Agij =

°

~ gik@gkj, k=I

A[eij] =

i

[eik]®[ekj],

k=l

and EndV(U) is the linear span of the elements [ eij ] as V r u n s over the representations of M.qe d W h e n G is a compact group, w e have a canonical m a p of coalgebras i : ~ End(Vx) ) EndV(U). k~ Gv

It is an i s o m o r p h i s m since t i is the i s o m o r p h i s m q of Section 1 Proposition 4. We shall use this m a p to identify

EndV(U) with ~xEnd(Vx). With this convention, the following

result is the dual of Section 3 Corollary 6. Corollary 5. For any compact group G, the Fourier cotransform provides an isomorphism of coalgebras: :

~

End(V~)

- )

R(G)

~-e Gv

It is of interest to see the description of the inverse cotransform jr-l:

R(G)

)

~

End(V;~).

~.eG v

The orthogonality relations i m p l y that, for any f~ R(G), w e h a v e yv-l(f)(~,) = d;~ fG/~;~(x)-I f ( x ) d x . Recall that a (right) comodule over a coalgebra E is a vector space V e q u i p p e d with a coaction

441

(~ : V ~V®E which is associative and unitary; that is, the following diagrams commute. V

• V®E

V®E

V®8

V

~ V®E®E

~'-V ® E

"

~

V

W h e n V is finite dimensional, there is a bijection between comodule structures (~ : V

) V ® E and coalgebra maps : End(V)

If e I , . . . ,

e n is a basis for V then we have

) E .

n

(x(ei) = 2 e j ®

j=l

where

o:ij

(o~ij) is a matrix with entries in E. Putting eij = e i ® e j ,

we then see that the

coalgebra map determined by (x is given by the equations (eij) = o~ij • W e have the formulas

n

~(~®V) = ((~®l)o~(v),

Offv) = E e i ® ~ ( e ] ® v ) i=1

valid for all v ~ V and (~V*. Expressed in terms of the matrix (0~ij), the associativity and unitarity conditions are very simply: ~

= c~(~

and

ec~ = id

where

~(O~ij) = (~O~ij), (O~ij)~(O~ij) = ( ~ k O ~ i k ® f ~ k j ) , E(O~ij) = (¢Oqj), id = (~ij)' The vector space V is canonically a comodule over the coalgebra End(V) via the coaction c : V .... ~ V ® End(V), the corresponding coalgebra map : End(V)

c(x) = ~i ei ® (el ® x); )End(V)

is the identity m a p of End(V). W a r n i n g . The dual E* of a coalgebra is natura]Iy an algebra. However, the reader should note that a right comodule over E has a natural structure of a left E*-module (but not of a right E*-module). If cc:V ) V ® E is the coaction then, for all q~V* and v~V, w e h a v e • v = (1®~) (x(v).

Amorphism f : (V, (x) -----* (W,[3) of E-comodules is a linear m a p p i n g f : V - - - * W such that the square b e l o w commutes. The category of (right) E=comodules will be designated by Comod'E, a n d its full subcategory of finite d i m e n s i o n a l comodules will be denoted b y

ComadyE.

442

V

,,

0~

~V®E

fl

I f®E

w

~

~W®E

It is instructive to compute the finite dimensional comodules over the coalgebra R(M) for any topological monoid M. First, for any V~ ~ p ( M , C ) , if we compose the map ~/v :V---~ V ®EndV(U) with the Fourier cotransform, we obtain a m a p cv : V } V ®R(M) which is a coaction. To see this, pick a basis e 1, . . . , e n of V. Then a short computation gives Cv(e i) = ~ e j ® / ~ i j j=l where

(~ij) is the matrix of ~v in the basis e 1. . . . , %. The identities FI

/l;ij(xy)= £ ~ i k ( x ) r c k j ( y ) , k=I mean that 8(~) = ~ t ~ , functor

c(~) = id, and therefore that Cv is a coaction. We have defined a U- : Rep(M,C)~

Proposition

categories.

ltij(e)=Sij

Comod/R(M).

6. For any topological monoid

M,

the functor

U - i s an equivalence of

P r o o f . It suffices to describe an inverse U ~-1. Let (V, (z) be a finite dimensional comodule over R(M). We have n

0~(ei) = ~ e j ® j=l where el . . . . .

% is a basis for V. The relations 6(o0 = (x~(z,

matrix (O~ij) determines a representation functions

Gij

~ of

M

e(c~) = id mean that the

which is continuous since the

0qj are. W e p u t U--I(V, rz) = (V,~).qe d

Let C be a category. An E-comodule structure on a functor X : C natural transformation ¢z:X ..... } X ® E

~ r/~Ctc is a

such that czc is a comodule structure on X(C) for all C~ C. Using (z, we obtain a functor (X, cz) : C

~ Comod(E)

by putting (X, 00(C) = (X(C), otc). This functor fits into a commutative triangle

Comod~E)

c ~

Vectc

443

where

t~

is the forgetful functor. W h a t is more, it is clear that a n y functor

X~ : C

; Comod(E) such that U X - = X is of the form (X,00 for a u n i q u e c o m o d u l e structure c~ on X. W h e n X takes its values in finite d i m e n s i o n a l vector spaces, the natural transf o r m a t i o n 7 : X ............. ~ X ®EndV(X) defines an Endv(X)-comodule structure on X. This follows from the c o m m u t a t i v i t y of the triangles

X(C)®End~X)

x(c)

/~)®[

]

X(C) ® End(X(C)) w h e r e c is the End(X(C))-comodule structure on X(C) ---* Endv(X) is the canonical m o r p h i s m of coalgebras.

and where

[ ] : End(X(C))

P r o p o s i t i o n 7. Let X : C ~

q/ectc be a functor whose values are finite dimensional } X ® E, there is vector spaces. For any coalgebra E and any comodule structure n : X precisely one morphism i~ : E n d v ( X ) ~ E of coalgebras such that the following triangle commutes. X

7

v

.... _~ X ® End(X)

X@E Proof. To p r o v e this P r o p o s i t i o n , it suffices to verify that the linear m a p ~, w h o s e existence a n d uniqueness are assured b y Proposition 3, is a m a p of coalgebras. The verification is left to the reader.qe d

§5. Tannaka duality for homogeneous spaces. In this p a r t w e s h o w h o w T a n n a k a d u a l i t y can be u s e d to o b t a i n results on h o m o g e n e o u s spaces. The proof given here is i n d e p e n d e n t of the p r o o f in Section 1. As in Section 2, the results p r o v e d here can be used to s h o w that a n y h o m o g e n e o u s space over a c o m p a c t Lie g r o u p is a (real) algebraic variety. This is the basis for the construction of q u a n t u m h o m o g e n e o u s spaces such as q u a n t u m spheres [Pd~ Let G be a c o m p a c t group. By a (left) G-space w e m e a n a topological space X e q u i p p e d with a continuous (left) action G x X----~ X. A G-space is homogeneous w h e n it is n o n - e m p t y a n d the action is transitive. For each closed s u b g r o u p H c G , the space G / H of orbits for the action of H on the right of G is a h o m o g e n e o u s G-space. If K c G is another s u b g r o u p , w e shall write K \ G / H for the space of orbits for the action of K on the left of G / H . Let X be a h o m o g e n e o u s G-space. For any representation V of G, w e shall write V x for the vector space HomG(X,V) of G-invariant m a p s from X to V. If w e choose a b a s e p o i n t x0e X a n d H is the stabilizer of x0, then the m a p g ~ g x0 induces an i s o m o r p h i s m between G / H a n d X, while the m a p a ~ a(x0) defines an i s o m o r p h i s m

444

between

Vx and the subspace

VHcV

consisting of the H-invariant vectors. For all

V~ Re/fiG, C), putting U x ( v ) = v x and U H ( V ) = V H, we obtain two isomorphic functors U x, U H : ~ p ( G , C ) ~ '/4ctc. Let Y be another homogeneous G-space. We define the Fourier transform of a Ginvariant function f~ C(YxX, C) to be the natural transformation Ff : U x ..... -~ U Y whose value at V~ Rep(G, C) is the linear map (Y-f)v : VX given by the formula

* VY

(Ff)v(a)(y) = ~ f ( y , x ) a ( x ) d x where dx is the normalised Haar measure on X. As in Section 1, it can be proved that the projection H ° I ( U K , U¥) ) I ] H o m (V x, V Y) is an isomorphism, so that we can view the Fourier transform as a map F : C ( Y x X , C) G

X

Y

) I~Hom(Vx,V~). K~G v

W e n o w want to show that y defines an isometry of Hilbert spaces. Let L2(Yx X) denote the space of square integrable functions on Y x X with respect to the product dx dy of the normalised Haar measures on X and Y. Also, let L2(YxX/G) denote the space of square integrable functions on product measure

YxX/G

with respect to the projection onto

YxX/G

of the

dx dy. The space L2(YxX/G) is isometric to the subspace L2(YxX) G of

G-invariant elements of L2(yx X) in which C(YxX, C) G is dense. On the other hand, for any K~G v, there is a canonical Hitbert space structure on Hom(V~ x ,V~.¥) that we shall n o w describe. More generally, any G-invariant inner p r o d u c t on X induces an inner product on V x given by (a,b) = (a(xo),b(xo)) where the value of the right hand side does not depend on the choice of the basepoint Xo~ X since we have (a(ox0),b(ox0)) = (Ga(x0),ob(x0)) = (a(x0),b(x0)) . Using inner products on V~x and Vxx which are induced by some G-invariant metric on V x , we can define the adjoint f ' : V~Y---* Vxx of any map f : V~x

...., V~Y. This adjoint

does not d e p e n d on the particular choice of the G-invariant metric on V;v O n the vector space Hom(V;~x ,VKY) we shall use the inner product (f,g) = d~Tr(f*g) where d~. denotes (as before) the dimension of V~. Theorem 1. The Fourier transform extends continuously to an isometry

445

Y: : L2(X × Y / G )

~ )

--- h~lb,, H o m ( V x;~, V ~) x . ~ Ke G

Proof. W e can s u p p o s e that X = G / H and Y = G / K . Let PH : V---o V H be the averaging operator defined for a n y Va Rep(G, C) b y the formula PH(V)

=

fHhVdh

w h e r e d h is the n o r m a l i s e d H a a r measure on H. If i K : V K ~ V denotes the inclusion, it is e a s y to see that the m a p f ~ iK f PH defines an i s o m e t r i c e m b e d d i n g of Hom(V~. H ,V~ K) into Hom(V x,V~.). We obtain in this w a y an isometric e m b e d d i n g

0~ :

E h i l ~ ' * H o m ( V H, V I~)

hilbert

,

kEG v

H o m ( V ~,V ~)

k ~G"

w h o s e i m a g e consists of the elements which are fixed u n d e r the actions of K on the left a n d H on the right. On the other hand, the m a p g t - o (Y0 • g x0) induces a m e a s u r e p r e s e r v i n g h o m e o m o r p h i s m b e t w e e n K \ G / H and Y x X / G (the m e a s u r e on K \ G / H obtained b y projecting the H a a r m e a s u r e on G ). This defines an isometric e m b e d d i n g j3 : L2(Y x X / G ) :, L2(G)

is

whose image consists of the elements fixed b y K (on the left) and b y H (on the right). The T h e o r e m then follows from the relation a n d the fact that the Fourier transform of Section 1 Theorem 14 respects the actions of G on the left a n d o n the right on both its d o m a i n and codomain.qe a When

Y =G

acting on itself b y left translation, we have Y x X / G

-= X. We can

define the Fourier transform of a function f e C(X,C) to be the natural transformation Ff : U x ~ U w h o s e value at

VE R e p ( G , C ) is given by the formula (Ff)v(a) = Jxf(X)a(x)dx.

C o r o l l a r y 2 . The Fourier transform can be extended continuously to an isometry Y : L2(X/G)

~ )

~ h i l b ~ H o m ( v X , v;~) . KeG v

At this p o i n t w e w a n t to h a v e s i m i l a r r e s u l t s for the F o u r i e r t r a n s f o r m of generalised distributions on X. We shall let R(X) denote the set of elements of C(X,C) w h o s e orbits generate a finite d i m e n s i o n a l s u b s p a c e of C(X,C). A c c o r d i n g to Section 2 Proposition 1, this R(G) is the set of representative functions on G (as before). W h e n X -G / H , w e have R ( G / H ) = R(G)H. Let us see that, for all Va Rep(G, C), aa V x, and ~a V*, the function x ~

~(a(x)) belongs to R(X). But, if x0a X, then the function

g ~-o ~a(gx0) ) = ~(~v(g) a(x0)) belongs to R(G) a n d is invariant u n d e r right translation of the stabilizer H of x0; this p r o v e s the claim. W e n o w define the Fourier transform y h of an element h e R(X)*. For V~ P~cp(G,C), let e 1 . . . . . en be a b a s i s for V. Every element a ~ V x can be written as

446 I1

a(x) = ~ ai(x) e i i=1 where ai(x)eR(X) since ai(x)=ei*(a(x)).

We put n

(Fh)v(a) P r o p o s i t i o n 3.

=

~(h,ai)e i=1

i.

The Fourier transform defines an isomorphism of topological vector spaces F : R(X)*

- )

H H°m(vX,v~)' keG"

Proof. We can suppose that X = G / H and consider the averaging operation PH : R(G) ) R(G/H) defined by p H ( f ) =JHf(Xh) d h . Composition with PH determines an embedding [3 : R(G/H)* ) R(G)* whose image is the set of (generalized) distributions which are invariant under right translation by the elements of H. Similarly, if we compose with the averaging operators PH : Vk ~ V kH, we obtain a map a :

YI H ° m ( v H , Vk) ) l--I H ° m ( V k , V k ) keG v k eG" whose image consists of the elements which are invariant under right translation by elements of H. The result then follows from the identity

F°~ - - ~ ° F and Section 3 Corollary 6.qe a The next thing to do is to define the Fourier transform of G-invariant generalized distributions on Y x X. We shall do this without choosing basepoints in X and Y. Definition. Let G be a compact group acting continuously on a topological vector space V. An element ve V is G-finite when its orbit generates a finite dimensional subspace of V. By definition, R(X) is the set of G-finite elements of C(X,C). Let us see that R(X) is also equal to the G-finite elements of R(X)*. To do this, let j : R(X) ~ R(X)* be the map which associates to a function fe R(X) the measure f(x)dx on X. For any ge R(X), we have (j(f), g ) = Sxf(X)g(x)dxThe image of j consists of G-finite elements since j is G-equivariant and every element of R(X) is G-finite. Let R(X)° denote the set of G-finite elements of R(X)*. P r o p o s i t i o n 4.

The mapping j determines an isomorphism R(X)

Moreover, R(X) ° is dense in R(X)*.

~ ) R(X) °.

447

Proof. W e shall prove the result when X = G; the general result is proved similarly. If we use the i s o m o r p h i s m y : R(G)* - ~ 1-[ H o m ( V ~ , V ~ ) . k~G v This reduces the problem to proving that, if an element g e I ~ End(V~) is G-finite (for the left action of G) then all except a finite n u m b e r of its components are zero. For each ~ G v, let p;~ = (px(6) I 8e G v ) be the projection operator, where P~(~) =

otherwise •

Then the orthogonality relations imply that p~ ; F(d~. ~.) where Zx is the character of V~: We have the formula P~g = d~, Sc~--~x( x ) ~ ( x ) g d x which shows that, if g is G-finite, then all its components pxg belong to the finite dimensional subspace generated by the orbit Gg. This proves that pxg = 0 except for a finite n u m b e r of k e GV.qed Let R(YxX) be the s e t o f G x G - f i n i t e elements of C(YxX, C). We have a canonical isomorphism R(Yx X) ~ R(X) ® R(Y). For all elements he R(x)*Y the pairing (g,f) is a G-invariant bilinear form distribution C(Y, R(X)* ).

te R(YxX)*

[ p(h)

> Syg(y) ( h ( y ) , f) dy on

R(Y) x R(X).

Let us say that a generalized

is right regular w h e n it is of the form

p(h)

for s o m e

he

Proposition 5. Every G-invariant generalized distribution is regular. More precisely, the map f) defines an isomorphism HomG(Y,R(X)*) ~ HomG(R(YxX),C). Proof. W e describe an inverse to p. We compose the canonical isomorphisms H o m G ( R ( Y x X ) , C ) ~_ H o n ~ ( R ( Y ) ® R ( X ) , C ) _= HomG(R(X),R(Y)*) HomG(R(X),R(Y) °) --- HomG(R(X),R(Y)) with the i s o m o r p h i s m k : HomG(R(X),R(Y)) ~ HornG(Y,R(X)*) defined as follows. For any G-invariant operator u : R(X) ---* R(Y), we put (k(u)(y), f) = u(f)(y). W e leave the reader to check that k is bijective and we have defined an inverse to P-qed W e can n o w describe the Fourier transform of

t e R ( Y x X ) *c.

According to

Proposition 5, t is right regular, so that t = p(h) for some heHomG(Y,R(X)*).

For all

448

V e Rep(G, C), we w a n t to define the Fourier transform (Ft)v : Vx ~ V Y. Let e 1 . . . . , e n be a basis for V. For all a~ V x, we can write a Then w e have

=

~ i ai ei where ai~ R(X).

n

(Ft)v(a)(y) = ~(h(y),ai}ei. i=1

Proposition

6. The Fourier transform defines an isomorphism of topological vector spaces R(X x y)*G _= 1 - I v H o m ( V X , vY ) . KeG

Proof. If we e x p o n e n t i a t e the Fourier transform of Proposition 3 b y isomorphism R(X)* Y~ I ~ v H ° m (Vx,V x Y ~) • ~.EG

Y, we obtain an

Composing this with the isomorphism of Proposition 5, we have the Fourier transform as defined above.qe a To complete these results, we should describe the Fourier cotransform 5rv : H o m v ( U x , U Y) W e shall s u p p o s e that cotransform

Y = G

a n d p o s t p o n e the general case. We w a n t to describe the

yv: For all Ve Rep(G,C), all ~ V *

~ R(YxX) G.

HomV(Ux,U)

~ R(X).

and all ~eV x, we put y~([~®a]) = 9 ° a .

Proposition

7. The Fourier cotransform

yv is an isomorphism

Hom"(uX,u)

~ , R(X).

W e n o w w a n t to use these results to study the spectrum of the algebra R(X). Let us first remark that, for all V, W e ~ p ( G , C), we have a canonical pairing ® : VXxW x

~ (V®W) x

(a,b) ~-+ a®b

defined by (a®b)(x) = a(x)®b(x). We also have a unit element l e C x = C. Definition. A n a t u r a l transformation u : U x

~ U is tensor preserving w h e n , for all V,

W e ~ p ( G , C), ae V x, be W x, the following equations hold: Uv®w(a®b) = uv(a)®uw(b) a n d uI(1) = 1. Let Hom®(U x , U )

denote the set of tensor-preserving n a t u r a l transformations. There is

also the notion of self-conjugate natural transformation u : ux----+ U ; this m e a n s that u ~ ( a ) = Uv(a) for all VEfi~ep(G,C) a n d all a ~ V x. Let T(X) denote the set of self-conjugate tensorpreserving natural transformations U x

~ U.

449

For all xE X, w e have an element ~ (x)e T(X) defined by K (x)v (a) = a(x) for a e V x. There is also the canonical m a p j : X ~

R(X)* defined b y (j(x),f) = f(x).

Proposition 8. The Fourier transform topological spaces:

F induces the following isomorphisms of

Spec(R(X))

~ ~ Hom®(ttx,t4)

SpecR(R(X))

.~ , T(X).

Moreover, the following triangle commutes.

J

X

~- Spec R(X)

T(X) Proof. Let heR(X)* a n d let V, W e ~ p ( G C ) . Let us choose a basis e 1 . . . . . % for V and a basis fl . . . . . fn for W. For any a ~ V x and b e W x, w e have a = Y',iaiei and b = ~ j b 1 fj. The equation (Fh)~® w(a®b) = ( F h ) v ( a ) ® ( F h ) w ( b ) means exactly that w e have (h, ai~ } = (h, ai } (h, bj }. Similarly, the self-conjugacy cor.dition for F h means exactly that w e have ( h,-~i} = ( h , ai} . This proves the Proposition since the coefficients a i linearly generate R(X).qe a Let Ve R e p ( G , C ) a n d let g : V ® V

~

C be a positive definite invariant hermitian

form. Let us see that, for any u e T(X), the m a p u v : V x * V is an isometric e m b e d d i n g , w h e r e the metric on V x is the one i n d u c e d from g on V. We can view g as a C-linear pairing h : V ®V ~ C and w e can r e p e a t mutatis mutandis the p r o o f of Section 1 Proposition 5. We obtain that g(uv(a), u v (b)) = g(a, b), which m e a n s that u v is an isometric e m b e d d i n g . The collection of these e m b e d d i n g s is compact, since it is a h o m o g e n e o u s space over the u n i t a r y g r o u p U(V, g). It follows that T(X) is a closed subspace of a p r o d u c t of compact spaces. This proves:

Proposition 9.

T(X) is compact,

W e next w a n t to p r o v e that X

- ~ T(X).

For this w e n e e d s o m e preliminaries.

S u p p o s e that G acts c o n t i n u o u s l y on s o m e c o m p a c t space S. Let A c C ( S , C ) subalgebra which separates points of S and which is d o s e d u n d e r conjugation.

be a

L e m m a 10. If A is stable under" the action of G and every element of A isG-finite then A G separates the orbits of G in S. ProoL Consider the averaging operator E : C(S, C)

~ C(S, C) defined b y

450

E(f)(x) = jcf(g x ) d g . If f E A then E(f)~A since Eft) belongs to the convex hull of the orbit Gf whose linear span is a finite dimensional subspace of A. Let p : S ~ S / G be the projection, and let c, c' ~ S / G be any two distinct orbits of G in S. The space S / G is compact and so there is a continuous function ~ : S / G ~ R such that ~(c) = 0 and #(c') = 1. It follows from the Stone-Weierstrass Theorem that A is dense in C(S,C); so we can find f~ C(S,C) such that 1 Hf-(~°P II < where we use the uniform norm. We have IJE(f)-(~op] I = liE(f- qboP)l] < IIf - * o p l l

1

This implies that, for all x~ c and y~ c', 1 [ E ( f ) ( x ) - E(f)(y) I > ~ . So we have found an element E(f)~ A G which separates the orbits c, C'.qed T h e o r e m 11. The maps g,j provide

two homeomorphisms

:X . . . . . ~ T(X)

and j :X

....~" ~ SpecR(R(X)).

ProoL To prove that j is injective is equivalent to proving that R(X) separates the points of X. But this follows from Lemma 10 since we have X = G / H , R(X) = R(G) H and R(G) separates the points of G (Section 1 Corollary 13). Let S = Specg(R(X)). Then S is compact since T(X) is (Proposition 9) and Fourier transform gives a h o m e o m o r p h i s m S ~ T(X). The group G acts on S since it acts on R(X). This action is continuous since S c R(X)* and G acts continuously on R(X)'. The algebra A = R(X) becomes a subalgebraof C(S,C) i f w e p u t f ( g ) = g ( f ) for all f~R(X) and x~S. Clearly, A separates the points of S and the hypotheses of L e m m a 10 are satisfied. This proves that A c separates the points of S/G. But we have A c = R(X)G = C, and so S / G must reduce to a singleton. This implies that j : X ~ S is surjective since its image j(X) is an orbit of G in S. We have proved that j is a homeomorphism. It follows that g is a homeomorphism, since g = F°j .qed

§6. Minimal m o d e l s .

A model of a representative function f on a topological m o n o i d M is a pair (q~,v)~V*xV such that f(x) = q)(~v(X)V) where V is some representation of M. Such a pair is far from being unique. A model is minimal if dimV is minimal. It is easy to see that a given representative function has a minimal model which is u n i q u e up to a (unique) isomorphism. In this part, w e shall extend the concept of minimal model b y giving a description of the elements of the tensor product of any two functors which satisfy some exactness conditions. This description is the key technical tool for p r o v i n g the Representation Theorem of Section 7. Recall that a category C is C-additive when each of its homsets C(A,B) has the structure of a complex vector space and composition is C-bilinear. Recall also that a functor F : C ~ D between C-additive categories is C-additive when the maps

451

F : C(A,B) ) D(FA, FB) are C-linear. Let C be an abelian C-additive category. We shall suppose that C is artinian, meaning that each decreasing sequence of subobjects in C terminates. Let T : C ~ q&ctc be a left exact C-additive functor. We shall need the category el(T) of elements of T. An object of d(T) is a pair (A,x) consisting of A e C

and x~T(A). A arrow f : (A,x)

(B,y) in ed(T) i s a n a r r o w f : A ) B s u c h t h a t T ( f ) ( x ) = y . We shall say that x~T(A) is contained in a subobject B~-) A when x is in the image of the map T(B) ~ T(A). From the left exactness of T we see that, if x is contained in the subobjects B >---) A and B'~-~ A, then it is contained in their intersection B^B'>---) A. Using the artinian property of the poser of subobjects of A, it follows that x is contained in a smallest subobject Span(x) of A. When Span(x)= A, we say that x generates A. We also need the full subcategory Span(T) of eE(T) whose objects are the pairs (A, x) in which A = Span(x). Observe that, between any two objects of Span(T), there is at most one arrow. To see this, suppose f, g : (A, x) ---) (B,y); so x is contained in Ker(f - g) since T(f)(x) = y = T(g)(x). Therefore, f = g since A = Span(x). Note also that, if f : (A,x) ) (B,y) is an arrow in Span(T), the map f : A ) B is epimorphic. This is because T(f)(x) is contained in the image of f and therefore Im(f) = B since T(f)(x) = y is generating. W e can n o w give a first description of the elements of the tensor product S ® c T for any contravariant C-additive functor S. We suppose that the abelian category C is artinian and that the covariant functor T is left exact. We shall write [0 ® x] for the image of a pair (0, x) by the canonical map S(A) x T(A)

~ S ®C T.

Proposition1. Under the above hypotheses, any element of

S ® c T is of the form [0® x]

for some (0,x) ~ S(A) x T(A) for which A = Span(x). Moreover, if (0,x) ~ S(A) x T(A) and (xg,y)~ S(B) xT(B), then the equality [ 0 ® x ] = [ ~ ® y ] holds if and only if there exists an object (C,z)~ Span(T) and arrows (A,x) < f

(C, z)

g > (B,

y)

such that S(f)(0) = S(g)(~). Proof. W e shall use some standard results from category theory [ML]. Since T is left exact, the category el(T) has finite limits and is therefore filtered (or codirected). If p : el(T) ----) C is the projection functor and y : C°P ----) Vectc c is the Yoneda embedding then w e have the canonical isomorphism T ~

li....~m(e/(T) op

P

~

C°P

Y

-- V e c ~ )

Tensoring this isomorphism with S and using the isomorphisms S @cY(A) = S(A), we obtain a canonical isomorphism S@cT~

~(

el(T)°P

p

~

cOP

S

_ Vectc )

The result is then a consequence of the standard description of colimits of directed diagrams, and the fact that Span(T) is initial in e/(T).qe d

452

To o b t a i n a m o r e c o m p l e t e description of the elements of S ® c T, w e shall m a k e a d d i t i o n a l a s s u m p t i o n s on the category C a n d the functor S. More precisely, w e shall s u p p o s e that C is noetherian, m e a n i n g that each i n c r e a s i n g s e q u e n c e of subobjects terminates; or equivalently, that C°P is artinian. W e shall also s u p p o s e that S is left exact, m e a n i n g that S transforms right exact sequences in C into left exact sequences of linear maps. As before, for a n y ¢eS(A), there is a smallest quotient object Cospan(¢~)

supporting ~ (we a d o p t this t e r m i n o l o g y to a v o i d confusion w i t h the p r e v i o u s case of a covariant functor T ). If Cospan(¢) = A, we say that ~ cogenerates A. We shall say that a pair ( ~ , x ) e S(A) x T(A) is a model of an element z c S ® c T w h e n z e [~)®x]. A m o d e l (~,x) is min imal when Cospan(¢) = A = Span(x). A n isomorphism between (¢,x) e S(A) x T(A) and ( ~ , y ) e S(B) x T(B) is an invertible m a p f : A ~ T(f)(x) = y. In this case w e have

B such that S ( f q ) ( ¢ ) = ~ a n d

[¢®x] = [S(f)(w)®x] = [~®T(f)(x)] = [~@y]. If t w o m i n i m a l m o d e l s and y are generating.

(¢,x), (V,Y) are isomorphic, the i s o m o r p h i s m is u n i q u e since x

T h e o r e m 2. Suppose that the abelian C-linear category C is artinian and noetherian, and

that the functors

S and T are left exact. Then every element of

S ® c T has a minimal

model which is unique up to a unique isomorphism. Moreover, for any ( ¢ , x ) e S ( A ) x T ( A ) , there is a minimal pair ( 0 ' , x ' ) e S ( A ' ) x T ( A ' ) such that [~' ®x'] = [~@x] w h e r e A' is a subquotient of A. Proof. W e begin b y p r o v i n g the last statement. Let q Span(x)

J ~ A' ~

~ Cospan(~)

be the i m a g e factorization of the composite i

p

Span(x) ~

, A

~ Cospan(~).

By definition, there exist x 1 a n d (~1 such that S(p)(~h)=( ~ a n d T(i)(x~)=x. I f w e p u t

(~'=

S(j)(¢1) a n d x' = T(q)(x l) then we have [¢' ® x'] = [¢' ® T(q)(x~ ) ] = [S(q)((~' ) ® x~] = [S(jq)(O,) @x~] = [S(pi)(¢,)®x~] = [S(i)(~)@x~] = [¢@T(i)(x,)] = [¢@x]. M o r e o v e r , x' generates A' since xl generates Span(x). Similarly, (~' cogenerates A'. This proves that the pair (~',x') is a m i n i m a l m o d e l of [(~@x]. This proves the existence of a m i n i m a l m o d e l for each element of S ®c T since all elements of S ® c T

are of the

form [ ~ ® x ] b y Proposition 1. To p r o v e the uniqueness, s u p p o s e that ( ¢ , x ) e S(A) x T(A) a n d (~, y) e S(B) x T(B) are both minimal models of z. Let (C, z), f, g be as in Proposition 1. The m a p f is e p i m o r p h i c since T(f)(z) = x a n d z, x are both generating. If c~ = S(f)(¢) then

Cospan(c~)~ A

since

f is e p i m o r p h i c a n d

e p i m o r p h i c a n d C o s p a n ( o ) = B since isomorphism

i :A ~

~

¢ is cogenerating. Similarly, g is

is cogenerating. This p r o v e s that w e have an

B such that i f = g a n d S(i)(~¢) = ¢~. We have also

453

T(i)(x) = T(i)T(f)(z) = T(i f)(z) = T(g)(z) = y. Hence i is an i s o m o r p h i s m between the pairs (~),x) a n d (~,y). W e h a v e a l r e a d y r e m a r k e d that such an i s o m o r p h i s m m u s t be unique.qe d C o r o l l a r y 3 . For all (q~,x) ~ S(A) x T(A), the equality [q~®x] = 0 holds if and only if there exists a monomorphism j : B ~ - - * A such that x ~ I m ( T ( j ) ) a n d S(j)(qb)=0. Proof. Let (qY,x')eS(A') x T(A') be a minimal m o d e l of [ ~ ® x ] w h e r e A' is the i m a g e of the m a p Span(x) ~ A ---~ Cospan(~)) as in the proof of the Theorem. If [~' ®x'] = 0 then A' = 0 b y u n i q u e n e s s of the m i n i m a l model. This s h o w s that the c o m p o s i t e Span(x) A ---,7 Cospan(#) is zero, and the result follows with B = Span(x).qe d To e n d this Section, w e shall p r o v e that, for a n y c o a l g e b r a E, w e h a v e an qYectc is the forgetful functor. We i s o m o r p h i s m E = EndV(U) w h e r e U : ComodfE ~ h a v e an o b v i o u s coaction

cc : U

(Xv: V ) V ® E where m o r p h i s m of coalgebras

(V,~t)~ Coma(dyE. U s i n g Section 4 P r o p o s i t i o n 5, w e obtain a

~ U®E

obtained by p u t t i n g together all the coactions

: End"(U)

) E.

In o r d e r to p r o v e this m o r p h i s m is invertible, w e need the following L e m m a w h o s e proof is left to the reader. L e m m a 4. Let (V, c0 be a finite dimensional comodule over a coalgebra E. For each

~)~.V*t

the comodule Cospan((~) is obtained by taking the image factorization of the map V

c~ ) V ® E

~®1 ) E .

P r o p o s i t i o n 5. For each coalgebra E, the map : End V(U)

~E

is an isomorphism of coalgebras. Proof. W e easily see that, for all x~ V, ~E V* a n d (V, ~)E ComodyE, [~®x] = (~)®1) 0~v(X). To p r o v e surjectivity, take x~ E. There is a finite dimensional s u b m o d u l e V c E such that x~ V. Let ~) be the restriction of the counit ~ : E - - ~ C to the subspace V. Then w e have [~® x] = (~)®1)8(x) = (¢® 1) 3(x) = x . To p r o v e injectivity, take z~ Endv(U) in the kernel, a n d let (~, x)~ V*x V be a m i n i m a l m o d e l of z. We h a v e (q~®1) 0~v(X) = ~ (z) = 0. But L e m m a 4 shows that the m a p (~®1) o~v is injective since shows that x = 0 and hence z = 0.qe d

0 is cogenerating. This

454

§7. The representation theorem. In this Section, w e shall p r o v e a r e p r e s e n t a t i o n t h e o r e m for an abelian category C e q u i p p e d w i t h a n exact faithful functor U w i t h v a l u e s in finite d i m e n s i o n a l vector spaces. W e p r o v e that C is equivalent to the category of finite dimensional E-comodules, w h e r e E is the coalgebra Endv(U) constructed in Section 4. This result is basic in the t h e o r y of Tannakian categories in which the c a t e g o r y comes first, or is given, a n d the g r o u p comes second, o r is constructed. There are m a n y e x a m p l e s of T a n n a k i a n categories in nature, especially from algebraic g e o m e t r y in G r o t h e n d i e c k ' s theory of motives [DM]. The representation t h e o r e m p r o v e d here is not n e e d e d in the rest of this paper. W e first recall a few properties of the category of comodules over a coalgebra C. For any vector space V, the tensor p r o d u c t V ® C has a c o m o d u l e structure obtained b y left comultiplication V®8 : V®C-----* V®C®C It is the cofree c o m o d u l e over the vector space

V. If (z : V ~

V ®C

is a c o m o d u l e

structure o n V then (~ is also a m o r p h i s m of comodules from V to the cofree c o m o d u l e V ® C. The set of s u b c o m o d u l e s of V is closed u n d e r intersections a n d sums. If f : V W is a m o r p h i s m of c o m o d u l e s then the direct a n d i n v e r s e i m a g e s u n d e r f of s u b c o m o d u l e s are s u b c o m o d u l e s . For completeness w e i n c l u d e a p r o o f of the following classical result.

Proposition 1. Every comodule is the directed union of its finite dimensional subcomodules. Every coalgebra is the directed union of its finite dimensional subcoalgebras. ProoL Let E c V b e a finite d i m e n s i o n a l s u b s p a c e of the c o m o d u l e (V, (z). There exists a finite d i m e n s i o n a l s u b s p a c e F c V such that c~(E)c F ® C . We have E c c~-I(F®C) a n d ~ - I ( F ® C ) is a s u b m o d u l e since it is the inverse i m a g e of the s u b c o m o d u l e F ® C

of the

coffee c o m o d u l e V ® C. Moreover, w e have c~-I(F ® C) c F since F ® C c (1 ® E)-1 (F) a n d o¢-1 (1 ® r) -1 (F) = ((1 ® ¢)00 -I (F) = E This p r o v e s the first sentence of the Proposition, and the second sentence follows once we a d d this remark: the s u b c o m o d u l e s of C, as a c o m o d u l e o v e r C°P ® C via left a n d right comultiplication, are exactly the subcoalgebras of C (where C°P is the o p p o s i t e coalgebra of C).qed W e n e e d to p r o v e s o m e results on the c a t e g o r y comodules over a coalgebra C.

ComadyC o f finite d i m e n s i o n a l

Definition. A full s u b c a t e g o r y of an abelian category is replete if it is closed u n d e r finite direct sums, subobjects a n d quotients. In this definition it is explicitly a s s u m e d that a n y object i s o m o r p h i c to an object of the r e p l e t e s u b c a t e g o r y also belongs to it. For a n y s u b c o a l g e b r a C' c C, w e h a v e an

ComoaC/C'c ComodyC since every c o m o d u l e o v e r C' is n a t u r a l l y a c o m o d u l e over C. N o t e that CamodfC' is a replete subcategory of Comod/C. inclusion

Proposition 2. For any coalgebra C, the assignment C': ~ ComodyC' defines a bijection between the subcoalgebras of C and the replete subcategories of Camo~C.

455

Proof. We shall produce the inverse assignment. For any C-comodule (V, ~), let Im(V) be the image of the coalgebra morphism : End(V)

) C.

So Im(V,(x) is a subcoalgebra of C. For any replete subcategory D c Como~C, let Im(D) be the sums of the subcoalgebras Im(V, c~) as (V, c~) runs over the objects of D. For any subcoalgebra C' c C, we obviously have Im(D) c C' if and only if D c ComadfC'. This shows that Im(Como~C') c C' and D c Comodf(Im(D)). To prove that C' c Im(Como~C') we can suppose that C' = C. Using Proposition 1, it will be enough to show that every finite dimensional subcoalgebra V of C is contained in Im(Caraa~C). N o w the map : End(V)

) V

corresponding to 8 : V ----* V @ V is surjective since ~(e®v)

=

( e ~ l ) 8(v)

=

v.

This shows that Ira(V, a) = V and hence Im(Como~C) D V. To prove that Comodf(Im(D)) c D, we can suppose that Ira(D) = C. Then our task is to prove that D c CamodfC. Let us first remark that, for any comodule (V, (z), the subcoalgebra Im(V, a) is generated by the images of the maps V

a >V®C

~®1 > C ,

where # runs over V*. This is a direct consequence of the identity (#~x)

=

(~1)

c~(x).

If Ve D then the images of the maps (#® 1) cz belongs to D since D is replete. Therefore Im(V,(z) belongs to D. By hypothesis, C is generated by the Im(V,00 for (V,c~)e D. This shows that every finite dimensional submodule of C belongs to D. Let us prove that each (W,[3)e ComodfC belongs to D. If #1 . . . . . #n is a basis for W* then the images Wi of the maps W ~ ," W ® C #1®1 ) C belongs to D since they are subcomodules of C. Combining the maps W " Wi together, we obtain a map from W into the direct sum of the W i . This map is easily seen to be monomorphic. So We D.qed Suppose now that C is a C-additive category and U : C , q~ctc is a C-additive functor with values in finite dimensional spaces. In Section 4 we saw that the natural transformation 7: U , U @ Endv(U) determines a lifting

U~ : C

, Como~Endv(U)

of U up into the category of Endv(U)-comodules.

456

Theorem

3.

If C is abelian and U is exact and faithful then U - is an equivalence of

categories. P r o o f . Obviously U ~

is faithful. Before p r o v i n g that U -

is full, let us see that, for any

object AE C a n d any subcomodule E c U-(A), there exists a subobject j : B ~ A such that E = ImU(j). Let e 1 . . . . . e n be a basis for the vector space U(A) chosen in such a way that e 1 . . . . . e k generate E. For any x~U(A), the coaction (x : U(A) ~ is given b y the formula

U(A)®Endv(U)

n

(X(X) = Z e i i=1

@

*

[el®x].

The hypothesis 0fiE) c E ® EndV(U) means that, for all i < k and j > k, we have [ e j ® e i] = 0 . W e can n o w apply Section 6 Corollary 3 since Endv(U) = U * ® c U a n d both U a n d U* are exact ( C is both artinian a n d noetherian since U is faithful). This m e a n s that, for all i < k and j > k, there is a subobject Bij ~

A such that ei~ U(Bij) and ej (U(B i j)) = 0. If we p u t

B i = ("IB i. j>k l

and

B

=

~,JB i i k. This shows that E cU(B), a n d U ( B ) c E since E To prove that U -

=

C'~Ker(ej) . j>k

is full, let f : U - ( A )

~ U-(B) be a m o r p h i s m of comodules.

Then the graph Ff of f is a subcomodule of U-(A)~BU~(B) ~- U~(A~B B) . Therefore there is a subcomodule C c A @ B whose image can be identified with I f . We d a i m that C is the graph of an arrow u : A ~ B whose image u n d e r U is f. To see this, let i be the composite of the inclusion C ~-~ A~B B a n d the first projection Pl : AfB B

~ A. Then

U(i) is the i s o m o r p h i s m Ff ~ U(A)@U(B) ~ U(A). Since U is exact a n d faithful, it follows that Ker(i) = Coker(i) = 0; so i is an isomorphism. The arrow u is defined as the composite .-I P2 A I >C c >A~BB >B. Obviously U(u) = f since U transforms this description of u into the description of f. To finish the proof, we have to show that every object of ComodfEndv(U) is isomorphic to a c o m o d u l e in the i m a g e of

U-.

If we let

D

be the full subcategory

consisting of the comodules isomorphic to those in the image of U ~, we have to prove that D = ComodfEndv(U). But we have seen that the image of U - , and therefore D, is closed u n d e r subobjects. A similar a r g u m e n t shows that it is d o s e d u n d e r quotients. Thus, D is a replete subcategory, and, from Proposition 2, we have D = ComodfC' for some subcoalgebra C' c Endv(U). This implies that, for all A e C, the vector space U(A) has the s t r u c t u r e of a C ' - c o m o d u l e T'A: U(A)---~ U ( A ) @ C ' w h i c h 'lifts' the coaction of Endv(U). More precisely, we have the c o m m u t a t i v e triangle of n a t u r a l transformations below.

457

U

-

U®C'

U ® End (U) Using Section 4 Proposition 5, w e have a u n i q u e coalgebra m a p 7' : E n d " ( U )

> C'

such that the following triangle commutes. U

Y

~ U ® Endv(U)

U®C' But then w e have i 7'=id b y the uniqueness p r o p e r t y of Section 4 Proposition 5. This proves that i is surjective, and therefore C ' = Endv(U) since i is an inclusion.qed For any coalgebra C, the dual algebra C* acts on the left of a n y (right) C-comodule. For any V~ Comacf(C), ~0~C* and x~ C, we write -OAx for the action of ~) on x. We have Aflx = ( l v ® ¢ ) CZv(X). Each element (~a C* defines a natural transformation

_~()

:U

~U

w h e r e U : Comad(C)---* q/ectc is the forgetful functor.

Proposition 4. For any coalgebra C, the map ¢~ ~ C*

~ ( ) is an isomorphism of algebras

~ ~ Hom(U,U).

Proof. U s i n g t h e fact t h a t a n y c o m o d u l e is a d i r e c t c o l i m i t of finite d i m e n s i o n a l comodules, w e can replace the category Coraad(C) b y its subcategory Comodf(C). In this case, w e have defined a coalgebra i s o m o r p h i s m : End"(U )

>C .

The r e s u l t will f o l l o w if w e v e r i f y t h a t the n a t u r a l t r a n s f o r m a t i o n u : U

~U,

c o r r e s p o n d i n g to the c o m p o s i t e of (~ with the above coalgebra i s o m o r p h i s m , is equal to ~kl ( ) .

Let V~ Camacff(C), ~g~V* and x~ V. By definition of u, w e have ( u, [~®x] ) = ~( ~ (~®x)) = ~((~¢®1C ) O~v(X))

= ~( (1 v®(~) O~v(X)) -- W(_~ x) = ( ~ ( ) ,[V®x] ) which shows that u = ~1 ( ) "qoa

458

§8. The bialgebra Endv(X) and tensor categories. The tensor p r o d u c t M ® N of two m o d u l e s over an algebra A is an A ® A - m o d u t e , but it is not in general an A-module. However, when A is a bialgebra, w e can define an Am o d u l e structure o n M ® N by restricting the A ® A - m o d u l e structure along the d i a g o n a l (= comultiplication) m a p

8 :A

* A®A.

Similarly, the tensor p r o d u c t

M®N

of two

c o m o d u l e s o v e r a coalgebra C is a C ® C - c o m o d u l e , and, w h e n C is a bialgebra, w e can corestrict this c o m o d u l e structure along the multiplication m a p

~ :C®C

" C to obtain a

C - c o m o d u l e structure o n M®N. The category Camad(C) of comodules over a bialgebra C is therefore a tensor category (also called a m o n o i d a l category). The m a i n p u r p o s e of this section is to reverse this process: starting with a p a i r (C, X) consisting of a tensor category C a n d a t e n s o r - p r o d u c t - p r e s e r v i n g functor X : C ~ Vectf, w e shall s h o w that the coalgebra EndV(X) can be enriched with the structure of a bialgebra. In w h a t follows, w e let Vectf d e n o t e the category of finite d i m e n s i o n a l vector spaces. In Section 4 w e saw h o w to construct a coalgebra Endv(X) from a pair (C, X) where C is a category a n d X : C ~ Veery is a functor. It is easy to see in a d d i t i o n that a c o m m u t a t i v e triangle of functors F C ~'D

V~9 gives rise to a m a p of c o a l g e b r a s

Endv(X)

~ Endv(y)

w h i c h w e m i g h t call the ~ End(X) ). W h e n the functor F is an equivalence of categories, the corestriction m a p along F is an i s o m o r p h i s m of coalgebras. It is useful to formalise this process b y introducing the category Cat/Veery of categories over Vecty. A n object of Cat/V¢ty is a pair ( G X) w h e r e C is a

corestriction along F (it is p r e d u a l to the usual restriction m a p End(Y)

(small) category a n d X : C ~

Vecty is a functor. A m o r p h i s m

(F, c~) : (C, X)

-* (D, Y)

consists of a functor F : C ~ D a n d a natural i s o m o r p h i s m (z : X ~ YF. C o m p o s i t i o n of m o r p h i s m s is the obvious one. We have a covariant functor EndV: Cat/Vectf ~ Coalg with values in the category Coa~ of coalgebras. W e define the (external) tensor product of X : C'----* Vectf with Y : D ~ Vectf to be the functor X ® Y : C x D---~ Vectf where, for (A,B)e C x D, (X ~ Y)(A, B)

=

X(A) ® Y(B).

Proposition 1. There is a canonical isomorphism 0 : EndV(X) ® Endv(y)

~ ~ EndV(X ~ Y).

Proof. For a n y (A, B)~ C x D, S~ End(X(A)), T~ End(Y(B)), w e have S®T~ End(X(A)®Y(B)). W e p u t 0([S]®[T]) = I S ® T ] . The best w a y to p r o v e that 0 is w e l l d e f i n e d a n d an i s o m o r p h i s m is to see t h a t it is a special case of the f o l l o w i n g canonical i s o m o r p h i s m

459

between tensor products of functors (HOcX)@ (K®~y)

~ ) (H~K)OcxD(X®Y)

where H, K are contravariant functors on C, D, respectively. W h e n H = X* a n d K = Y*, we have H ® c X = Endv(X) and K ®oY = Endv(y).qea Recall [ML] that a tensor (or "monoidal") category C = (C, ®, I, a, ~ r) consists of a category G a functor ® : C x C ) C (called the tensor product), an object Ie C (called the unit object) a n d natural isomorphisms a = aA,B,C : ( A ® B) ® C

~ ." A @ (B ® C ) ,

[=[A: I O A ~)A, r=rA : A@I --)A (called the associativity, left unit, right unit constraints, respectively), such that, for all objects A, B, C, D e C, the following two coherence conditions hold: aA,B,C®D ° aA®B,C,D = (AOaB,C, D) ° aA,BOC,D ° (aA,B,cOD) a n d (A®/S) ° aA,i,B = rOB. It follows [ML] that all objects obtained by c o m p u t i n g the tensor product of a sequence A I ® A2O . . . @A m b y bracketing it differently a n d b y cancelling units are coherently isomorphic to each other. A tensor category is called strict w h e n all the constraints aA,B,c , EA , rA are identity arrows. Each tensor category C is equivalent to a strict tensor category st(C). The objects of st(C) are words w = A I A 2. .. A m in objects of C. A n arrow f : w : [w]

) w' is an arrow f

: [w'] in C where we define [f~] = I,

The tensor square

[A] = A,

and

[A1 A2- • • Am +1] = [A1 A2- • • Am ] ® Am+l • ® for st(C) is given by v ® w = v w and by commutativity of the following f®g

Iv] ® [w]

, [v'] ® [w']

'!

'1

[vw]

, [v'w'] fog

A n example of a tensor category is the category Cat/Vectf with the external tensor product described above. The u n i t object I in Cat/Vecty is the functor C : 1 ---~ Vecty where 1 is the category with a single object * and a single arrow (the identity of *) a n d C denotes the functor assigning to * the one-dimensional vector space CE Veery. Let C, D denote tensor categories. Recall [ML] that a tensor (or "strong monoidal") functor F = (F, (~, Co) : C---~ D consists of a functor F : C---~ D, a n a t u r a l isomorphism ---- O A , B

a n d an i s o m o r p h i s m

;

FA @ FB

~ ) F(A @ B),

00 : I ~ ) F I , such that the following three equations hold (where

we write as if C, D were strict): 0AOB,C ° (OA,B@FC) = (~A,BOC ° (FAOOB,C), %@FA = (~,A° F(%OA)

and

FAO00=0A,I°F(A@%).

460

The tensor functor is strict w h e n all the i s o m o r p h i s m s e x a m p l e of a tensor functor is the equivalence example is the functor End v : Cat/Vectf Recall that a monoid M e C a n d arrows It : M ® M M®M®M

I1

are identifies. One

~ Coa[g.

M = (M, It, 11) in a tensor category

~ M, rl : I ~ I~M > M®M

M®M

~A,B, ¢0

C ........* st(C) t a k i n g A to [A]. A n o t h e r C consists of an object

M such that the following d i a g r a m s commute. n®M M¢~ M ~ M ® M ~M

- M

A comonoid is a m o n o i d in C°P. For example, algebras are m o n o i d s in the category of vector spaces, coalgebras are c o m o n o i d s in the same category, and bialgebras are m o n o i d s in the c a t e g o r y of coalgebras. M o n o i d s in Cat (where the tensor p r o d u c t is cartesian product) are strict tensor categories. Monoids in Cat/Vectf are t h e p a i r s (C,X) for which C is a strict tensor category a n d X : C

~ Vectf is a tensor functor (not necessarily strict). It

follows from Proposition 1 that the coalgebra Endv(X) c o r r e s p o n d i n g to such a pair has the s t r u c t u r e of a b i a l g e b r a since it enherits a m o n o i d s t r u c t u r e in the c a t e g o r y of coatgebras. M o r e generally, if w e h a v e a pair (C, X) w h e r e C is a tensor category, not necessarily strict, a n d w h e r e X is a tensor functor, w e also have a bialgebra structure on EndV(X). To see this, w e can replace the pair (C,X) by a pair (C',X') w h e r e C ' = st(C) is a strict tensor category equivalent to C. We use the fact that the corestriction m a p Endv(X ') Endv(X) along an equivalence C ' ~ C is an i s o m o r p h i s m of coalgebras. However, w e shall directly describe the algebra structure on Endv(X) w i t h o u t recourse to ( C ; X ' ) . For a n y A, Be C, S e End(X(A)) a n d Te End(X(B)), let us w r i t e S ® T for the dotted arrow in the square X(A ® B)

--:,-- X(A @ B)

X(A) @ X(B)

S@T

~ X(A) ® X(B)

Also, let us write l e X(I) for the element c o r r e s p o n d i n g to l e C u n d e r the i s o m o r p h i s m C -= X(I). These notational abuses are harmless, not o n l y because the context will dissipate the ambiguity, b u t also because tensor functors satisfy a coherence t h e o r e m [Le]. W e can n o w specify the algebra structure on EndV(X). The p r o d u c t of the elements [S] a n d [T] of Endv(X) is given b y the simple f o r m u l a [S][T] = [ S ® T ] .

The unit e l e m e n t of Endv(X) is equal to [1]. When structure on result.

C = ~p(M,C) EndV(tt).

and

X is the forgetful functor

U, w e obtain a bialgebra

The m e a n i n g of the bialgebra structure is e l u c i d a t e d b y the next

Proposition 2. For any topological monoid M, the Fourier cotransform y v : EndV(U) R(M) is an isomorphism of bialgebras.

461

Proof. It r e m a i n s to verify that ~p(M,C)

F v is a h o m o m o r p h i s m of algebras. For all

V, W e

a n d all AeEnd(V), BeEnd(W), we have

y v ([A][B]) = F v ([A®B]) = Tr (nv®w ° (A®B)) = Tr ((~VO~W) ° (AOB)) = Tr (TtvA®TrwB) = Tr ( n v A ) T r (Uw B) = y:v ([A]) y v ([B]) and y:v(1) = Tr(x I) = 1-qea The next thing we shall do is to characterize the algebra structure on EndV(X) by a u n i v e r s a l property. More precisely, for all algebras A, w e shall p r o v e that the correspondence n ~ fi is a bijection between tensor-preserving n a t u r a l transformations X ----* X ® A a n d algebra h o m o m o r p h i s m s Endv(X)

* A. Our first task is to define the

former. A coaction of the algebra A on a vector space V is a linear m a p 0t : V or equivalently, a linear m a p : End(V)

* V ® A,

> A.

W e define the trace of a as the value of the last linear m a p at the identity e n d o m o r p h i s m of V. If e~ , . . . , e F i is a b a s i s of V and I1

a ( e i) = Y, e j ® o t i j j=l then Tr(00 =

~ o~ii. i=I

W e have the formula (S) = Tr(ccS) which is valid for all S e End(V). The tensor product o¢~[~ of coactions ct : V -----* V ® A , :W

~ W @ A is defined to be the composite

VOW

~®~ ) V ® A ® W ® A

where ~ : A ® A ~

~ >V®WOA®A

vow®~

>V®W®A

A is the multiplication of the algebra A, a n d the m i d d l e i s o m o r p h i s m

in the composite uses the symmetry m a p A® W ~

W ® A. If

m

s=]

gives a matrix for ~ t h e n Fi r r f l

( o ~ ~)(e i ®fr) = j,s=l

(ej ®fs) ®(Otij~r s)

gives a matrix for 0 ~ . W e shall say that a natural transformation u : X ..........; X® A is tensor when, for all C, D e C, we have Uc® D = U c ~ U D

and

Let C be a tensor category a n d X : C ~ Proposition 3, we defined a natural transformation

preserving

u I = 1.

qdecty be a tensor functor. In Section 4 I : X~

X® Endv(X).

462

Proposition 3. The natural transformation y is tensor preserving. Moreover, for all algebras A and all tensor-preserving natural transformations n : X .....* X ® A , there is precisely one algebra homomorphism gi:EndV(X) ~ A such that the following triangle commutes. X

7

~ X ® Endv(X)

Proof. To p r o v e y tensor preserving, take C, De C. Choose a basis e I . . . . . e m of X(C) and a basis f l , - - - , fn of X(D). By definition of YC~YD, w e have the equality (~fc~yD)(x®y) = ~ei®fj®[ei®x][fi®y]. i,j O n the other hand, w e have ~/C®D(x®y) = Z e i @ f j ® [ e i ® f i ® x @ y ] . i,j The equality TC~YD = YC®D is n o w a consequence of the identity [~®x] [~/® y] = [q)®V® x®y] which h o l d s in EndV(X). To p r o v e the rest of the Proposition, let n : X ~ X ® A be a t e n s o r - p r e s e r v i n g n a t u r a l t r a n s f o r m a t i o n . A c c o r d i n g to Section 4 P r o p o s i t i o n 3, there exists precisely one linear m a p fi : Endv(X) ~ A such that the triangle of the Proposition commutes. A s t r a i g h t f o r w a r d c o m p u t a t i o n shows that w e h a v e fi([S]) = T r ( n c S ) , w h e r e w e are using the trace introduced above. Using this w e have n ([S] [T]) = n ([S ® T]) = T r (n c® D(S® T)) = T r ( n c® nDo (S ® T)) =Tr(ncS®nDT)=Tr(ncS)Tr(nDT)

= n([S]) n([T]).qed

W h e n A is a bialgebra, a coaction cc : V - - ~ V ® A defines a c o m o d u l e structure on V if a n d only if it is associative a n d unitary. Also, the tensor p r o d u c t of t w o coactions % ~ is a c o m o d u l e structure if b o t h of 0c, ~ are. The category Comady(A) of finite d i m e n s i o n a l c o m o d u l e s is a tensor category. Clearly, if a n a t u r a l t r a n s f o r m a t i o n n : X X® A is tensor p r e s e r v i n g a n d defines a c o m o d u l e structure o n X then w e obtain a functor X ' = (X,n) : C ~ Como~(A) which is tensor p r e s e r v i n g a n d r e n d e r s c o m m u t a t i v e the following triangle, w h e r e the forgetful functor.

cx--v L

U is

463

Proposition 4. Let C be a tensor category and X : C

~ q/ect/ be a tensor functor. For all bialgebras A, there is a bijection between tensor-preserving functors X ' : C ~ Comocf/(A) such that U,X' = X (i.e. X' lifts X) and bialgebra homomorphisms Endv(X) ~A.

Proof. Just combine Proposition 4 with Section 4 Proposition 7.qe d §9. Duality and Hopf algebras. W e begin by recalling the basic concepts of duality theory in a tensor category C. G i v e n A, Be C, we shall say that a pair of maps TI : I ----, B®A, ¢ : A®B ----, I form an adjunction between A and B if the following two composites are identities: A

A®'~ ) A ® B ® A

~A

)A,

B

~®B ) B @ A

®B

B®E "B

W e call q the unit and E the counit of the adjunction. We say that A (respectively, B ) is left adjoint or left dual to B (respectively right adjoint or right dual to A). We also write (~, e) : A -q B to indicate that the pair (~, ~) is an adjunction between A a n d B. The u n i t a n d counit of an adjunction determine each other. More precisely, let us say that a m a p e : A®B

~ I ( a 'pairing') is exact when, for ail objects X, Y, the function

E# : C(X,B®Y)

~ C(A®X,Y),

f I

~ (eeY)(Aef)

is bijective.

Proposition 1. A pairing ~ : A®B ~

I is exact if and only if there exists a map

"q : I

B®A such that the pair ('q, ~) is an adjunction between A and B.

Proof. If E is exact then take ~ to be the u n i q u e m a p such that ¢#('q) is the canonical isomorphism

A®I ~

A. Conversely, if (T1,~) is an a d j u n c t i o n then the function h

(Be h ) (rl e X) is an inverse

for

Edt.qed

In the category of finite dimensional vector spaces, a pairing ~ : V®W ---* C is exact if a n d only if the corresponding m a p x ~

¢(x,-) from V to W* is an isomorphism.

W h e n the p a i r i n g is exact, we can describe

~ : C

~W®V

b y g i v i n g the value

"q(1)~W®V. To any basis e I . . . . . e n of V there corresponds a dual basis fl . . . . . fn of W such that ¢(e i , fj)

=

~ij" We have TI(1) = flee1 + . . . + f n e e n •

In a tensor category C, let E : A®B -----+ I a n d ¢' : A'®B' ~ I be two exact pairings. W e shall say t h a t a m a p f : A - - - - , A ' is left adjoint to g : B ' - - - - + B (or that g is right adjoint to f ) w h e n we have ¢'(feA) = ¢(Beg). For all f, there is a u n i q u e right adjoint g given by g = 0 l ' e B) (B'e f e B ) (B'e e). Similarly, for all g, there is a unique left adjoint f given by f = (A'e -q) ( A ' e g e A ) ( E ' ® A ) . A p p l y i n g this to the case where A = A', we see that two right adjoints B, B' of A are canonically isomorphic. Similarly for left adjoints.

464

Definition. A tensor category C is a right adjoint. When

autonomous

w h e n e v e r y object of C has both a left a n d

C is autonomous, w e can choose, for each Ce C, a pair of adjunctions

(Tlc, ec) : C ¢--4 C and (~'c,e'c) : C~ W e obtain in this w a y a pair of contravariant functors ( )¢ : C°P

" C

and

( )r : COp

C ¢. ~ C.

Obviously, for all CE G w e have canonical i s o m o r p h i s m s (Cr) ¢ = C -=- (C~) r m a k i n g the functors ( );, ( )r m u t u a l l y quasi-inverse (i.e. they give an equivalence of categories). It is instructive to w o r k o u t an e x a m p l e of an a u t o n o m o u s tensor category w h e r e right a n d left adjoint are different. For any algebra A, let us write Car(A) for the category whose objects are the coactions cz : V ---* V ® A on finite d i m e n s i o n a l vector spaces. In Section 8 w e defined a tensor p r o d u c t of coactions, a n d so this category becomes a tensor category. W e first identify the adjunctions within this category. Let (V, a), (W, ff)e COd(A) and let (~,~) : (v, cz) -~ (W,~) be an exact pairing. Clearly the p a i r i n g (1"1,~) defines an exact p a i r i n g b e t w e e n the vector spaces V a n d W (since the forgetful functor Cad(A) ----* Vectf preserves tensor product). Let e 1 , . . . , e n be a basis of V and let fl ,- • •, fn be a dual basis. We have 0fie i) = ~•e . ®l

• [~(fi)= ~. f

0~ji

J

I ®[~ji

'

l

~ ( e i ® f j ) = ~ij

'

~(1) = ~ f i ® e i i

Expressing that e is a m o r p h i s m in Cod(A), w e obtain

~Otki[~kj k

----- ~ij '

Similarly, expressing that 11 is a m o r p h i s m , w e obtain

~ik k

O~jk = ~ij

If cz = (Gij) and ~ = (~i j), these equalities can be formulated as the matrix equations (tG)

right adjoint czr of the m a t r i x cx is equal to (tcz)-l. leftadjoint cz~ of o~ is the matrix t(a-1). If the algebra A is

= ~ (too = i d . In other words, the Similarly, w e obtain that the

not c o m m u t a t i v e , there is in general no r e l a t i o n s h i p b e t w e e n t (0t-l) a n d (tcc)-l. One m i g h t exist a n d not the other. We can inductively define a (°) = a , a(n+I) = (&n))~ for n > 0 , and a(n-1) = (a(n))r for n < 0. Let us say that a matrix 0~ is totally invertible w h e n aCn) exists for all n e Z . Clearly the coactions w i t h totally invertible matrices form an e x a m p l e of an a u t o n o m o u s category for which left and right duals do not coincide in general. W e n o w give a brief review of the basic theory of H o p f algebras. Recall that, for any coalgebra C and any algebra A, the convolution product defines an algebra structure on the vector space Horn(C, A), w h e r e the convolution ~* ~g of ¢ : C ~ A with ~g : C ....~ A

465

is the composite C ~ )C®C ~,®v > A ® A The unit of Hom(C, A) is the composite C where

Tl(K) = K.1 . If f : C' ~

~>C

~ >A.

~)A

C is a m o r p h i s m of coalgebras a n d

g : A ---o A' i s a

m o r p h i s m of algebras then the assignment (~ ~ g ~ f is a m o r p h i s m of algebras Hom(C, A) ~ Hom(C',A'). W h e n C = A = H is a bialgebra, w e obtain an algebra structure on Horn(H, H).

An

antipode

v on a bialgebra H is a t w o - s i d e d inverse for the identity m a p 1 H : H ---* H with respect to the convolution product. More explicitly, this means that the following two d i a g r a m s commute. v®l HS.~ "H®H

~ H ® H ~ H

H~H®H

l®v

~--H®H 1 be an integer and let h e (C ®n )* be a linear form. For any nsequence (V 1 , . . . , Vn) of C-comodules, we can define an operator ALl : V l ~ - . . ( ~ V n

....

) Vl~-.-~Vn

by using the fact that V I ® . . . ® V n is a comodule over the coalgebra C ®n = C ® C ® . . . ® C. More precisely, for all xle V I . . . . . xn~ V n, we have _ ~ (xl ® . . . ® x , )

= (1®¢) a(xl~ ...®x,)

where w e have used the c ® n - c o m o d u l e structure a : Vl®...®V

If U ®n : Comad(C) n

n

.~ c ® n @ ( v l @ . . . ® V n ) .

~ qdectc denotes the functor

(V1 . . . . . V n ) : .~ V I ® . . . @ V then we have defined a natural transformation _~1( ) : t t ~n

Proposition 2. The

assignment

n

; t~®n

q~ t---* ALl ( ) is an algebra isomorphism (c®n) *

~- ~ H o m ( U ® n , u ® n ) .

Proof. This is a consequence of Section 7 Proposition 4 and Section 8 Proposition 1-qea For a n y bialgebra A,

the category

category. If c is a braiding on C o m o d ( A ) ,

Comod(A)

of (right) comodules is a tensor

we obtain a linear form

473

7

=

(A@A

CA'A > A @ A

E®E " C ) .

TO state the next Proposition 'we shall use the insertion operators

Sij : (A@A)*

----* (A@A@A)* defined by s12 = t(A®A®E), Proposition

on

s2s = t(E®A@A),

s13 = t ( A @ e ® A ) .

3. The assignment c b-o T described above is a bijection between braidings c

Comod(A) and linear forms T~(A®A)* which are invertible for the convolution

product

* and satisfy the following identities: ~t'*~ = y*~t,

T(~t@IA) = S13(~)*S12(~) , T(1a®bt)

= S13(~)*S23('~).

P r o o f (Sketch). The braiding is obtained from y by the formula

c ( x @ y ) = ( .X]x@y)21 = _X21Jy@x . The situation is then entirely dual to that of Example 9. We can obtain these equations by rotating the pictures in that Example through 180°.qe~l Recall [J,D] that a Yang-Baxter operator on a vector space V is a linear isomorphism R : V®V suchthatthe~llowinghexagoncommutes.

V@V@V

"~ V®V

R®I

V®V®V

V®V®V

V®V®V V@V@V

I®R

- V®V®V

The equation (R@I) (I@R) (R®I) = (I®R) (R®I) (I®R) is called the Yang-Baxter equation. The translation into pictures is as below.

R

=

The rule of the g a m e is to replace (whenever possible) these planar d i a g r a m s by 3dimensional ones in which crossings replace the nodes labelled by R or R -1 , as indicated in the following picture. The Yang-Baxter equation is then depicted as the equality shown after that.

474

A n e x a m p l e of a Yang-Baxter operator [J, T, JS3] is the following. Let e i , . . . , e n be a basis for V, and let q~ C be a n o n - z e r o complex number. W e define R = Rq : V ® V ~ V ® V as follows: ej®e i R(ei®ej) =

for i > j

lej®ei+(q-q-l)ei®ej [ q e i ®e i

for i = j

fori<j .

The o p e r a t o r R satisfies the equation (R-q)

( R + q -1) = 0.

M o r e o v e r , the inverse of R is given b y Iej®e i R -1 ( e i ® e j) =

for i < j

l e j ® e i + (q-1 _ q ) e i ® e j for i > j [q-lei®e i

fori=j

.

One can check directly that this R is a Yang-Baxter operator. G i v e n a n y Yang-Baxter o p e r a t o r R on V, w e can define, for e v e r y n > 0, a representation ~R of the b r a i d g r o u p B n in the general linear g r o u p G L ( V ®n ) b y putting i-1 /~R (Si)

=

I

"1®... ®1®R@1@...®1

for each generator s i of Bn . Putting together these representations, w e obtain a tensor functor ~R : B

Proposition 4.

The

correspondence

R ~

~ ~Ctc • 1~R is a bijection between the Yang-Baxter

operators on V and the (isomorphism classes of) tensor functors

~ : B

~ Vectc s u c h

that ~ (1) = V. More generally, let T : A ~

V be a functor from a category A to a tensor category

D e f i n i t i o n [JS3]. A Yang-Baxter operator on T is a natural family of i s o m o r p h i s m s

V.

475

Y = YA,13 : T A ® T B " ' such that the following hexagon commutes. y®l TA@ TC@TB l@y J

~ ~ TB®TA

TC®TA®TB

/

TA ® TB ® TC

TC ® TB ® TA TB ® TA ® TC

1Ny

~'~-~-TB ® T C ® TA

A n y functor T : A ~ V into a braided tensor category V comes equipped with a Yang-Baxter operator obtained from the braiding of V: YA,B = CTA,TB : T A @ T B N, TB@TA. The i m p o r t a n c e of Yang-Baxter operators is partly explained by the following considerations. For any category A, there is a braided tensor category BJA of braids having their strings labelled by arrows of A. (The notation B JA is intended to indicate that it is a wreath product in a generalized sense [K, JS3].) The objects of BJA are finite sequences of objects of A . An arrow ( ° t , f l . . . . . f n ) : (A1 . . . . . An) ~ (B1 . . . . . Bn) consists of Ix~B n and fi ~ A ( A i , Bc~(i)) where i ~-* 0t(i) is the permutation defined by cz. Composition of labelled braids is performed by composing the label on each string of the composite braid. The operation of addition of braids extends in the obvious w a y to labelled ~ B~A, yielding a tensor structure on B~A. There is an obvious braids B j A x B~A braiding on B JA obtained from the braiding on B. We have an inclusion functor i:A

~ BJA

identifying A with the labelled braids with a single string. The braiding on B JA defines a (formal) Yang-Baxter operator z on the functor i . The next Proposition explains the sense in which this z is universal. Proposition 5 [JS3]. The braided tensor category BJA is free on A . Moreover, for any tensor category V and any pair ( y, T), where y is a Yang-Baxter operator on T : A - - - ~ V,

there exists a unique (up to a unique isomorphism) tensor functor T' : B JA that T ' ( z ) = y and the following triangle commutes. i A ~ BJA

q~ Example 10. For any algebra A, a Yang-Baxter operator on the forgetful functor U : Mod(A) , Vectc is completely determined by the element 7 = YAA (1®1) E A @ A . The operator is given by

V such

476

YM, N (x®y) = y ( y ® x ) . A p a r t from invertibility, the only condition on y is the equation S23(~/) S13(~') $12(Y) = S12('y) S13(7) S23(~) w h e r e sij : A ® A

~ A ® A ® A are the insertion operators. We shall say that a n invertible

e l e m e n t y~ A ® A satisfying these equations is a Yang-Baxter element of the algebra A. It s h o u l d be d i s t i n g u i s h e d from the o p e r a t o r y that it defines. M o r e precisely, w h e n End(V) w h e r e

A =

V is a finite d i m e n s i o n a l v e c t o r space, a Yang-Baxter e l e m e n t ?~

End(V)®End(V) = End(V®V)

defines a Yang-Baxter o p e r a t o r R = ? o c w h e r e c : V®V

- - 4 V®V is the usual s y m m e t r y operator.

Example

11. For any coalgebra C, a Yang-Baxter operator y on the forgetful functor

U

:

Comod(C)

~ Vectc

is d e t e r m i n e d b y the linear form y

=

(C@C

Yc,c ) C ® C

~®E ; C ) .

y ( x ® y ) = (.X.lx®y)21 =

.~21jy®x •

W e have the f o r m u l a The linear form T is invertible in the algebra (C ® C)* a n d satisfies the following equation in the algebra (C@C®C)* : S12(7) * S13(~/) * S23 (~/) = S23(~/) * S13(~/) * 812(7) w h e r e the sij are the insertion operators defined before Proposition 3. D e f i n i t i o n [JS3]. S u p p o s e q / i s a b r a i d e d tensor category. A (full) twist for 'I7 is a natural family of i s o m o r p h i s m s 0=0A:A ~,A such that 01 = 1 and the following diagram (T) commutes.

CA,B A @ B ...............

~ B@A

OA@BI A®B

! 0130OA ~

CB,A

B®A

A tensor c a t e g o r y e q u i p p e d with a b r a i d i n g a n d a twist is called a balanced (or

ribbon) tensor category. For any b r a i d e d bialgebra ( A , y) (Example 9), the twists on 9dad(A) are in bijection w i t h invertible central elements z~ A satisfying the equations ¢(z)=1

and

8(z)21 = ~/21(I®'~) "y.

W e have z = 0a(1) and 0 (x) = z x . Similarly, for any cobraided bialgebra ( A , y), the twists on Comod (A) are in bijection

477

with the invertible central elements "¢~A* satisfying the equations z(1)=1

and

zog

=y*('c®z)*Y21 •

D e f i n i t i o n [JS3, Sh]. A tensor category is said to be tortile w h e n it is b a l a n c e d a n d each object A has a left d u a l A* satisfying

0A. ----0A* . Definition. A tortile Hopf algebra (H, y, I;) is a Hopf algebra H e q u i p p e d with a b r a i d i n g y a n d a twist "c such that v (1:) = "~ where v is the antipode. Definition. A cotortile Hopf algebra ( H , y, z) is a Hopf algebra e q u i p p e d with a cobraiding y a n d a cotwist "~ such that z o v = v . The category of comodules over a cotortile Hopf algebra is a tortile tensor category. Proposition 6. Every tortile tensor category is autonomous. Proof. In any braided tensor category, if ( TI, e ) : A* ~ A is an adjunction then ( ~ , , e~ ) : A --t A* is also a n adjunction where e~ = E o CA, A" and 1"1,= (CA*,A)-1 o TI • TO see this, we can use the following abstract argument. O n any tensor category (C, ®) there is a reverse tensor product C ® ' D = D ® C .

Clearly, if (~, e) : A*-~ A in (C,®) then (11, e): A-~ A*

in (C, ®' ). W h e n C is braided, we have a natural i s o m o r p h i s m C=Cc, D : C ® D ~ ~ C®'D which is coherent : it is an isomorphism of the two tensor structures [JS3, JS4]. Using c we can transport the adjunction (TI , e ) : A--I A" in (C,®') to an adjunction (~h,e~) : A--t A* in (C,®). That the formulas for ~ , , ~ are as claimed is n o w clear.qe a Proposition 7. In a tortile tensor category, the square of the twist 0 A is given by A 2 (0 A )

A Proof. W e have the following c o m m u t a t i v e diagrams: I

01

~A

I

A

®A*

1o

CA, A*

v A*®A

0A.® 0A

®A*

I"---"~ ~A

A ® A*~

A* ® A

CA*,A

,oo1

A®A*

A®A*

CA, A*

-'~ A * ® A

.'~A* ® A

CA, A*

478

which s h o w that CA,A.-1 CA.,A-1 1] A -- (0A®0A.) 1] A and therefore, tensoring this equality on the right with A a n d c o m p o s i n g on the left with A ® E A , w e have (A@EA)((CA, A*-1 CA*,A-1 T I A ) ® A ) = ( A O E A ) ( 0 A ® 0 A , ® A ) ( ' Q A ® A ) . The left-hand side is equal to the value of the picture in the Proposition, so it r e m a i n s to show that the right-hand side is equal to (0 h)2. But w e have a(0A.®i ) = e(A®0A) since 0A, = 0 A* - The following sequence of pictures finishes the proof. A

A

A 0A 0A

A The adjunction

A

A

qed

(Tll , ¢q) : A -~ A* is not the a p p r o p r i a t e one in a tortile tensor

category. The reason is that, if (T12,g2 ) denotes the pair obtained b y a twofold application of the assignment (rl, e) t---* (rl:, eq), then we do not have (T12, g2 ) = ('q • e). Let us analyse the situation. For these pictures we shall use ribbons rather than strings. The twist 0 w i l l be represented by a full right h a n d screw turn of the ribbon. First, the pictures for e and TI are: A A*

V A*

A

The adjunction equations are:

%: A

A

A

A*

A*

H: 9 A

A*

A*

The pictures for eq, ~11 a n d the adjunction identities for them are s h o w n below. By viewing the adjunction d i a g r a m s in 3-dimensional space, w e can m o v e the t w i s t e d ribbon on the left of these pictures a n d p u t it in the u n t w i s t e d p o s i t i o n on the r i g h t (the motion, technically called an isotopy, is restricted to a left-right sliding of the attached parts at the

479

top a n d bottom). A*

A

A

A*

A

A

A

a •

A*

A*

A*

A

W e w o u l d like to e l i m i n a t e the l o o p i n g in the p a i r (TI~, ¢q ). O n e w a y to do this is to cancel the l o o p i n g b y a twist. If w e p u t e' = e~ o ( 0 A ® A * ) = ~ o cA,, a o (0 A ®A* ),

TI' = (A* ®0A-I) o'q~ = (A* ®0A-1)oT~ o (CA.' A)-1

then the pictures for e' a n d "q' are: A*

A

A

A*

w h i c h s h o u l d be r e d e s i g n e d to look like: A*

A A

A

V A*

A f o r m a l verification that this is correct is as follows. If (TI", e") is the p a i r o b t a i n e d b y a t w o - f o l d a p p l i c a t i o n of the t r a n s f o r m a t i o n

(T1 , e ) ~

(rl', e') t h e n w e h a v e

('q", ~") =

(T1, e ). To see this, look at the p i c t u r e of TI" a n d c o m p u t e ( s o m e steps of this calculation are m i s s i n g a n d w e invite the r e a d e r to fill in the gaps):

480

A*

A

A*

°2J

=

A

A*

O

A

=

Suppose we have two adjunctions 01A, eA) : A*-4 A , in a tensor category. The m a t e o f a m a p

01B ,eB) : B*-4 B f: X®A

*B®Y

is the m a p

f~ : B*®X

Y ® A * described by the diagram y

B*

A*

X

Equationally, this means that f~ = ( e B ® Y ® A * ) ( B * ® f ® A * ) ( B * @ X ® T 1 A ) . Proposition 8. In a braided tensor category, the mate of (CA,B,)-1 : B* ® A

Ca, B : A ® B

B®A

is

; A @ B*.

Proof. It suffices to prove that

(CA,B.) (CA,B)o

=

1, which is d o n e by the following

sequence of diagrams.

\/

B*

A

B*

A

For the benefit of the reader, we also give the same p r o o f written in the usual sequential notation:

481

(CA,B,) (CA,B) O = (CA,B*) (EB ® A ® B * ) ( B° ®CA,s ®B*) ( B* ®A ®¢lB ) = ( 8B O CA,B.) ( B* O CA,BOB*) ( B*oA®11B ) = (eBOB*®A) (B*® BOCA,B.) (B*OCA,BOB*)( B*OAOT1B) = (~B ®B*OA) (B* ® ((BOCA,B. ) o ( CA,B ® B*)) ( B* ®A ® vlB ) = (eB OB*®A ) (B*® (CA,BOB*)) ( B* OA ®'qB ) = (eB ®B*OA ) (B+® (CA,B®B. (A®TIs ))) = ( £B ® B* ®A) (B* ® ((~qB® A) o CAr l )) = ((8B OB*) o (B*® ~B)®A = B * ® A "qed

Corollary 9. If 01 • ¢) : A* -~ A in a braided tensor category then the following equations

hold: B

A*

B

B

A

A* B

A

B

A

A*

A

B

B

B

A*

Proof. According to Proposition 8, we have B

A*

A*

B

B

A*

A*

B

which can be composed on the top with B o ¢ to yield B

A*

B

B

A

A*

B

A

which proves the result.qe d D e f i n i t i o n [JS3, JS4] Let T : A ---+ V be a functor from a category A to a tensor category qZ.

A Yang-Baxter operator y on T is called dualisable when, for all A~ A, the object TA has a left dual (TA)* and, for all A, B~A, the mates of YA,B, (YB,A) -I : T A ® T B ---+ T B ® TA are invertible. It was shown b y Proposition 8 that a braiding on a tensor category is a dualisable Yang-Baxter operator if every object has a left dual.

482

A dualisable Yang-Baxter operator y on a functor T : A ----* V can be extended by duality to a Yang-Baxter operator y' on a functor T' : A' ----* V where A' is the disjoint union A + A °p of the category A and its opposite A °p. To avoid ambiguities, w e shall write A ° and f° for the object and arrow in A°P corresponding to A and f in A. The extension of T is given as follows: T'(A) = T(A), T'(A °) = T(A) °, T ' ( f ) = T(f), T'(f °) = T(f)*. The extension of y is given as follows: Y'A,B = YA,B,

y,A,BO = (YA,Bo)-I,

Proposition 10 [JS3] The extension

Y'A°,B = (YB,A-I )~,

Y'A*,B* = (YB,A)*'

y' of a dualisable Yang-Baxter operator is aYang-Baxter

operator. The example of a Yang-Baxter operator R = Rq : V ® V

~ V ® V previously given,

on a finite dimensional vector space V and involving a non-zero q~ C, is dualisable. A tedious straightforward calculation gives the following formulas, where we write Yv,v for R and Yv,v* for Yv,v* : ej®e i for i > j

I

Yv,v(ei®ej) = lej®ei+(q-q-1)ei®ej Lqei®ei • Yv'v'(ei®ej)

for i < j

for i = j

[ e ~ ® e i for i ~ j = lq-le~®ei+

~q2(i-k)ek®

ek

for i = j

k>i

ej® e~ y v . , v (e~ ®ej)

=

for i ~ j

q-lei®ei+

~(q-l-q)ek®e

k

for i = j

k

Ie*®e~ Yv.,v.(e~e~)

for i< j

= lei®e ~ +(q-q-1)e~®e; [qe;®e~

for i = j

for i> j

.

When R : V ® V ~ V ® V is a dualisable Yang-Baxter o p e r a t o r on a finite dimensional vector space, we can use the extension R' to define a Yang-Baxter operator on V@V*. pairing

The vector space

V@V*

is e q u i p p e d with a non-degenerate symmetric

and also with a non-degenerate simplectic form c 0 ( x ~ O l y ~ g ) = ~(y)-~g(x). Definition. Let R be a Yang-Baxter operator on an object Z in a tensor category, and let E : Z ® Z ~ I be an exact pairing. We say that R respects e when w e have the equations (in which the crossings are labelled by R and R -1 according to the convention previously explained):

483

A Proposition 11. If 0 1 , ¢ ) : Z ~ Z and if R respects ¢ then the equations below hold:

v

y

Proof. Exercise for the reader.qe d

Proposition 12. A Yang-Baxter operator pairing is dualisable.

R : Z® Z

,Z@Z

which respects an exact

Proof. The mate of R is equal to R q since we have:

Similarly, the mate of R -I is equal to R. This proves that these mates are invertible.qe d Proposition 13. If R is a dualisable Yang-Baxter operator on a finite dimensional vector space V then its extension R' to V S V* respects both the canonical symmetric and the

canonical symplectic pairings on V ~ V * . Proof. Exercise for the reader.qe d T(A)

T(A)

484

•,~/~

Suppose n o w that y is an arbitrary dualisable Yang-Baxter operator on a functor T ~ The picture above defines a canonical natural transformation 0 ' = 0' A : T(A)

~T(A)

called the double twist. In the picture we use the extended Yang-Baxter operator y' to label the crossings.

Proposition 14. For any dualisable Yang-Baxter operator (y, T), the double twist natural isomorphism O' : T -=--* T. Moreover, the following equations hold: y(0'®T) = ( T ® 0 ' ) y ,

O' is a

y(T®0') = ( 0 ' ® T ) y .

Proof. The picture for the inverse of 0' is obtained from that for 0' by rotating through 180 ° and changing all the crossings; the rest is left to the reader.qe a

DefInition [JS3, JS4 ] Let y be a Yang-Baxter operator on a functor T : : ,~t ~

~ A twist o n

y is a natural isomorphism 0 : T - * T such that y (0 ®T) = (T®0) y , y (T®0) = (0®T) y . A tortile Yang-Baxter operator is a pair (y, 0) where y is a dualisable Yang-Baxter operator

and 0 is a a twist on y such that 0 2

=

0' where 0' is the double twist defined by y .

Example 11. In a tortile tensor category, the pair (c, 0 ) is a tortile Yang-Baxter operator since we have proved that 02 = 0' (Proposition 7). Example 12. A short calculation gives that the double twist on the operator Rq is equal to the map x ~

q2nx on V of dimension n. If we put 0(x) = q n x ,

we obtain a tortile

Yang-Baxter operator (Rq, 0 ).

§11. Knot invariants. In this Section w e p r o v i d e a brief introduction to the m e t h o d used by N. Yu. Reshetikhin and V.G. Turaev for obtaining knot invariants. We describe h o w Yang-Baxter operators can be used to produce tensor functors from the category of tangles of ribbons to vector spaces. If we apply the Tannaka duality machinery to these functors, we obtain quantum groups. This will be the subject of Section 12. Let P be a Euclidean plane. A geometric tangle T is a compact 1-dimensional oriented submanifold of [ 0,1] x P which is tamely embedded and whose boundary 3T is equal to T ~ 3( [0,1] x P). We suppose that T meets 3 ( [ 0 , 1 ] x P) transversally. The

target of T is the subset 3T c~ ({ 1} x P) as an oriented submanifold. The source of T is the subset 3T c~ ({0} x P), but with orientation reversed. A geometric tangle can be pictured as shown at the top of the next page.

A tangle is an isotopy class of geometric tangles where the isotopies keep the boundaries fixed. The source and target of a tangle are regarded as signed subsets of P. Let 1, 2, 3 , . . . denote equally spaced collinear points in the plane P.

485

N o w we can define the autonomous braided tensor category T of tangles [FY, Y]. The objects are functions A : { 1, 2, 3. . . . . n } ~ { +, - } for n > 0, called signed sets. The arrows are the tangles which have these signed sets as sources a n d targets. Composition a n d tensor are as for braids. The braiding is illustrated by the following figure.

+

--

+

+

--

The left dual A* of a signed set A is given by reversing the order a n d the signs of the points. The arrows ~IA a n d cA are illustrated in the following figure: +

A=(+,-,+)

+

+

A

11A --

+

-

+

-

+

Let S 1 be the unit circle in the tangent space of the Euclidean plane P. A f r a m i n g on a geometric tangle T is a continuous function f : T ~ S 1. If ~ > 0 is small enough then the set T~ = {x+0~f(x) : x~T, 0_