Locally Presentable and Accessible Categories (London Mathematical Society Lecture Note Series)

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 ISB, England The books in the series listed below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 34 36 39

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Representation theory of Lie groups, M.F. ATIYAH et a! Homological group theory, C.T.C. WALL (ed) Affine sets and affine groups, D.G. NORTHCOTT p-adic analysis: a short course on recent work, N. KOBLITZ Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds) Commutator calculus and groups of homotopy classes, H.J. BAUES Techniques of geometric topology, R.A. FENN Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Several complex variables and complex manifolds II, M.J. FIELD Representation theory, I.M. GELFAND et a! Symmetric designs: an algebraic approach, E.S. LANDER Spectral theory of linear differential operators and comparison algebras, H.O. CORDES Isolated singular points on complete intersections, E.J.N. LOOIJENGA Probability, statistics and analysis, J.F.C. KINGMAN & G.E.H. REUTER (eds) Introduction to the representation theory of compact and locally compact groups, A. ROBERT Skew fields, P.K. DRAXL Surveys in combinatorics, E.K. LLOYD (ed) Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE Topological topics, I.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS Polytopes and symmetry, S.A. ROBERTSON Classgroups of group rings, M.J. TAYLOR Representation of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Representations of general linear groups, G.D. JAMES Low-dimensional topology 1982, R.A. FENN (ed) Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Groups and geometry, ROGER C. LYNDON Surveys in combinatorics 1985, I. ANDERSON (ed) Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Syzygies, E.G. EVANS & P. GRIFFITH Compactification of Siegel moduli schemes, C-L. CHAI Some topics in graph theory, H.P. YAP Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, H. GONSHOR Analytical and geometric aspects of hyperbolic space, D.B.A. EPSTEIN (ed) Lectures on the asymptotic theory of ideals, D. REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG An introduction to independence for analysts, H.G. DALES & W.H. WOODIN Representations of algebras, P.J. WEBB (ed) Homotopy theory, E. REES & J.D.S. JONES (eds) Skew linear groups, M. SHIRVANI & B. WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Non-classical continuum mechanics, R.J. KNOPS & A.A. LACEY (eds) Lie groupoids and Lie algebroids in differential geometry, K. MACKENZIE Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE Van der Corpus's method of exponential sums, S.W. GRAHAM & G. KOLESNIK New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK Model theory and modules, M. PREST

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Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S. PUTCHA Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M. PEINADOR and A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) The geometry of jet bundles, D.J. SAUNDERS The ergodic theory of discrete groups, PETER J. NICHOLLS Introduction to uniform spaces, I.M. JAMES Homological questions in local algebra, JAN R. STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO Continuous and discrete modules, S.H. MOHAMED & BJ. MOLLER Helices and vector bundles, A.N. RUDAKOV et al Solitons nonlinear evolution equations & inverse scattering, M. ABLOWITZ & P. CLARKSON Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds) Number theory and cryptography, J. LOXTON (ed) Classification theories of polarized varieties, TAKAO FUJITA Twistors in mathematics and physics, T.N. BAILEY & R.J. BASTON (eds) Analytic pro-p groups, J.D. DIXON, M.P.F. DU SAUTOY, A. MANN & D. SEGAL Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER (eds) Groups St Andrews 1989 volume 1, C.M. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (eds) Lectures on block theory, BURKHARD KULSHAMMER Harmonic analysis and representation theory for groups acting on homogeneous trees, A. FIGA-TALAMANCA & C. NEBBIA Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE Groups, combinatorics & geometry, M.W. LIEBECK & J. SAXL (eds) Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) Stochastic analysis, M.T. BARLOW & N.H. BINGHAM (eds) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK Squares, A.R. RAJWADE Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M.P. FOURMAN, P.T. JOHNSTONE, & A.M. PITTS (eds) Lower K- and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD, C. PESKINE, G. SACCHIERO & S.A. STROMME (eds) Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory II, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of finite groups, D.J. BENSON Finite geometry and combinatorics, F. DE CLERCK et al Symplectic geometry, D. SALAMON (ed) Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI W. METZLER & A.J. SIERADSKI (eds) The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN

London Mathematical Society Lecture Note Series. 189

Locally Presentable and Accessible Categories

Jiri Addmek Czech Technical University, Prague

Jill Rosicky Masaryk University, Brno

CAMBRIDGE UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia

© Cambridge University Press 1994 First published 1994 Library of Congress cataloguing in publication data available British Library cataloguing in publication data available

ISBN 0 521 42261 2 paperback

Transferred to digital printing 2004

We dedicate this book to the memory of our excellent colleague and very dear friend Jan Reiterman

Contents Preface

ix

Introduction

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0

Preliminaries

1

1

Locally Presentable Categories

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1.A Locally Finitely Presentable Categories . . . . 1.B Locally Presentable Categories . . . . . . 1.C Representation Theorem . . . . . . . . . . 1.D Properties of Locally Presentable Categories . 1.E Locally Generated Categories . . . . . . . . .

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2 Accessible Categories

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2.G Accessible Embeddings Into the Category of Graphs 2.H Limits of Accessible Categories . . . . . . . . . . .

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3 Algebraic Categories 3.A Finitary Varieties . . . 3.B Finitary Quasivarieties

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131 132 151

CONTENTS

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3.C Infinitary Varieties and Quasivarieties 3.D Essentially Algebraic Categories . . . .

Exercises

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4 Injectivity Classes

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4.A Weakly Locally Presentable Categories . . . 4.B A Characterization of Accessible Categories 4.C Locally Multipresentable Categories . . . . Exercises . . . . . Historical Remarks 5

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Categories of Models 5.A Finitary Logic . 5.B Infinitary Logic Exercises . . . . . .

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Appendix: Large Cardinals Historical Remarks

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6.A Vopenka's Principle . . . . . . . . . . . . . . . . . 6.B A Characterization of Locally Presentable Categories . 6.C A Characterization of Accessible and Axiomatizable Categories . . . . . . . . . . 6.D A Characterization of Reflective Subcategories . . . . . . . 6.E A Characterization of Accessible Functors . . . . . . . . . 6.F Colimit-dense Subcategories . . . .

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6 Vopenka's Principle

Exercises

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254 256 264 268 277 278

281 294

Open Problems

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Bibliography

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List of Symbols

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Index

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Preface The basic theme of our monograph is the syntax and semantics of a categorical theory of mathematical structures (as used in algebra, model theory, computer science, etc.). The semantics part is the study of properties of the categories of structures. We concentrate on two kinds of categories: locally presentable categories, which are rather close to quasivarieties of algebras, and the broader (and less "pleasant") accessible categories, which are close to classes of structures axiomatizable in first-order logic. The syntax part describes categories of structures by means of sketches: a sketch is a small category with specified limits and colimits, and a model of the sketch is a set-valued functor preserving the specified limits and colimits. Locally presentable categories are precisely the categories of models of limit-sketches (i.e., no colimit is specified), and accessible categories are precisely the categories of models of sketches. A different approach to syntax is by means of first-order logic: we characterize theories needed to axiomatize locally presentable and accessible categories. The fundamentals of the theory of locally presentable categories are exhibited in Chapter 1, and those of accessible categories in Chapter 2. The rest of our monograph is devoted to some related topics: algebraic categories, injectivity, categories of models, and Vopenka's large-cardinal principle. The book is completely self-contained: we expect the reader to be familiar with the basic category-theoretical concepts (such as adjoint, limit, etc.) which we mention briefly in the Preliminaries, but all the more advanced concepts are carefully explained in the text. Facts about large cardinal numbers, used in the last chapter, are presented in the Appendix.

Organization Every chapter is concluded by a set of easy exercises which illustrate some of the features of the text. References appear in the historical remarks. Open problems are listed at the end of the book. ix

x

PREFACE

Acknowledgements We are grateful to Hans-Eberhard Porst, Walter Tholen and Jiri Velebil for their suggestions on improvements of the text. Miroslav Dont undertook the extremely difficult task of turning our almost illegible manuscript into a perfect AMS-I&TEX file, which we greatly appreciate. For the diagrams he used the macro package ItMS-TEX of M. D. Spivak. April 1993

J. A. & J. R.

Introduction Locally presentable categories The concept of a locally presentable category is one of the most fruitful concepts of category theory. The definition, generalizing the concept of an algebraic lattice, is natural and simple. The scope is very broad: varieties and quasivarieties of (many-sorted) algebras, Horn classes of relational structures, and functor-categories are all locally presentable. Furthermore, locally presentable categories enjoy a number of important properties: they are complete and cocomplete, wellpowered and co-wellpowered, and they have a strong generator. The definition of a locally presentable category is due to P. Gabriel and F. Ulmer. Their lecture notes [Gabriel, Ulmer 1971], by now classical, are a profound, but by no means easily readable, treatise on the topic. One of

the aims of our monograph is to make the fundamentals of the theory of locally presentable categories more accessible to readers who work in category theory, computer science, and related areas. We have collected these fundamentals in Chapter 1, where the basic properties of locally presentable categories are proved and several equivalent ways of introducing these

categories are exhibited. For example, we show that locally presentable categories are precisely the categories sketchable by a limit sketch (i.e., the categories of all set-valued functors preserving specified limits).

Accessible categories These generalize locally presentable categories by weakening cocompleteness to the existence of some directed colimits. The collection of all categories obtained by this generalization is much broader than that of all locally presentable categories, and it includes categories such as fields and homomorphisms,

Hilbert spaces and linear contractions, xi

xii

INTRODUCTION

linearly ordered sets and order-preserving functions, sets and one-to-one functions.

An important special case: for each sketch in the sense of C. Ehresmann the category of set-valued models of the sketch (i.e., set-valued functors preserving specified limits and specified colimits) is accessible. It turns out that this is actually no special case: each accessible category is sketchable, i.e., equivalent to the category of all models of some sketch. This fundamental relationship between accessible and sketchable categories was discovered by [Lair 1981], who called accessible categories "categories modelables". The name "accessible" is due to [Makkai, Pare 1989], whose book

is a comprehensive treatise devoted to accessible categories. Our prime aim in presenting (in Chapter 2) the fundamentals of the theory of accessible categories is to make this theory easy to grasp. Thus, for example, our proof of the equivalence of accessible and sketchable categories is conceptually simpler than any previously published proof, being based on the concept of a pure morphism (a concept "borrowed" from model theory). Unlike M. Makkai and R. Pare, we do not stress the 2-categorical aspects of the theory, although some of the basic results (e.g., on limits of accessible categories) are addressed.

Algebraic categories Locally presentable categories are closely related to varieties and quasiva-

rieties of many-sorted algebras. We devote Chapter 3 to this interrelationship. We present J. R. Isbell's characterization of quasivarieties (i.e., implicational classes) of algebras as precisely the locally presentable categories with a regularly projective regular generator. Varieties are precisely the quasivarieties with effective equivalence relations. We also introduce the Lawvere-Linton concept of algebraic theory, and prove that varieties are precisely the categories of models of algebraic theories (= product sketches). We finally study the concept of essentially algebraic theory due to P. Freyd, which is an equational theory of partial operations in which the domain of definition of each operation is determined by equations in the "preceding" operations. We prove that locally presentable categories are precisely the categories of models of essentially algebraic theories-this is a folklore result whose proof has not been published before.

Injectivity and generalizations of locally presentable categories Some natural generalizations of locally presentable categories are studied in Chapter 4: weakly locally presentable categories, i.e., accessible cate-

INTRODUCTION

xiii

gories with weak colimits (equivalently: with products) and locally multipresentable categories, i.e., accessible categories with multicolimits (equivalently: with connected limits). These concepts are closely related to those of orthogonality or injectivity w.r.t. a morphism, or w.r.t. a cone. Weakly locally presentable categories are precisely the full subcategories of locally presentable categories which are specified by injectivity w.r.t. a set of morphisms. And locally multipresentable categories are precisely the full subcategories of locally presentable categories specified by orthogonality w.r.t. a set of cones.

Categories of models Chapter 5 deals with first-order logic. We call categories of models of first-order theories axiomatizable. In the finitary, many-sorted first-order logic L, we prove that categories axiomatizable by so-called limit theories are precisely the locally finitely presentable categories. This is a result in [Coste 1979]. More generally, in the A-ary logic LA categories axiomatizable by limit theories are precisely the locally A-presentable categories. We also exhibit a full characterization of accessible categories by means of (more general) basic theories: a category is accessible if it is axiomatizable by a basic theory in some of the logics LA. A crucial difference between the above two characterization results is that in the case of accessible categories we cannot restrict to a given A: we show an example (1) of a finitely

accessible category which cannot be axiomatized in L,, and (2) of a basic theory in L,,, whose category is not finitely accessible.

Vopenka's principle Some results on locally presentable and accessible categories depend on the existence of certain large cardinal numbers. Notably, the large-cardinal Vopenka's principle turns out to be equivalent to important properties of locally presentable categories. Vopenka's principle implies e.g. that (*)

a category is locally presentable if it is cocomplete and has a colimit-dense set of objects.

This is quite surprising because it is thus possible to define "locally presentable" without a reference to the concept of presentable object. (The explanation is that, under Vopenka's principle, in each category with a colimit-dense set all objects are presentable.) Conversely, the statement (*) implies Vopenka's principle. Thus, large cardinal numbers turn out to have a close link to locally presentable categories.

xiv

INTRODUCTION

We devote Chapter 6 to the role of Vopenka's principle in the theory of locally presentable and accessible categories. All concepts concerning large

cardinal numbers which are needed in that chapter are explained in the Appendix.

0.

Preliminaries

This monograph is devoted to a theory of locally presentable categories and accessible categories. We assume that the reader has basic knowledge of categories, functors, and adjoints, but we are careful to explain all the necessary concepts of model theory, logic, and set theory, as well as all the more advanced categorical notions in the text. We have concentrated all the required facts concerning cardinal numbers in the Appendix. We now recall some conventions and facts of category theory necessary for avoiding later misunderstandings. The proofs of the (standard) statements presented here can be found e.g. in [Adamek, Herrlich, Strecker 1990]"`.

0.1 Set Theory. We distinguish, as in the Bernays-Godel set theory, between sets and classes. Until Chapter 6 this is all that need be said-in other words, we just use naive set theory with a distinction between "small"

and "large". But we use transfinite induction frequently; thus, the axiom of choice (for classes) is assumed without mention. The first infinite cardinal is denoted by w or Ho, the next one by w1 or R1. Categories K are understood to be locally small, i.e., objects and morphisms form classes K°bi and Kmor respectively, whereas hom(A, B) is a set (for any pair A, B of objects). A class of objects of a category is called essentially small if it has a set of representatives w.r.t. isomorphism.

0.2 Composition is written from right to left, that is, if f : A -* B and g : B -> C are morphisms, then g f [or g f] is their composite.

0.3 Comma-categories. For each object K of a category K we form the comma-category K 1 K of all arrows with the domain K, whose morphisms

from K - A to K -b + B are the K-morphisms f : A --+ B with b = f a. Dually, K j K denotes the comma-category of all arrows with the codomain K. *References to the literature listed at the end of our book are denoted by square brackets. 1

0.

2

PRELIMINARIES

0.4. By a diagram in a category K is meant a functor D : D -> K from a small category D (called the scheme of the diagram D). The diagram D is said to be finite if D has finitely many morphisms. A category is called (co)complete provided that every diagram in it has a (co)limit.

Definition. Let A be a small, full subcategory of a category K. For each object K in K the canonical diagram of K (w.r.t. A) is the diagram of all arrows A --> K where A lies in A; more precisely, the canonical diagram is the natural forgetful functor D: A J. K -> K. We say that K is a canonical colimit of A-objects provided that the canonical diagram has a colimit with the colimit-object K and the colimit maps D(A -+ K) °-> K.

0.5 Hierarchy of Monomorphisms. A monomorphism m : A -+ B is called

(1) regular if m is an equalizer of some pair fl, f2: B --f C; (2) strong if each commuting square

peQ fl A

19

rn

B

such that e is an epimorphism has a diagonal fill-in (i.e., a morphism

d:Q-+Awith f (3) extremal if every epimorphism e : A -+ A' through which m factorizes is an isomorphism. Every regular monomorphism is strong, and the converse is true in each category with (epi, regular mono)-factorizations of morphisms. Every strong monomorphism is extremal, and the converse is true in each category with pushouts. Every complete, wellpowered category has (epi, extremal mono)-factorizations as well as (extremal epi, mono)-factorizations of morphisms.

0.6 Generators. A set C of objects of a category is called a generator provided that for each pair fl, f2: K , K' of distinct morphisms there exists an object G E G and a morphism g : G -+ K with fl - g # f2 g. The dual concept is cogenerator. A generator G is called strong provided that for each object K and each

proper subobject of K there exists a morphism G - K with G E 9 which

0.

PRELIMINARIES

3

does not factorize through that subobject. A shorter definition is possible in a cocomplete category: 9 is a strong generator if every object is an extremal quotient of a coproduct of g-objects. (It would be more reasonable, but unfortunately less standard, to call C an extremal generator.) Every category K with a strong generator 9 is wellpowered, i.e., each object has only a set of subobjects.

0.7 Adjoint Functors. A functor F: K -> .f is right adjoint to a functor G : G --* K provided that there exists a natural isomorphism

hom(G-, -) = hom(-, F-). Notation: G -i F. We often use Freyd's adjoint funclor theorem: if K is a complete category, then a functor F : K --> G is a right adjoint iff F preserves

limits and satisfies the solution-set condition (which says that for each object L in G there exists a set of arrows L FK= in L such that every arrow L L FK factorizes as f = Fk f; for some i). We also have Freyd's special adjoint functor theorem: if K is a complete, wellpowered category with a cogenerator, then a functor F : K -* G is a right adjoint if F preserves limits. 0.8 Reflective Subcategories. A subcategory A of a category K is called (1) isomorphism-closed provided that for each isomorphism i : A -> A' in K with A in A we have A' and i in A too,

(2) closed under limits if every limit cone in K of a diagram in A lies in A,

(3) reflective provided that the inclusion functor A - K is right adjoint. The latter means that each object K of K has a reflection map rK : K -+ A,

A E A, with the universal property that each morphism from K into an A-object uniquely factorizes through rK by an A-morphism. If each rK is an epimorphism, A is said to be epireflective in K. For complete, wellpowered, and co-wellpowered categories K the following holds: a full, isomorphism-closed subcategory of K is epireflective if it is closed in K under products and extremal subobjects.

0.9 The Yoneda Lemma. For each small category K the Yoneda embedding is the functor Y: K -> Seth " assigning to each object K of K the contravariant hom-functor hom(-, K) : K°P -f Set, and to each morphism

f : K -* K' of K the natural transformation hom(-, f ): hom(-, K) --+

4

0.

PRELIMINARIES

hom(-, K') defined via composites with f. The fact that Y is a full embedding follows from the Yoneda lemma: for each functor F in and each natural transformation r: hom(-, K) -+ F there exists a unique element x E FK with 7A(h) = Fh(x) for all h E hom(A, K). SetK.P

0.10 Cones. A set of morphisms with a common domain A is called a cone with domain A. Special cases: every morphism f : A B is considered to be a cone with domain A, and every object A is considered to be the empty cone with domain A.

0.11 Cofinal Subdiagrams. A functor H : Do --+ V is said to be cofinal provided that for each object d in D (a) there exists a morphism f : d - Hdo for some object do in Do, and

(b) given two such morphisms f : d Hdo and f: d -+ Hdo, there exist morphisms g: do -, do and g : do -> do in Do such that the square d

f

F1 Hdo

Hdo

JHg'

Hg

y Hdo

commutes.

Observation. For each cofinal functor H: Do -+ V the categories Do and D are "equivalent as diagram schemes" w.r.t. colimits: (1) a category has colimits over D if it has colimits over Do, and (2) a functor preserves colimits over D if it preserves colimits over Do. In more detail: let D: D - K be a diagram. There is a bijective correspondence between compatible cocones of D and those of D D. H (hence,

a bijective correspondence between colimits of D and D H). In fact: (1) Given a compatible cocone (Dd-`- C)dEDobj for D, then the cocone (DHdo

cHdo

C)doEDpb'

is compatible for D H.

(2) Given a compatible cocone (DHdo c- C)d.EDob, for D H, then choose a morphism f : d , Hdo for each d in D. It is clear that ad = cd,, D f : Dd -+ C is independent of the choice off and do, and that the cocone (Dd - C)dEDoby is compatible for D.

0.

PRELIMINARIES

5

Remark. In particular, a cofinal subdiagram of a diagram D: V -* K is a subdiagram Do: Do --+)C (i.e., Do is a subcategory of D and Do = D/D0) such that the inclusion functor Do y D is cofinal. 0.12 Equivalence of Categories is a full and faithful functor E : K -> ,C which is isomorphism-dense, i.e., each object of C is isomorphic to an object

of E(K). We call K and C equivalent; the notation for this is K : G. The notation for isomorphic categories is K = L.

0.13 The Quasicategory of all Categories. We denote by Cat the category of all small categories and all functors. On two occasions we also refer to the quasicategory CAT of all categories and all functors. A quasicategory is defined as a category except that it lives beyond the universe of our set theory-thus, all objects may form a collection which fails to

be a class, and the same is true about any hom(A, B). (We are vague here because the use is so infrequent and unimportant that an effort for axiomatization would be wasted. The reader may consult the monograph [Adamek, Herrlich, Strecker 1990].)

Chapter 1

Locally Presentable Categories The first chapter is devoted to an important class of categories, the locally presentable categories, which is broad enough to encompass a great deal of mathematical life: varieties of algebras, implicational classes of relational structures, interesting cases of posets (domains, lattices), etc., and yet restricted enough to guarantee a number of completeness and smallness properties. Besides, locally presentable categories are closed under a number of categorical constructions (limits, comma-categories), see also Chapter 2. The basic concept, a finitely presentable object, can be regarded as a generalization of the concept of a finite (or compact) element in a Scott

domain, i.e., an element a such that for each directed set {di i E I} with a < ViEI di it follows that a < di for some i E I. Now, an object A is I

finitely presentable if for each directed diagram {Di I i E I} every morphism A -i colimiEr Di factorizes (essentially uniquely) through Di for some i E I. More generally, an object A is A-presentable (for a cardinal A) if every morphism from A to a A-directed colimit colimiEr Di factorizes (essentially

uniquely) through some Di. A category is locally A-presentable if it has colimits and is generated (in some strong sense) by a set of A-presentable objects. We will see that there are many equivalent ways in which locally A-presentable categories can be introduced: they are precisely (1) the cocomplete categories in which every object is a A-directed colimit of A-presentable objects of a certain set (Definition 1.17);

(2) the cocomplete categories with a strongly generating set of)-presentable objects (Theorem 1.20); 7

CHAPTER 1. LOCALLY PRESENTABLE CATEGORIES

8

(3) the full, reflective subcategories of categories of relational structures closed under A-directed colimits (Corollary 1.47); (4) the categories of A-continuous set-valued functors (Theorem 1.46); (5) the A-free cocompletions of small categories (Theorem 1.46); (6) the A-ary essentially algebraic categories (Theorem 3.36); (7) the categories of models of A-small limit-sketches (Corollary 1.52); (8) the categories of models of A-ary limit-theories (Theorem 5.30). The role that the cardinal A plays here is analogous to, say, an upper bound on the arity of operations in universal algebra. We begin with the important case of finitely presentable categories, i.e., with A = Ro. The concept of a locally presentable category is closely related to that

of a small-orthogonality class, i.e., the class M' of all objects orthogonal to morphisms of a given set M. We prove that (a) each locally presentable category is equivalent to a small-orthogonality class of some functor-category SetA, and (b) every small-orthogonality class in a locally presentable category is locally presentable [Theorems 1.46 and 1.39]. A concept often easier to work with than A-directed colimits is that of A-directed unions. We show that locally presentable categories can equivalently be introduced by means of A-directed unions-this is the local generation theorem 1.70 (which is, somewhat surprisingly, technically rather difficult).

1.A

Locally Finitely Presentable Categories

The concept of a locally finitely presentable category can be viewed as a direct generalization of the concept of an algebraic lattice. Recall that a non-empty partially ordered set is called directed provided that each pair of elements has an upper bound. An element a of a partially ordered set (K, C)iE1 be

a directed colimit in Set. For each function f : K --> C and each element x E K there exists i., E I such that f (x) lies in the image of ci..

Since K is finite, and I is directed, there exists an upper bound i E I of all i,, (x E K); thus, f(K) C ci(Di). This implies that f factorizes through ci. To show that the factorization is essentially unique, use the following property of directed colimits in Set (see Exercise 1.a): whenever elements y, y' E Di fulfil ci(y) = c2 (y'), then there exists

j E I with i < j such that D(i - j)(y) = D(i -+ j)(y'). (2) For each set S (of sorts) let Sets denote the category of S-sorted sets, i.e., collections X = (X,),ES of sets X, indexed by S, and Ssorted functions f: X --> Y, i.e., collections f = (f,),ES of functions f,: Xs , Y, indexed by S. For each S-sorted set X we call the cardinal

#X = E card X, sES


10

the power of X. An S-sorted set is finitely presentable in Sets if it has finite power. The proof is analogous to (1) since (directed) colimits in Sets are computed coordinate-wise.

(3) In the category Pos of posets (partially ordered sets) and order-preserving functions the finitely presentable objects are precisely the finite ones. The proof is analogous to (1) since directed colimits are computed on the level of Set.

Analogously in the category Gra of graphs (i.e., sets endowed with a binary relation) and homomorphisms (i.e., functions preserving the binary relation), the finitely presentable objects are precisely the finite graphs.

(4) Let S be a set of sorts, and let E be a (finitary, S-sorted, relational) signature. That is, with each symbol a E E we are given an arity ar D' =

(sl, ... , sn) E Sn. (If n = 1, then ar o = s means that o- is a unary symbol of sort s, if n = 2, then o- is a binary symbol of sorts s1, s2; etc. The case n = 0 is denoted by ar a = 0; this is a nullary symbol a.) A relational structure A of type E consists of an (underlying) S-sorted set JAI = (A$),Es and, for each o E E, of a relation 0A C A,, x A,2 x

x A,,,, where ar a = (sl, ... , sn). (If n = 0, then 0A C_ A0, where A0 is a terminal object. Thus, we just distinguish between two cases: O'A = 0 or QA # 0.) Let Rel E denote the category of relational structures of type E, where morphisms f : A --> B are the homomorphisms, i.e., S-sorted functions f : JAI , JBI such that for each o E E of arity (sl, ... , s/,) with n > 0 we have that (x1,...,xn) E o'A implies

(f,i(x1),...,f3, (xn)) E OB

and, for each o of arity 0, if vA # 0 then o-B # 0. A relational structure A is finitely presentable in Rel E if it has finitely many vertices (i.e., JAI has finite power) and finitely many edges (i.e., EIEE card oA is finite). The proof is analogous to (1) above. (5) A group A is finitely presentable in Grp, the category of groups and homomorphisms, if it can be presented by finitely many generators and finitely many equations in the usual algebraic sense. (That is, if A is isomorphic to the quotient group of the free group F{xi}L 1 generated by {x1,. .. , xn} modulo a congruence generated by finitely many equations on F{xi} 1.) For example, (Z, +) is finitely presentable and (118, +) is not.

In general, in each variety of finitary algebras, an algebra is finitely presentable if it can be presented by finitely many generators and finitely

1.A. LOCALLY FINITELY PRESENTABLE CATEGORIES

11

many equations in the usual algebraic sense. A full proof will be presented later (see Theorem 3.12).

(6) Let Aut be the category of (deterministic, sequential) automata: objects are sixtuples A = (Q, I, 0, qo, 6, 3) where Q is a set of states, I is a set of input symbols, 0 is a set of output symbols, qo E Q is the initial state, 6: I x Q Q is the next-state map, and

/3: Q --+ 0 is the output map.

Morphisms from A to A' _ (Q', I', O', qo, 6', 3') are triples (f, i, o) of functions f : Q --+ Q', i : I -- I', and o : 0 -+ O' satisfying (i) f(qo) = qo,

(ii) f (s(q, x)) = 6'(f (q), i(x)), (iii) 0'(f (q)) = Q(o(q)),

for all states q E Q and all inputs x E I. Composition is defined coordinate-wise, and the identity morphisms are (idQ, idi, ido).

An automaton is finitely presentable if each of the sets Q, I, and 0 is finite. The proof is analogous to that in (1) since directed colimits in Aut are computed coordinate-wise.

(7) Let A be a small category. By the Yoneda lemma (see 0.9), every hom-functor is a finitely presentable object of SetA.

(8) Let CPO denote the category of CPO's, i.e., complete posets (posets in which every directed set has a join) and continuous functions (i.e., functions preserving all directed joins). No non-empty object is finitely presentable in CP0. In fact, consider the following directed diagram D of inclusions of linearly ordered CPO's: {0} C {0, 1} C {0, 1, 2} C .... A colimit in CPO can be described by the inclusions of those CPO's into wT = {0, 1, ... , n,... } U {T}. Now let K be a non-empty CPO and

let f : K --+ C be the constant map of value T. This is a continuous function which does not factorize through any of the colimit maps of D.

12


(9) In the category CSLat of complete semilattices ( = complete lattices) and join-preserving homomorphisms no object of more than one element is finitely presentable-the proof is analogous to that in (8) above.

(10) A topological space is finitely presentable in Top, the category of topological spaces and continuous functions, if it is finite and discrete. In fact, any topological space A with a non-open subset M C A fails to be finitely presentable: consider the sequence D of topological spaces for n E N = { 0, 1, 2.... }, where D is the space on the disjoint union

A + N of A and N such that a subset of D,1 is open if its intersection with A is open in A and its intersection with N is disjoint with 10, 1, ..., n - 11. The colimit of D = (Dn)nEN is the indiscrete space on the set A+ N. The canonical injection of A into colim D does not factorize through any of the colimit morphisms D ---* colim D.

1.3 Proposition. A finite colimit of finitely presentable objects is finitely presentable. PROOF. Let D: V K be a finite diagram with colimit (Dd -°+ K)dED°bi Then we will prove that K is finitely presentable provided that each Dd is finitely presentable. Suppose D* : (I, D?1 with D(io -> ii) 9d = g kd (d E D°bi). Consequently, kd

for all d E D°bi, which implies f = c;, g. To prove that f factorizes through c;, essentially uniquely, consider g' : K --> D;1 with c;, g = c=, g'. For each d E D°bi we have two factorizations of f kd:

f kd=C,1 .g.kd=cil - g' - kd which (since Dd is finitely presentable) implies that there exists j(d) with D(ii -+ j(d)) g kd = D(ii --> i(d)) g' kd. Finally, let j be an upper bound of all j(d), d E D°bi, then D(il j) g kd = D(il --+ j) g' kd (for all d). This implies D(ii -> j) g = D(ii --> j) g'. Remark. Consequently, a split subobject (or a split quotient) of a finitely presentable object A is finitely presentable: we can express it by a coequalizer of two endomorphisms of A. In contrast, a quotient (or, dually, a subobject) of a finitely presentable object need not be finitely presentable: consider an algebraic lattice as a category, then every object is a quotient of the initial object. Even a regular subobject can fail to be finitely presentable. For example, in the category of lattices the free lattice on three generators contains sublattices which are not finitely presentable, see [Whitman 1941].

Directed and Filtered Colimits A number of authors prefer working with filtered rather than directed, colimits. (The obvious reason is that canonical diagrams are often filtered, but not directed.) In this subsection we will show that those two concepts are equivalent.

1.4 Definition. A category D is called filtered provided that every finite subcategory of V has a compatible cocone in D. In other words, (1) D is non-empty, (2) for each pair D1, D2 of objects there exists an object D and morphisms

fl: D1-->Dand f2: D2-+ DinV, (3) for each pair g, g': D1 -> D2 of morphisms in V there exists a mor-

phism f : D2 --+D in D with f g = f g'.

14


Observe that (2) and (3) imply that every pair of morphisms with a common domain can be completed to a commuting square. Every directed poset, considered as a category, is filtered. The category

with one object and two morphisms id, f satisfying f f = f is filtered. A filtered diagram in a category K is a diagram D : V --. K whose scheme V is filtered. Colimits of such diagrams are called filtered colimits. We will

show how to "reduce" every filtered diagram to a directed diagram. To make the concept of reduction precise, we use cofinality (see 0.11):

1.5 Theorem. For every (small) filtered category D there exists a (small) directed poset Do and a cofinal functor H: Do --r D. Remark. The following proof is somewhat more technical than the reader might expect at first sight. To realize the difficulty, consider the simple

case where D has just one object d and two morphisms id and f = f f. Here Do has to be an infinite directed category such as d --f-+ d f

d L d ....

PROOF. I. Let us first suppose that V has the following property: every finite subcategory of V can be extended into a finite subcategory with a unique terminal object. Then the set I of all subcategories of D with a unique terminal object, ordered by inclusion, is obviously directed: given two such subcategories Al, A2, we extend Al UA2 to a subcategory A with a unique terminal object, and we have an upper bound of Al, A2 in I. The functor H : (I, C) --+ V defined by

H(A) = the terminal object of A H(A --* A') = the unique A'-morphism from H(A) to H(A') is cofinal. In fact, for each object d we have idd : d - H{d}. Given fi : d -> HA, (i = 1, 2), let A E I contain Al U A2 U If,, f2}, then H(Al --* II. Let V be an arbitrary filtered category. Then the category D xw (where w is the linearly ordered category of natural numbers) is also filtered, and it has the property required in I above. In fact, if a subcategory A of D x w has a compatible cocone with codomain (A, n), then the object (A, n + 1)

is the unique terminal object of the following extension of A: add to A the object (A, n + 1), its identity morphism and all composites of the given cocone and the canonical morphism (A, n) -> (A, n + 1). The projection functor D x w -* D is, obviously, cofinal. By I, we have a cofinal functor

from a directed poset into D x w. It is clear that the composite of two cofinal functors is cofinal.

LA. LOCALLY FINITELY PRESENTABLE CATEGORIES

15

Corollary. A category has filtered colimits if it has directed colimits. For such categories K, a functor F: K --> C preserves filtered colimits if it preserves directed colimits. 1.6. We have reduced filtered colimits to directed colimits. We will make a further step, and reduce directed colimits to colimits of chains (= wellordered diagrams, or, diagrams whose schemes are ordinals).

Lemma. Every infinite directed poset (I, K)iEI. Since K is finitely presentable, idA factorizes through

some ki, i.e., there exists m: K -. Di (i E I) with ki m = idK. Thus, K is a split subobject of Di E A.

1.10 Examples (1)

Set is locally finitely presentable. In fact (i) every set is a directed colimit of the diagram of all of its finite subsets (ordered by inclusion), and (ii) there exists, up to isomorphism, only a (countable) set of finite sets.

Analogously, Pos, Rel E, Grp, and Aut are locally finitely presentable categories. (2) Every variety of finitary (many-sorted) algebras is locally finitely presentable, as will be proved in Chapter 3.

(3) CPO and Top are not locally finitely presentable. (4) The category of finite sets is not locally finitely presentable since it is not cocomplete.

(5) A poset, considered as a category, is locally finitely presentable if it is a complete lattice which is algebraic (i.e., each element is a directed join of finite elements).

A Criterion for Local Finite Presentability In the definition of locally finitely presentable category K an important weakening is possible: instead of a set A of finitely presentable objects which "generates" all of K via directed colimits, it is sufficient to require that A be a strong generator (see 0.6): 1.11 Theorem. A category is locally finitely presentable if it is cocomplete, and has a strong generator formed by finitely presentable objects. PROOF. The necessity is clear. To prove the sufficiency, let K be a cocomplete category with a strong generator A formed by finitely presentable

objects. Let A be a closure of A under finite colimits (i.e., the smallest subcategory of K closed under finite colimits and containing A). It is clear


18

that A is essentially small and, by Proposition 1.3, objects of A are finitely presentable. (In fact, the collection of all finitely presentable objects is closed under finite colimits and, since it contains A, it must contain A.) It is sufficient to prove that every object of IC is a filtered colimit of objects of A. For each object K_ we can form the canonical diagram D w.r.t. A (see Definition 0.4). Since A is closed under finite colimits, D is clearly filtered.

Put K* = colim D and for each morphism f : A -* K with A in j denote the corresponding colimit morphism by f*: A -+ K*. Let m: K* , K be the unique morphism with f = m f * for each f . We are going to show that m is an isomorphism. It is sufficient to verify that m is a monomorphism: since A is a strong generator and each morphism from an A-object into K factorizes through in, it then follows that m is an isomorphism.

Given p, q: B -> K* with m p = m q, we will prove that p = q; it is sufficient to prove this in the case B E A, since the general case follows from the fact that A is a (strong) generator. The diagram D is filtered, and B is finitely presentable. Thus, there exists f : A -> K, for A in A, such

that both p and q factorize through f*. That is, we have p', q': B --+ A with p = f * p' and q = f * q'.

Let c: A -> C be a coequalizer of p' and q'. Since A and B lie in A, it follows that C also lies in j, thus, the unique morphism g : C --+ K with f = g c belongs to the diagram D. Since f* = (g c)* = g* c, we have

0

1.12 Example. For each small category A the category SetA of all functors from A to Set is locally finitely presentable: it is cocomplete, and the set of all hom-functors (which are finitely presentable objects) is a strong generator.

1.B

Locally Presentable Categories

Analogously to the transition from finitary to infinitary algebras, we now generalize the concept of a locally finitely presentable category. We use

1.B. LOCALLY PRESENTABLE CATEGORIES

19

a regular cardinal A, i.e., an infinite cardinal which is not a sum of a smaller number of smaller cardinals. More precisely, A is a regular cardinal if it is infinite and cannot be expressed as A = Eii A.

1.14 Examples (1) A set is A-presentable in Set if it has cardinality smaller than A. An S-sorted set is A-presentable in Sets if it has power smaller than A.

(2) Analogously to finitary relational structures (see Example 1.2 (4)) we can introduce A-ary relational structures: Let S be a set of sorts and

let E be a A-ary S-sorted relational signature. That is, with each symbol a E E we are given an arity which is a collection (si)iE' of

sorts si E S with card I < A. A relational structure A of type E consists of an underlying S-sorted set JAI = (A,),Es and of relations 0'A C _ R EI A,; (for each a E E of arity (si)iEi). The category Rel E of relational structures of type E has as morphisms all homomorphisms, i.e., S-sorted maps preserving the corresponding relations.

For each A-ary signature E, an object A of Rel E is A-presentable if it has less than A vertices, i.e., #JAI < A, and less than A edges, i.e., EIEE card UA < A.

20


(3) By a convergence space on a set X is meant a relation --> ("converges") between w-sequences in X and elements of X such that (a) for each x E X we have i -> x for the constant sequence i =

(x, x, x,

... );

(b) if (xn)n<w of (xn).

x then (xnk)k<w --> x for each subsequence (xnk)

The category Con of convergence spaces is a full subcategory of Rel E

where E consists of one one-sorted w-ary symbol o (provided that (xo, x1) x2, ...) E o is rewritten as (x1, x2, 53, ...) -+ xo). That is, morphisms of Con are the continuous functions, i.e., functions preserving convergence.

A convergence space is A-presentable, where A > w1, if its cardinality is smaller than A.

(4) No non-empty CPO is presentable (in the category CPO); the proof is analogous to Example 1.2 (8). Analogously, no complete semilattice of more than one element is presentable in CSLat (cf. Example 1.2(9)).

Let w CPO denote the category of wCPO's (i.e., posets in which every increasing w-chain has a join) and w-continuous maps (i.e., maps preserving joins of w-chains). Then each finite wCPO is w1-presentable, and an infinite wCPO is A-presentable if it has cardinality smaller than A. (5)

For each variety V of finitary algebras and each uncountable regular cardinal A larger than the number of basic operations, an algebra is A-presentable in V if it has power less than A. (See Corollary 3.13.) In contrast, let V be the variety of algebras of one w-ary operation with no equations. There exist arbitrarily large regular cardinals A such that a A-presentable algebra in V can have A or more elements. (In fact, let a be a cardinal cofinal with w, and let A be a successor of a. Then the V-free algebra A on a generators is A-presentable, but since aw > a, the cardinality of A is larger than a and, thus, larger or equal to A.)

(6) No non-discrete space is presentable in Top; the proof is analogous to that in Example 1.2(10).

1.15 Definition. A diagram D : V -+ /C is called A-small provided that its scheme D has less than A morphisms.

1.B. LOCALLY PRESENTABLE CATEGORIES

21

1.16 Proposition. A colimit of a A-small diagram of A-presentable objects is A-presentable.

The proof is analogous to that of Proposition 1.3. Remark. Consequently, a split quotient of a A-presentable object is A-presentable. In contrast, neither a subobject nor a quotient of a A-presentable object is A-presentable in general, see Remark 1.3.

1.17 Definition. A category is called locally A-presentable (A a regular cardinal) provided that it is cocomplete, and has a set A of A-presentable objects such that every object is a A-directed colimit of objects from A. A category is locally presentable if it is locally A-presentable for some regular cardinal A.

1.18 Examples (1)

Locally w-presentable categories are precisely the locally finitely presentable ones.

(2) w CPO is locally w1-presentable but not locally finitely presentable (see Example 1.14(3)).

(3) A poset, considered as a category, is locally presentable if it is a complete lattice. The chain LA (A regular) of all ordinals smaller or equal to A is locally A-presentable, but not locally A'-presentable for

any A' A, and that every object of K is presentable. (See Proposition 1.16.) 1.21 Remark. Analogously to Theorem 1.5, A-directed diagrams can be substituted by A-filtered diagrams in the definition of A-presentability. A category V is called A-filtered provided that each subcategory of less than A morphisms has a compatible cocone in D. This means that (1) D is non-empty,

(2) for each collection Di, i E I, of less than A objects of D there exists an object D and morphisms fi : Di --> D, i E I, in V, (3) for each collection gi : Dl --> D2, i E I, of less than A morphisms in D there exists a morphism f : D2 --> D in D with f gi independent of i.

For every (small) A-filtered category D there exists a (small) A-directed poset Do and a cofinal functor H : Do D. Thus A-filtered and A-directed colimits are "equivalent", see 0.11. In particular, an object is A-presentable if its hom-functor preserves A-filtered colimits. All this is quite analogous to Corollary 1.5. In contrast, the step from directed colimits to colimits of chains (see Corollary 1.7) does not generalize to A: a category with wi-directed colimits of chains need not have wi-directed colimits in general, see Exercise l.c. The reason why A-filtered diagrams are preferred by some authors to A-directed ones is obvious: whereas in the definition of locally A-presentable category each object is presented by some (unspecified) A-directed colimit

of A-presentable objects, one can be more specific and use the canonical colimit (see 0.4), as we shall see now. However, the canonical colimit is A-filtered, not A-directed.

1.22 Proposition. For each object K of a locally A-presentable category the canonical diagram w.r.t. Presa K is A-filtered, and K is its canonical colimit.

PROOF. The first statement is an immediate corollary of Proposition 1.16: for A = Presa K, any subcategory D of A f. K of less than A morphisms has a compatible cocone, e.g., a colimit of the forgetful functor D , K.

I.B. LOCALLY PRESENTABLE CATEGORIES

23

Next, for each object K there exists a A-directed diagram D of )-presentable objects with a colimit (Di --'> K);EI. To prove that the canonical diagram w.r.t. A has the canonical colimit K, it is sufficient to observe that the objects Di - K of A I K form a cofinal sub diagram (see 0.11) of the canonical diagram: each arrow A -°-- K in A I K factorizes essentially uniquely through some ki (since hom(A, -) preserves the A-directed colimit of D). Now use Exercise 1.o(3).

1.23 Definition. A small, full subcategory A of a category K is called dense provided that every object of K is a canonical colimit of A-objects (see 0.4).

Remark. Every dense subcategory is a strong generator. The converse does not hold, e.g., the vector space 118 forms a (singleton) strong generator in Vec which is not dense (see the Example 1.24(4)).

1.24 Examples (1)

In a locally A-presentable category we have seen that Presa K is a dense subcategory.

(2) In the category Pos the single-object subcategory consisting of the two-element chain is dense.

(3) No single-object subcategory of Gra is dense: given a graph A = (JAI, a), then either A is discrete (i.e., a = 0), hence no non-discrete object is a colimit of copies of A, or A is non-discrete, and then no non-empty discrete object is a colimit of copies of A. The full subcategory consisting of A = ({0},0)

and

B = ({0,1},{(0,1)})

is dense in Gra. (4) The difference between canonical and arbitrary colimits is well illustrated by the vector space 1[8 in the category Vec of real vector spaces: although every object is a coproduct of copies of 1R, it is not true that JR is dense. To verify this, consider a map f : A -, B between vec-

tor spaces which is homogenous (i.e., f(rx) = r f (x) for all r E

IR

and x E A) but not additive. The passage R

A

--*

TR - + B

is a natural transformation of the canonical diagrams. (In fact, if a is

24


linear, then each f a is homogenous and, thus, linear.) However, since f is not linear, the natural transformation above is not induced by any morphism from A to B, which demonstrates that A is not a colimit of its canonical diagram w.r.t. On the other hand, the space 118 x 118 is dense in Vec. (5)

In a variety of (one-sorted) algebras where all arities are smaller or equal to n the free algebra on n generators forms a singleton dense subcategory.

(6) In the category of complete lattices and complete homomorphisms no small subcategory is dense. (This follows easily from the fact that for each cardinal A there exists a A-complete lattice which is not complete.) (7) The category Top of topological spaces and continuous maps also has no dense subcategory. This follows easily from the fact that for each cardinal A there exists a topological space which is not discrete (i.e., not every subset is open) but whose subspaces of cardinality smaller than A are all discrete.

1.25 Notation. Given a small, full subcategory A of a category K we define the canonical functor E: K -> SetA°P

analogously to the Yoneda embedding (0.9): E assigns to each object K the restriction EK = hom(-, K)/Aop of its hom-functor to A°p, and to each morphism f : K -+ K' the domaincodomain restriction of the Yoneda transformation hom(-, f).

1.26 Proposition. Let A be a small, full subcategory of a category K. The canonical functor E: K -->

SetA-P

(i) is full and faithful iff A is dense,

(ii) preserves A-directed colimits iff every object of A is A-presentable in K.

PROOF. (i) is trivial since morphisms from EK to EK' in SetA°P are precisely the compatible cocones of the canonical diagram of K with the codomain K'.

I.B. LOCALLY PRESENTABLE CATEGORIES

25

(ii) Let E preserve A-directed colimits. Given a morphism f : A --> K with

A in A and given a A-directed colimit (Ki K)iEI in K, then since f E (EK)A and EK is a A-directed colimit of EKi, there exists i E I with f E (Eki)A; i.e., f factorizes through ki. The essential uniqueness of such a factorization follows from the description of directed colimits in Set (see Exercise 1.a). Conversely, let each object of A be A-presentable. Given a A-directed colimit (Ki -4 K)iEI in K and given an element f E (EK)A = hom(A, K), there exists an (essentially unique) i c I such that f factorizes through ki, (Eki)A

i.e., f e (Eki)A[(EKi)A]. This proves that the cocone ((EKi)A (EK)A) iEI is a (A-directed) colimit in Set, see Exercise l.a. Since colimits' in SetA are computed object-wise, we conclude that (EKi "' EK)iEI is a colimit in SetA .

1.27 Proposition. For each small, full subcategory A of a cocomplete category K the canonical functor E: K - SetA*P is a right adjoint. PROOF. Given an object F: A°P -+ Set of SetAOP, let V be the category of pairs (A, a) where A is an object of A and a E FA, with morphism f : (A, a) -a (A', a') those K-morphisms f : A -> A' which fulfil a = F f (a'). The diagram given by the forgetful functor D : D --r K has a colimit (A kA K)AEA,aEFA This defines a natural transformation r: F -* E(K) by rA(a) = kA,a E hom(A, K)

for A E A, a E FA.

r is universal since, given a natural transformation r': F -- E(K'), then the cone (A r"W K')AEA,aEFA is compatible with D, and the unique K' with r'A(a) = k rA(a) (A E A, a E FA) is the morphism k : K unique K-morphism with r' = E(k) r. 1.28 Corollary. Every locally presentable category is complete. In fact, by Proposition 1.26 every locally presentable category is equivalent to a full subcategory of SetA0 , and by Proposition 1.27 this subcategory is reflective (thus, complete).

1.29 Corollary. For each category K the following are equivalent: (i) K is equivalent to a full, reflective subcategory of a locally presentable category,

26


(ii) K is equivalent to a full, reflective subcategory of SetA for some small category A,

(iii) K is cocomplete and has a dense subcategory. In fact, (iii) = (ii) follows from Propositions 1.26 and 1.27, (ii) (i) is clear, and to prove (i) = (iii), observe that if a category L has a dense subcategory Go, then for each full, reflective subcategory K of L the reflections of Go-objects in K form a dense subcategory of K. 1.30 Remark. Let K be a locally A-presentable category. For each regular cardinal p > A we know that (1) a p-small colimit of A-presentable objects is p-presentable (see Proposition 1.16).

The converse is also true:

(2) every p-presentable object is a p-small colimit of A-presentable objects. The proof (which is rather technical) can be found in [Makkai, Pare 1989], pp. 35-37. However, the following weaker statement is trivial: each p-presentable object K is a split quotient of a p-small colimit of A-presentable objects. In fact, express K as a A-directed colimit (Di - K)iEI of A-presentable objects Di. Consider the poset 1* of all subsets J C I with card J < p, ordered by inclusion. For each J form a full subdiagram (Di)iEJ of D, and

let D) be a colimit of this subdiagram. We obtain a natural p-directed diagram D* of all Dg's and all the canonical arrows D* D*, for J C P. It is clear that K is a colimit of this diagram, and since K is p-presentable, idK factorizes through some of the colimit maps dj : D* -> K. Thus, dJ is a split epimorphism.

1.31 Example. For each small category A and each regular cardinal A > cardA°bi, a functor F is A-presentable in SetA if EXEAob; card FX < A. In fact, if F satisfies that inequality, then it is a colimit of less than A homfunctors (by the Yoneda lemma 0.9) and since A > card A°bi, that diagram is A-small-thus, F is A-presentable by Proposition 1.16. Conversely, if F is A-presentable, then it belongs to the (iterated) closure of hom-functors under A-small colimits, see Remark 1.30. Each hom-functor satisfies the above inequality, and a A-small colimit of functors satisfying that inequality satisfies it too.

1.C. REPRESENTATION THEOREM

1.C

27

Representation Theorem

The aim of the present section is to prove that the following classes of categories coincide: (1) locally A-presentable categories;

(2) full, reflective subcategories of SetA (or of Rel E) closed under A-directed colimits;

(3) categories Conta A of A-continuous set-valued functors defined on a small category A.

(The list will be continued in Section 1.D and Chapter 3, where we will also show that locally presentable categories are precisely the sketchable categories and the essentially algebraic categories.) We begin with the concept of a small-orthogonality class, since this is the basic technical tool used to prove the representation theorem below.

Orthogonality Classes 1.32 Definition (1) An object K is said to be orthogonal to a morphism in: A -+ A' provided that for each morphism f : A -> K there exists a unique morphism f: A' -+ K such that the triangle

K commutes.

(2) For each class M of morphisms in a category K we denote by .Ml the full subcategory of K of all objects orthogonal to each A --. A' in M. Conversely, a full subcategory of K is called a (small-) orthogonality class provided that it has the form Ml for a (small) collection M of morphisms of K.

1.33 Examples (1) Every full, reflective, isomorphism-closed subcategory of a category K is an orthogonality class in K. In fact, given such a subcategory A, we

28


choose, for each object K in K, a reflection map mK : K --+ K' in A. Then A = { MK

IK1EK.bi

.

In fact, every object of A is, obviously, orthogonal to each rK. Conversely, every object K orthogonal to its reflection map MK lies in A: since idK factorizes through mK we know that mK is a split monomorphism, hence, an isomorphism (see Exercise 1.g), and A is closed under isomorphisms.

The fullness is substantial here, see Exercise 11. (2) Let Frm denote the category of frames (i.e., complete lattices in which joins distribute over finite meets: (V2EI ai) A b = ViEI(ai A b)) and frame homomorphisms (i.e., maps preserving joins and finite meets). Complete Boolean algebras form a small-orthogonality class of Frm: they are precisely the frames orthogonal to the following inclusion:

-+

a

This is an example of a small-orthogonality class which is not reflective

(in fact, Frm is a cocomplete category and the category of complete Boolean algebras is not, see Exercise 1.q).

(3) Let Alg 2 be the category of algebras of one binary operation, let A = F{x, y, z} be the algebra of all terms in variables x, y, z, and let A' be the quotient algebra F{x, y, z}/- under the smallest congruence - with (xy)z - x(yz). An algebra is orthogonal to the quotient map m : A A' iff it is associative. Thus, {m}1 is the class of all semigroups.

(4) Pos is a small-orthogonality class of Gra, viz, Pos = {ml, m2i m3}1 for the following morphisms:

0

M1

M2

0

(reflexivity)

(antisymmetry)


/I

m3=id i

29

(transitivity)

(5) An example of an orthogonality class which is not a small-orthogonality class: complete join-semilattices in the category Pos* of all posets and

all functions preserving (all existing) joins. Recall that each poset P has a reflection in the subcategory CSLat of complete join-semilattices:

the reflection is the poset I(P) of all ideals (i.e., downwards closed sets I C P closed under all existing joins) ordered by inclusion. The reflection map P -* I(P) assigns to each element p the prime ideal J.p =

Ix EPI x F, such that the following triangle hom (Al , -) x hom (A2, -)

hom (Al x A2i -)

F

30

CHAPTER 1. LOCALLY PRESENTABLE CATEGORIES commutes. Here m is the canonical natural transformation whose components hom(Al x A2i -) -* hom(Ai, -) correspond to the projections

ai : Al x A2 - Ai. Thus {m}1 = the full subcategory of SetA formed by functors preserving the product (Al x A2 - " - ' + Ai).

(8) More generally, given a collection of limits limDi (i E I) in a small category A, the full subcategory of SetA formed by all functors preserving those limits is the orthogonality class M1, where M = { mi }iEI consists of the canonical maps mi :

hom(lidm Did, -) -> lidm[hom(Did, -)] .

A further generalization is straightforward: given a collection of diagrams Di with a compatible cone v(Di) for each i, the full subcategory of SetA formed by all functors F with F (o,(Di )) = lim FDi for each i is an orthogonality class in SetA.

1.34 Observation. Each orthogonality class is closed under limits. In fact, if M is a class of morphisms in K and if a diagram D : D -+ M 1 has a limit (L -- Dd)dEDob; in K, then L E M1: for each morphism f : A -+ L and each m : A --> A' in M we have the unique cone fd : A' Dd with

dED obj) (since Dd E M1). The unicity clearly guarantees that this cone is compatible. Thus, there is a unique f: A' /-> L with lrd

f' = fd

(d E Dobi).

The last condition is equivalent to f m = f. The question of whether a given orthogonality class is a reflective subcategory or not is sometimes called the orthogonal subcategory problem. In a locally presentable category the general answer is dependent on set theory, as we prove in Chapter 6, see Corollary 6.24. However, small-orthogonality classes are always "well-behaved", as we prove now: they are reflective and, moreover, as categories they are locally presentable.

Orthogonal-reflection Construction 1.35 Definition. A A-orthogonality class is a class of the form M1 such that every morphism in M has a A-presentable domain and a A-presentable codomain.

l.C. REPRESENTATION THEOREM

31

Remark. In a locally presentable category (a) each A-orthogonality class is a small-orthogonality class and (b) for each small-orthogonality class there exists a regular cardinal A such that this is a A-orthogonality class. This follows from the fact that each object is presentable, and that for each regular cardinal A there exists by Remark 1.19 essentially a set of A-presentable objects. Proposition. Each A-orthogonality class is closed under A-directed colimits.

PROOF. Let M = {Ai m'-> A;};EI be a set of morphisms in a category K such that each Ai and A; (i E I) is A-presentable. It follows that MI is closed under A-directed colimits. In fact, let (Kt K)tET be a colimit

of a A-directed diagram in Ml. For each morphism f : Ai --+ K there exists t E T and a factorization f = kt fl (since Ai is A-presentable), and there exists fi : A2 --* Kt with f = fl' mi (since Kt E M1). Thus f' = kt fi : A= -- K fulfils f = f' mi. To prove that f' is unique, let f": AE -* K also fulfil f = f" mi. Then there exists to E T such that both f' and f" factorize through kta, say, f' = kt,, fl and f" = kto f2. Since AE is A-presentable and since fl, f2: A; --> Kt,, fulfil kto fl = kto f2, there exists t > to such that the connecting arrow kt,,,t : Kta --> Kt satisfies kto,t

Ii = kto,t f2. Then f' = kt kt,,,t . fl = f".

1.36 Orthogonal Reflection. One of the crucial results for the development of the theory of locally presentable categories is that every smallorthogonality class in a locally presentable category K is a reflective subcategory of K. This can be proved abstractly, e.g., using the concept of pure subobject-we will exhibit such a proof in Chapter 2 (Theorem 2.48). In the present chapter we exhibit a constructive proof which, although less elegant than the abstract one, has the advantage of making the structure of the desired reflection quite lucid. If M consists of a single arrow A -m-, A' then for each object X we could proceed as follows:

(1) given a morphism f : A -+ X which does not factorize through m, form a pushout

A in

fl

A'

If'

X ro

X' and consider ro as the initial step of the desired reflection (i.e., in further steps work with X' in place of X);


32

(2) given a morphism f : A - X which factorizes non-uniquely through

m, say, f = p m = q m, form the coequalizer ro : X -* X' of p, q and consider it as the initial step of the desired reflection. The idea of the construction below is an iterative application of such steps. We proceed by transfinite induction; the limit steps are performed by forming the appropriate colimit. If K is locally A-presentable and if A and A' are A-presentable objects, we obtain the desired reflection in A steps.

1.37 Orthogonal-reflection Construction. Let K be a cocomplete category, and let M be a set of morphisms of K. For each object K of K we form a chain xij : Xi -+ X2 (i, j ordinals, i < j) by transfinite induction as follows:

1. First step: Xo = K. II. Isolated step: Given Xi, form a diagram in

A

f

A'

i

indexed by (i) all spans Xi Ff- A _2+ A' with in E M and (ii) all pairs p, q : A' -} Xi of morphisms for which there exists m : A -+ A' in M

with p m = q m. Denote by Xi..1 the colimit object of a colimit of that diagram; the colimit maps are denoted as follows: A'

m

AI

fl

y

1 =

xii+1

If'

for eachXi +L AA A', mEM.

i+1

Thus, xi,i+l is universal w.r.t. (i) commutative squares for all the spans above and (ii) the property that

p m = q m implies xi,i+1 p = xi,i+1

q

(for all m E M and all parallel pairs p, q with the codomain Xi).


33

III. Limit step: Form a colimit Xi of the previously defined chain and call the colimit maps xi,i: Xi --> Xi (j < i).

Proposition. The above construction stops after io steps (i.e., xi0,i0+1 is an isomorphism) if the object Xio is orthogonal to M. Then xo,io : K Xio is a reflection of K in M. PROOF. (1) If xio,io+1 is an isomorphism, then Xio E Ml. In fact, for each m : A -> A' in M and each f : A -+ Xio we have the commutative square above, which yields f-1 = xio,io+1

fi

M.

To show that this factorization is unique, consider f = p m = q m. Then one of the pairs coequalized by xi,i+l is p, q. Since xi,i+l is an isomorphism, we have p = q.

(2) If Xio is orthogonal to M, then the diagram defining Xio+1 has a compatible cocone formed by finding, for each span Xio +f A -+ A' with m E M the unique f*: A' Xio with f = f* in. Let g : Xio+1 --> Xio be the unique morphism with g xio,io+1 = id and g f = f* for each span above. Then xio,io+1 g = id because for each span we have (xi0,i0+1 9) . P = f'; this follows from the orthogonality of Xio, since [(xio,i0+1 g) f'] m = x%o,io+1 f* - m = xio,io+1 - f = f - m. Thus, xio,io+l = g-1. (3) If the construction stops after io steps, we will show that xo,i,, is a reflection map. Let h: K -+ L be a morphism with L E .Ml. Then we define a compatible cocone hi : Xi -> L by the following transfinite induction: First step: ho = h. Isolated step: we construct a compatible cocone of the diagram defining Xi+1 as follows: for each span Xi +L A -+ A', m E M, there exists

a unique f*: A' -+ L with hi f = f* m, and given p m = q m, then (hi p) in = (hi q) m implies hi p = hi q. Thus, there exists a unique hi+l : Xi+1 -+ L with hi+1 xi,i+l = hi and hi+1 f = f* for each span as above. Limit step: obvious.

Thus, if the construction stops after io steps, then each morphism

h: K - L with L E Ml factorizes as h = hi,, xo,io. The uniqueness of the factorization is easy to verify: if h = h' xo,io, then hi = h' xi,io for all i < io (which follows by transfinite induction on i).

1.38 Theorem. For each A-orthogonality class in a cocomplete category the reflection construction always stops in A steps.


34

PROOF. We will prove that Xa lies in M. In fact, for each morphism m: A -+ A' in M, since A is A-presentable, all morphisms f : A --+ XA factorize through m. This follows from the existence of a factorization f = xi,a .7 for some i < A: we have xi,i+l . f = f' m, thus f = xi+l,;k . xi,i+1 . f = (xi+l,a . f) . M.

To verify that the factorization is unique, consider p, q : A' --+ Xa with f = p m = q in. Since A' is A-presentable, there exists j < A such that p, q both factorize through xj,a, say, p = xj,,\ p and q = xj,,\ V. Then f = xj a p m = xj,,\ q m which implies (since A is A-presentable) that

there exists jl > j with xjJ, p m = xj j1 q m. Since m equalizes xi,.il p and xial q, it follows that xj,,.i,+1 coequalizes x.i,h .15 and xj,11 q; hence x.il,a also coequalizes them. In other words, p = q. Consequently, Xa E M1. Now apply Proposition 1.37.

1.39 Theorem. Let K be a locally A-presentable category. The following conditions on a full subcategory A of K are equivalent: (i) A is a A-orthogonality class in K; (ii) A is a reflective subcategory of K closed under A-directed colimits. Furthermore, they imply that A is locally A-presentable.

Remark. In the proof we will see that (ii) implies that every object A-presentable in K has a reflection rK : K --+ K* such that K* is A-presentable in A and

A={rKI KEPresAK} 1

PROOF. (i)

.

(ii) by Proposition 1.35 and Theorem 1.38.

(i). For each A-presentable object K of K let rK : K -+ K* be a reflection. Then K* is A-presentable in A because for each A-directed (ii)

colimit (Ai a`-+ A) in A and each morphism f : K* ---+ A, since A is closed

under A-directed colimits in K, the morphism f rK factorizes as f rK =

ai f for an (essentially unique) i and f': K Ai factorizes (uniquely) as f = f" rK. The collection M = { rK I K E Presa K } is small,

and A = M'. In fact, A C M J- is clear. Conversely, given an Morthogonal object K, express K as a A-directed colimit of A-presentable objects (Ki k`-+ K)iEi. For each i we have a reflection ri : Ki -+ Ki in A, and since K E M1 implies that K is orthogonal to ri, there is a unique k2 : Ki -+ K with ki = k2 ri. It is easy to see that the diagram

in A obtained by reflections from the above A-directed diagram has a colimit


35

k

(K; K)iEI. Since A is closed under A-directed colimits, we conclude that K lies in A. Furthermore, (ii) implies that A is locally A-presentable: the reflections of A-presentable objects of K in A form an (essentially small) collection of A-presentable objects in A, and each object of A is a A-directed colimit of these reflections.

1.40 Corollary. Every small-orthogonality class of a locally presentable category is locally presentable.

1.41 Example. For every small category A, there is a binary signature E such that the category SetA is equivalent to an w-orthogonality class in Rel E (thus, to a full, reflective subcategory of Rel E, closed under directed colimits) .

In fact, let A-objects be sorts and A-morphisms be relation symbols, i.e.,

S = A°bi

and E = .A"'°`

where the arity of each f : a , bin E is a x b. We will find a set M of morphisms with finitely presentable domains and codomains in Rel E such

that M' is equivalent to SetA. For each functor F: A -> Set consider the E-structure whose a-sorted underlying set is Fa (a E A°bJ) and whose relation corresponding to f : a -> b is the subset of Fa x Fb which is the graph of F f : Fa - Fb. Under this identification, Set-4 becomes a full subcategory of Rel E. To define the appropriate set M, we introduce the following simple E-structures:

Pa, where a E A°bi, has all underlying sets empty except the a-sort set {0}, and all relations are empty.

Qf, where f : a -> b is a morphism, has all underlying sets empty except the a-sort set {0} and the b-sort set {1} (in the case a # b), or the asort set {0, 1) (in the case a = b), and all relations are empty except f which consists of (0, 1) alone.

Now SetA is presented in Rel E by orthogonality to the following morphisms (with finitely presentable domains and codomains): (1) for each morphism f : a

b of A the inclusion Pa `-+ Qf

(which guarantees that the relations are graphs of functions);


36

(2) for each object a of A the quotient morphism

Q{d,. - Qida1where 0 . 1 (which guarantees the preservation of identity maps); (3) for each commutative triangle

C

in A the inclusion

(QJ + Qg)/ `- (Qf + Qg + Qh)/:.. where - is the smallest equivalence merging 0 in Qg with 1 in Qf, and . is the smallest equivalence merging 0 in QJ with 0 in Qh,

0 in Qg with 1 in Qf, and 1 in Qg with 1 in Qh. (This guarantees the preservation of composition.)

Free Cocompletions Recall that a diagram whose scheme has less than A morphisms is called A-small.

1.42 Notation. For each small category A and each regular cardinal A we denote by Cont,x A

the category of all set-valued functors on A preserving all A-small limits existing in A. (This is a full subcategory of SetA.)

1.43 Remark. Since hom-functors preserve limits, we have a restriction of the Yoneda embedding (see 0.9) Y : A --+ Conta A°p.

It is clear that Y is a full embedding. We are going to show that this extension of A is a free cocompletion preserving A-small colimits. A functor is called cocontinuous if it preserves (small) colimits.


37

1.44 Definition. Let A be a category. (1)

By a free cocompletion of A is meant a full embedding E: A -+ A* such that a. A* is a cocomplete category,

b. every functor F : A -> B with B cocomplete has an extension to a cocontinuous functor F*: A* -* B, unique up to natural isomorphism.

(2) Let A be a regular cardinal. By a A-free cocompletion of A is meant a full embedding E : A -+ A* such that

a. A* is a cocomplete category, and E preserves A-small colimits (which exist) in A, b. every functor F: A -- B with B cocomplete which preserves Asmall colimits has an extension to a cocontinuous functor

F* : A* - B, unique up to natural isomorphism.

1.45 Proposition. Let A be a small category. (i) The Yoneda embedding Y : A --+ SetAOP is a free cocompletion of A. (ii) For each regular cardinal A, the Yoneda embeddingY : A --> Cont,, A°P is a A-free cocompletion of A.

Remark. Explicitly, (i) states that for each functor F: A -* B, with B cocomplete, there exists a cocontinuous functor F*: SetAOP -> B with

F = F* Y. We call F* (determined up to a natural isomorphism) a left Kan extension of the functor F. PROOF. (i). Let F: A -* B be a functor with B cocomplete. We extend F to F* : SetA*P -+ B as follows. For each object H of SetA*P we denote by DH the category of all pairs (A, a) where A is an object of A and a E HA. The morphisms f : (A, a) -+ (A', a') of DH are those morphisms f : A --* A' of A which fulfil (H f) (a') = a. The forgetful functor DH : DH --* A defines

a diagram in A, and we put

F*H = colimF DH. Every morphism h : H -> H' in SetA°P leads to a compatible cocone of the diagram F DH with codomain F*H; we define F* h : F*H - F*H' as the unique factorization of that cocone.

38


(a) F* is cocontinuous. In fact, we prove more: F* is a left adjoint whose right adjoint is the functor F* : B --* SetAOP defined by

F*B = hom(-, B) F°P : A°' , Set

(for objects B) F*b = hom(-, b) F°P: F*B -> F*B' (for morphisms b: B --- B').

It is easy to verify that for each object B of ,6 and each functor

H : A°P - Set we have a bijection

F*H -*B H->F*B

(natural in H and B) since each of these arrows expresses a collection of morphisms dA,a : FA -+ B (A E A°bJ, a E HA) such that for each Amorphism f : A', A we have dA,a = dA,,H f(a) Ff. (b) F* is unique up to natural isomorphism. To verify this, we use the fact that each functor H : A°P

Set is a canonical colimit of hom-functors.

Since the canonical diagram of H w.r.t. hom-functors is Y DH : DH -> SetA'P, we have, for each cocontinuous functor F+ extending F (i.e., with

F+H = F+(colim Y DH) = colim F+ Y DH = colim F DH = F*H. (ii). The proof is analogous to (i), we just have to verify that Y preserves A-small colimits, and that if F preserves A-small colimits, then F. maps B into Conta A°P. The latter is obvious: F°P : A°P - ,6°P preserves Asmall limits, thus, so does F*B = hom(-, B) F°P. To prove the former, let D : D - A be a A-small diagram with a colimit (Dd C)dEDob; in A. Let fd :. hom(-, Dd) -+ H (d E D°bi) be a compatible cocone in ContA A°P. Since H preserves A-small limits in A°P, we know that (HC -i HDd) is a limit of H H. D°P in Set. The elements xd = (fd)Dd(id) form a compatible collection of H D°P, thus, there exists a unique x E HC with xd = Hcd(x) for each d. In other words, there exists a unique f : YC - * H with fd = f Ycd. Thus, (YDd -y-Cd_* YC) is a colimit of Y D in ContA, A°P.

1.46 Representation Theorem. Let A be a regular cardinal. For each category K the following statements are equivalent: (i) K is locally A-presentable;


39

(ii) K is equivalent to Cont, A for some small category A;

(iii) K is equivalent to a A-orthogonality class in SetA for some small category A;

(iv) K is equivalent to a full, reflective subcategory of SetA closed under A-directed colimits for some small category A; (v) K is a A-free cocompletion of a small category A.

Remark. The category A in conditions (ii)-(v) can be chosen as the dual of Presa K. (Thus, (v) generalizes the well-known fact that every algebraic lattice is a free completion of its subposet of all finite elements.) Consequently, the category cA- Cat

of all small categories with A-small limits and functors preserving A-small limits is dually equivalent to the quasicategory (see 0.13)

lp,\- CAT of all locally A-presentable categories and functors preserving limits and Adirected colimits. In fact, the functor R: (cA- Cat)°P --> lpa- CAT defined by

RA = Conta A°P and RH = F --. F HOP is an equivalence (see Exercise 1.s for details). PROOF. (i) (ii). Let K be a locally A-presentable category, and let A = Presa K. Then A is dense (see Proposition 1.22), and, by Proposition 1.26

the canonical functor E: K -> SetA°P preserves A-directed colimits, and K is equivalent to E(K). Let K' be the category of all functors in naturally isomorphic to some EK, K E K°bJ. We will prove that K', which is equivalent to K, is precisely Conta A°P. Every functor in K' preserves A-small limits. In fact, for every object K the functor EK = hom(-, K)/Aop preserves A-small limits in A°P SetAoP

because, by Proposition 1.16, A°P is closed under A-small limits in K°P (and

hom(-, K) preserves limits in K°P). Conversely, we prove that if a functor H : A°P -+ Set preserves A-small limits, then it is a A-filtered colimit of functors in K; then H lies in K' because K' is closed under A-filtered colimits (since E preserves A-directed colimits, thus (by Remark 1.21), A-filtered colimits). The Yoneda embed-

ding Y : A -* SetA*P fulfils Y(A) C K', and H is a canonical colimit


40

of objects of Y(A). It remains to prove that this colimit is A-filtered. By the Yoneda lemma (see 0.9), each A-small subcategory of Y(A) 1 H has the form YD(D) .. H for some A-small diagram D : D --+ A. If (Dd C) is a colimit of D in A, then (HC Hcd) HDd) is a limit of HD°P : D°P -+ A°P in Set. Thus, the elements xd E HDd representing the given maps hom(-, Dd) -+ H lead to a unique x E HC with Hcd(x) = xd. Then the subcategory YD(D) 1 H has a compatible cocone with the codomain YC --+ H (given by x E HC) formed by Cd. Thus, Y(A) 1 H is A-filtered.

(ii) #. (iii). In Example 1.33(8) we have seen how Conta A°P can be presented as the orthogonality class Ml in SetA*p, where M consists of the canonical natural transformations m : hom(lim Dd, -) - lidm [hom(Dd, -)]

for A-small diagrams D in A°P. Since the domain of each m is finitely presentable (see Example 1.2(7)) and the codomain is A-presentable (see Proposition 1.16), it follows that Ml is a A-orthogonality class in (iii) (iv) (i). See Theorem 1.39. (v) and (ii) are equivalent by Proposition 1.45. SetAop

1.47 Corollary. A category is locally A-presentable iffit is equivalent to a full, reflective subcategory of ReIE (for some finitary signature E) closed under A-directed colimits. This follows from the representation theorem and Example 1.41.

1.48 Example. The category Ban of (complex) Banach spaces and linear contractions is locally wi-presentable. To verify this, we introduce a larger category Tot of totally convex spaces. A totally convex space is an w-ary algebra A (i.e., an underlying set JAI together with operations which assign results to sequences in JAI) whose operations are indexed by all sequences (an) 000 of complex numbers

satisfying Z° o la, < 1, and which are denoted by 00

(xn)00

0

anxn

'-p

for sequences (xn) 0 in A,

n=O

subject to the following equations: 00

bnxn = xk

(1) n=O

where bk = 1 and bn = 0 for all n # k;

1.C. REPRESENTATION THEOREM co

n=0

00

i knxk) =

an 1

(2)

co

00

k=0

41

anNknJ xk.

k=0 n=0

A homomorphism f : X --> Y of totally convex spaces is a function satisfying

f (E anxn) = E an f(xn) for each (an) and each (xn). The category Tot of totally convex spaces and homomorphisms is a variety of w-ary operations; we will see in Chapter 3 that this implies that Tot is locally wl-presentable.

Ban is equivalent to a full subcategory of Tot: the unit ball U(B) of each Banach space B is a totally convex space. That subcategory is reflective since, given a totally convex space A, we have the following equivalence on the set C x A, where C is the set of all complex numbers:

(a, x) - (a', x')

if there exists ,l3 > max(Ia1, la'j) with ax =

x'.

Then the quotient algebra (C x A)/ -. , together with the function A --> (C x A)/- assigning to each a the equivalence class of (1, a), is a reflection of A in the subcategory U(Ban). The subcategory U(Ban) is clearly closed under wl-directed colimits. Thus, Ban is a locally wl-presentable category by Theorem 1.39.

Limit Sketches 1.49. We have seen above that locally presentable categories can be described as categories of set-valued functors preserving A-small limits. We will now show that, more generally, set-valued functors preserving specified limits (or, still more generally, turning specified cones to limits) always form a locally presentable category. Such categories are known as categories of models of limit sketches. Let us elaborate on this concept by specifying a collection of cones and working with set-valued functors turning the specified cones into limits.

Definition (1) By a limit sketch is meant a triple .9 = (A, L, o) consisting of a small category A, a set L of diagrams in A, and an assignment a of a compatible cone o(D) in A to each diagram D E L. (2) By a model of the limit sketch .9 is meant a functor F : A --> Set such that for each diagram D in L the F-image of the cone o,(D) is a limit of F F. D in Set. The full subcategory of SetA consisting of models of .9 is denoted by

ModY.


42

1.50 Examples (1)

Conta A is the special case of Mod 9' for the limit sketch S° where L is a set of diagrams representing all a-small diagrams in A, and a assigns to each diagram a limit cone.

(2) The category Alg 2 of algebras on one binary operation (o: X x X -* X) can be expressed as the category of models of the following sketch .':

A is the category of two objects a and a2 and, besides the identity morphisms, precisely three morphisms o, P1i P2 : a2 -* a;

L is the singleton set containing the discrete diagram which consists of two copies of a; o is the following cone:

a

a

Mod. Y. In fact, a model of the sketch Y is a functor F: A -> Set such that Fa2 = Fa x Fa with projections Fpl, Fp2. Thus, a model is given by a set X = Fa and a binary We have Alg2

operation Fo: X x X --* X. Morphisms in Mod .' are precisely the usual algebraic homomorphisms.

(3) To express the category of commutative binary algebras (i.e., the subcategory of Alg2 given by the equation o(x, y) = o(y, x)), we extend the above category A by a new morphism s : a2 --> a2 with the following composition

(4) Associativity can also be expressed by a finite-product sketch, see Exercise 1.r.

(5) The category Gra of graphs is the model category of the following

sketch .':


43

A has three objects a, a2, r and, besides the identity morphisms, three morphisms: pl, p2 : a2 - a (projections) and i : r -> a2 (expressing the inclusion of the binary relations).

L contains two diagrams: the discrete diagram consisting of two copies of a, and the span r 1-> a2 Set such that Fa2 = Fa x Fa (with projections Fp;) and Fi is a subobject of Fa x Fa, since the square

Fa x Fa is a pullback. Thus, a model is given by a set X = Fa and a subset o C

X x X representing the subobject Fi: Fr - X x X. Morphisms in Mod, are precisely the graph homomorphisms. (6) The full subcategory of Gra consisting of reflexive graphs (i.e., such that the relation contains the diagonal) can be obtained from the previous sketch by adding a new morphism j : a --> r to A satisfying Analogously, antisymmetry and transitivity can be sketched in order to express Pos as a category of models.

44


1.51 Proposition. For each limit-sketch .So the category Mod.' is locally presentable.

PROOF. This follows from the fact, established in Example 1.33(8), that Mod9 is an orthogonality class in the (locally finitely presentable) category SetA. Thus we can apply Theorem 1.39. 1.52 Corollary. A category is locally presentable if it is equivalent to the category Mod.' of models of a limit sketch Y. Remark. In more detail, a category is locally A-presentable if it is equiv-

alent to Mod 9' for a limit sketch V = (A, L, o) such that all diagrams in L are A-small. In fact, if diagrams in L are )A-small, then Mod .So is, as shown in Example 1.33(8), a A-orthogonality class of SetA.

1.53. For each sketch 9 = (A, L, a) and each category K we can introduce models of 9' in the category K: these are functors F: A - K such that for each diagram D in L the F-image of the cone o ,(D) is a limit of F D in K. The full subcategory of KA consisting of all models of 9 is denoted by

Mod(., K). Proposition. If K is a locally A-presentable category, then Mod(., K) is also locally A-presentable for each limit sketch Y with A-small diagrams.

PROOF. By representation theorem 1.46 we can assume that K = Cont), B

for a suitable small category B. To prove that Mod(9, K) is locally A-

presentable, we observe that it is equivalent to Mod9* where 9* = (A x B, L*, a*) is the following sketch: L* consists of all diagrams D x D' where D ranges through L, D' ranges through a set of representatives of A-small diagrams in B, and o.*(D x D') _ o-(D*) x limD'. 1.54 Corollary. For each locally A-presentable category K all functor-categories KA (A small) are locally A-presentable.

1.55 Examples (1) Let 2 be the two-chain 0 < 1, considered as a category. Then K2 is the category of K-morphisms. Thus, if K is locally A-presentable, then Ka is locally A-presentable (and A-presentable objects of K2 are, obviously, precisely the K-morphisms with a A-presentable domain and co-domain).

1.D. PROPERTIES OF LOCALLY PRESENTABLE CATEGORIES 45 (2) Consider the limit-sketch .9 consisting of the single pullback 0

0

in 2. Then Mod(.V, K) is the category of all K-monomorphisms.

1.D

Properties of Locally Presentable Categories

1.56 Remark. We know that every locally presentable category is (1) complete and wellpowered (since it is equivalent to a full, reflective subcategory of the complete and wellpowered category SetA, see Theorem 1.46), (2) cocomplete (by definition),

and we are now to going to prove the (non-trivial) fact that it is (3) co-wellpowered.

The last result can be derived from properties of accessible categories, as will be seen in Chapter 2 (Theorem 2.49), but we now present a direct proof. We first prove the local presentability of all comma-categories K I K and K I K (see 0.3):

1.57 Proposition. If K is a locally A-presentable category, then for each object K the comma-categories K J. K and K I K are locally A-presentable. PROOF. (1) The comma-category K I K is locally A-presentable by Theorem 1.20: K I K is, obviously, cocomplete, and its (directed) colimits are inherited from K. If .A is a strong generator formed by A-presentable objects of K, then UAEA hom(A, K) is a strong generator formed by A-presentable objects of K I K.

(2) To prove that K I K is locally A-presentable, let us consider K as a full, reflective subcategory of Rel E closed under A-directed colimits (see Corollary 1.47). Then K I K is, obviously, a full, reflective subcategory of K I Rel E closed under A-directed colimits; thus, it is sufficient to prove that K I Rel E is locally finitely presentable (see Theorem 1.39). If IKI = (K,),Es are the underlying sets of K, define a signature El = E + MES K, where each a E K, is a unary symbol of sort s. Consider

46


the full subcategory A of Rel El consisting of precisely those El-structures

A such that the relations corresponding to the elements of SKI all have cardinality 1, and the corresponding S-sorted function is a E-homomorphism from K to A. Then A is clearly isomorphic to K I ReIE, and since A is closed in Rel El under (i) products, (ii) extremal subobjects (thus, A is epireflective), (iii) directed colimits,

it follows that A is locally finitely presentable-see Theorem 1.39. 1.58 Theorem. Every locally presentable category is co-wellpowered.

f

PROOF. For each object A of a locally presentable category K we first choose an uncountable regular cardinal A such that K is locally A-presentable and A is a A-presentable object. We are going to find a set of representatives of all epimorphisms with domain A. For each morphism f : A -+ B in K we denote by A

,B i E I)

C2

the set of all factorizations of f through an object C; of Presa K. Let D(f) be the category whose objects are C2 (i E I) and whose morphisms are those morphisms h : C; - C3 for which el = he= and m; = mjh. Since Presa K is closed under A-small colimits (see Proposition 1.16), D(f) is A-filtered. From the fact that K is locally A-presentable it follows that B = colimD(f) C= in K, and since A is A-presentable, we also have (1)

f = colimei VU)

in A I K.

It is sufficient to prove that for every epimorphism f the full subcategory D(f) of D(f) formed by all factorizations in which ei is an epimorphism is cofinal in D(f) (see 0.11). In fact, we then know that (2)

f = colime2i D(f)

and it is clear that the collection of all possible categories D(f) is a set: if e2 is an epimorphism, then it determines the factorization f = m; e=

1.D. PROPERTIES OF LOCALLY PRESENTABLE CATEGORIES

47

uniquely, thus, D(f) can be considered as a subcategory of the (small) comma-category A 1 Presa K. Thus, for each i E I we want to find j E I such that ej is an epimorphism

and a morphism Ci -> Cj exists in D(f) (see Exercise 1.o(3)). First, let Pi denote a pushout of ei, ei, and P a pushout of f, f: A

P Since f is an epimorphism, we have r = q (and, given i E I, ei is an epimorphism if ri = qi). For each morphism h : Ci --* C,, of D(f) there exists a unique morphism with

h*:

(3)

h*

ri = r j h and

h*

qi = qj h.

These morphisms form a A-filtered diagram whose colimit is (Pi p'. P)iEI.

Since for each i E I the object Ci is A-presentable, and since pi pi

qi

(=r

ri =

mi), we conclude that there exists i* E I and a mor-

phism ci : Ci -* Ci. of D(f) with

ci ri=c, - qi.

(4)

We are ready to prove that for each io E I there is a morphism from Ci,, to C, such that ej is an epimorphism. Consider the following w-chain in D(f ), C'I Ci0

C` C=a - C2i

Cil = Ci2

C'2

Ci2 ...

and let (Cin kbe a colimit of this chain in K. Since A is uncountable, we know that C is A-presentable (see Proposition 1.16), thus, we can assume that C E Presa K. The unique morphism m : C -> B with

48


m k,, = mi. (n < w) yields a factorization f = m ko ei0 which is an object of D(f ), say, (ko ei0, m) _ (k5 , mj) for j E I. To prove that e1 is

an epimorphism, i.e., rj = qj, we use the fact that k = kn+1 ci. (thus, kn = kn+1 C!J; hence, from (3) and (4) we get

rj.kn=kn+i'c*,.'ri,.=k*

-c! -qi,.=qi kn

for each n < w.

Pi

Thus, rj = qj, which implies that ko: Cio -* C; is the required mor-

0

phism.

1.59 Proposition. In each locally A-presentable category IC all A-directed colimits commute with A-small limits. That is, given a diagram

D: I x J

IC

with (I, K and a natural (regular-)monotransformation 6: D -* D', then colim 6: colim D -+ colim D'

is a (regular) monomorphism.

1.61 Proposition. Every locally presentable category has both (strong epi, mono)- and (epi, strong mono)-factorizations of morphisms. PROOF. In a locally presentable category K the existence of pushouts guarantees that strong and extremal monomorphisms coincide, thus, since K is complete and wellpowered, it has (epi, strong mono)-factorizations, see 0.5. Dually, since K is cocomplete, co-wellpowered, and has pullbacks, it has (strong epi, mono)-factorizations.

1.62 Proposition. In a locally A-presentable category each A-directed colimit of (regular) monomorphisms has the property that (i) every colimit cocone consists of (regular) monomorphisms, and (ii) for every compatible cocone of (regular) monomorphisms the factorizing morphism is a (regular) monomorphism too.

PROOF. It is easy to verify that each of the categories SetA has the mentioned property. The proposition then follows from Theorem 1.46 (since the inclusion of a reflective subcategory preserves (regular) monomorphisms).

1.63. Recall that an object A is a union of subobjects mi : Ai -+ A (i E I) provided that A does not have a proper subobject containing all mi (i E I). Corollary. In a locally A-presentable category A-directed unions are A-directed colimits. More precisely, if A is a A-directed union of subobjects mi = Ai --> A, i E I, then the diagram of all Ai's and the factorizations of mi through mj for i < j has a colimit (Ai m' A)iEI PROOF. The factorizing morphism colimAi A is a monomorphism by Proposition 1.62, and it is an extremal epimorphism by the definition of union.

1.D. PROPERTIES OF LOCALLY PRESENTABLE CATEGORIES

51

1.64 Theorem. IfK and K°P are both locally presentable categories, then K is equivalent to a complete lattice. PROOF. We are to verify that hom(K, L) has at most one element for each pair of objects K, L of K. It is sufficient to show that hom(K, K) = { idK }: if we apply this to K' = K + K, we conclude that the two injections of K to K' are equal, hence, there exist no two distinct morphisms from K to L. Let A be a regular cardinal such that K is A-presentable, and both K and VP are locally A-presentable. For each morphism f : K K we will prove that f = idK. Let I be a set of cardinality A, choose io E I, and denote by I* the A-directed set of all subsets of I of cardinality smaller than A and containing io (ordered by inclusion). Consider the A-directed diagram D whose

objects are all powers KJ, where J E I*, and whose morphism from KJ to KJ' (J C J') is the unique morphism KJ -+ KJ' that composes with the jth projection as 7r5, if j E J, or as f aio, if j E J'- J (here 7rj denotes the jth projection of KJ). We define a compatible cocone rni : KJ -+ KI of the diagram D as follows: the composite of mJ with the jth projection

is aj (j E J) or f it 0 (j E I - J). Each mJ is a split monomorphism, since the projection pJ : K' -* KJ fulfils pj mJ = id. By Corollary 1.60, m = colim mJ : colim D -i KI is a regular monomorphism. Furthermore,

since pj mJ = id, the morphism pi m is a (split) epimorphism. The canonical A-directed limit (pi: KI --> KJ)JEI fulfils m = limJ(pJ m). By applying Corollary 1.60 to K°P, we conclude that m is an epimorphism,

thus, an isomorphism. It follows that (KJ -4 KI)JEI. is a colimit of D. Since K is A-presentable, the diagonal Li : K -+ KI factorizes through

some mi. Hence AI = mJ OJ, and by composing this with the jth projection for some j E I - J, we get idK = f.

1.65 Examples (1) Compact Hausdorff 0-dimensional spaces do not form a locally presentable category because the dual category is Bool (Stone duality) which is locally presentable.

(2) Unlike Banach spaces (see Example 1.48), Hilbert spaces do not form a locally presentable category. This follows from the observation that the category Hil of Hilbert spaces and linear contractions is self-dual. In fact, the functor E: Hil -> Hil°P which leaves objects unchanged and assigns to each linear contraction f : A --* B the unique adjoint contraction f * : B --f A defined by

(fx,y)=(x,f*y) is an equivalence of categories.

for all x E A and y E B

52


1.66 Adjoint Functor Theorem. A functor between locally presentable categories is a right adjoint if it preserves limits and A-directed colimits for some regular cardinal A.

PROOF. Let F: K --+,C be a functor, and let A0 be a regular cardinal such that K and C are locally Ao-presentable.

(1) If F is continuous and preserves A1-directed colimits, we will show that the solution-set condition of Freyd's adjoint functor theorem (0.7) is satisfied. Given an object L in C, let A be a regular cardinal larger or equal to both Al and A0, such that L is A-presentable. It is sufficient to show that each arrow f : L --* FK factorizes through an arrow f: L -* FK' such that

K' is A-presentable in K (i.e., f = Fh f' for some h : K' K). In fact, since K is locally A-presentable, there is a A-directed colimit (Ki -+ K) such that each Ki is A-presentable in K. Since A > A1, we have a A-directed colimit (FKi Fh'+ FK) in L. Consequently, each arrow f : L FK factorizes through some Fhi.

(2) Let F be a right adjoint with a left adjoint G. Let A be a regular cardinal such that for each Ao-presentable object L in C the object GL is A-presentable in K. Then F preserves A-directed colimits. In fact, let (Di ..* K)iEI be a colimit of a A-directed diagram Din K. Since L is locally Fk -;* Ao-presentable, in order to prove that FD has a colimit (FDi FK)iEI, it is sufficient, by Exercise 1.o(1), to verify that each Ao-presentable object L of L has the following properties:

(i) each morphism L --* FK factorizes through some Fki;

(ii) if a pair of morphisms h, h': L --+ FKi satisfies Fki h = Fki h', then there exists j E I with i < j such that FD(i -* FD(i -* The property (i) follows from the adjunction

L -->FK GL ->K since GL is A-presentable in K and K is a A-directed colimit of D. Analogously the property (ii) follows from the adjunction

L -*FDi GL - Di again via the A-presentability of GL.

I.E. LOCALLY GENERATED CATEGORIES

1.E

53

Locally Generated Categories

An important variant of the definition of locally presentable category works with A-directed unions (or, equivalently, A-directed colimits of monomorphisms, see Corollary 1.63) instead of general A-directed colimits.

1.67 Definition. Let A be a regular cardinal. An object A of a category is called A-generated provided that hom(A, -) preserves A-directed colimits of monomorphisms. An object is called generated if it is A-generated for some regular cardinal A.

1.68 Examples (1) In the categories Set, Sets, Pos, and Rel E the concepts of A-presentable and A-generated object coincide. (2) An algebra in a finitary variety is A-generated if it has a set of less than A-generators in the usual algebraic sense, as we will prove in Chapter 3. Thus, for example, in the category of algebras given by infinitely many constant symbols no finite algebra of more than one element is finitely presentable, but it can be finitely generated.

(3) No topological space which is not discrete is generated in Top (the proof is as in Example 1.2).

1.69 Proposition. In a locally A-presentable category (i) each strong quotient of a A-generated object is A-generated, (ii) each A-generated object is a strong quotient of a A-presentable object. PROOF. (i). This follows from the fact that the colimit cocone of a Adirected diagram of monomorphisms is formed by monomorphisms (see Proposition 1.62). Thus, if e : A -* B is a strong epimorphism and A is Agenerated, then for each f : B -> colim D, where D is a A-directed diagram of monomorphisms, we factorize f e through some of the colimit maps of D, and use the diagonalization property of strong epimorphisms. (ii). Let A be A-generated, and let D be a A-directed diagram of A-presentable objects with a colimit (Di - + A)iEI. Factorize each ai as a strong epimorphism ei : Di --> D; followed by a monomorphism mi : D; A. For i < j in I we see that mi factorizes through mj, thus, the objects D; form a A-directed diagram D' of monomorphisms with a colimit (D; =+ A)iEI. Since A is A-generated, idA factorizes through some mi, therefore, mi is an isomorphism. Then A is a strong quotient of Di. 0


54

Corollary. In a locally presentable category there exists, up to isomorphism, only a set of A-generated objects for each regular cardinal A. This follows from Theorem 1.58 and Proposition 1.69.

1.70 Local Generation Theorem. A category is locally presentable if it is cocomplete and has, for some regular cardinal A, a set A of A-generated

objects such that every object is a A-directed colimit of its subobjects from A.*

Remark. We do not claim that the category is locally A-presentable for the given cardinal A; in fact, a counterexample with A = w is presented below.

PROOF. I. Necessity. Let IC be a locally A-presentable category, and let A be a set of representatives for all A-generated objects. Each object K in K is a colimit of a A-directed diagram D of A-presentable objects. Factorize each of the colimit maps di : Di --> K as a strong epimorphism ei : Di -> D; with D= E A followed by a monomorphism mi : D; -+ K (see Propositions 1.61 and 1.69). Then K is a colimit of the obvious A-directed diagram of monomorphisms between the D; . II. Sufficiency. Let IC be a cocomplete category, and let A be a set as in the theorem. (i) We prove that the set A is dense in K, and that each A-generated object is isomorphic to an object in A. The latter is clear from the given property of A. Let K be an arbitrary object of K, and let D be a A-directed diagram of subobjects mi : Di -+ K (i E I) with Di E A and with a colimit

(D, - K)iEI. Then K is a colimit of the canonical diagram w.r.t. A because D is cofinal in it (see 0.11): each morphism f : A --+ K, A E A, factorizes through some mi (since A is A-generated).

(ii) K is equivalent to a full, reflective subcategory of SetA°P closed under A-directed colimits of monomorphisms. In fact, the canonical functor

E: K

SetA" is full and faithful (see Proposition 1.26) and it is a right

adjoint (by Proposition 1.27). It also preserves A-directed colimits of mono-

morphisms-this follows from the fact that for each A E A, hom(A, -) preserves these colimits (the proof is analogous to that of Proposition 1.26). K is equivalent to E(K), and E(K) has all the required properties. (iii) By (i) and (ii) it is sufficient to prove that for each locally finitely presentable category C (here L = SetA°P), if K is a full, reflective subcategory of L closed under A-directed colimits of monomorphisms in C, then `More precisely: of subobjects representable by monomorphisms with domains in A.

1.E. LOCALLY GENERATED CATEGORIES

55

K is locally presentable. Observe that since A is a set of representatives of all A-generated objects of K, every object of A has only a set of strong quotients L in K: by Proposition 1.69 each such quotient is represented by an arrow with a codomain in A. Let M denote the collection of all reflection arrows of objects X of G in K where X is (a) either a colimit of a finite diagram of A-objects in L, or

(b) the codomain of a multiple pushout (co-intersection) in G of a cone of strong epimorphisms in K with a domain in A. Since G is co-wellpowered (see Theorem 1.58), M is essentially small. By

Theorem 1.39, M' is a locally presentable category, closed in L under Ao-directed colimits for some regular cardinal A0. K is, obviously, a full, reflective subcategory of M1 closed under colimits of A-directed diagrams of monomorphisms for some regular cardinal A > A. Besides, the choice of M guarantees that K is closed in Ml under finite colimits of A-objects and under co-intersections of strong quotients of A-objects. (iv) We will prove that every morphism f : A -+ X in the category M 1 with A E A factorizes as a_strong epimorphism e f: A -+ A in K followed by a monomorphism m f : A --* X in M 1. In fact, let e f : A --> A be the co-intersection of all strong epimorphisms in K through which f factorizes. Then f = m f e f for some m f : A -- X. Since M 1 has (strong epi, mono)-factorizations (see Proposition 1.61), a co-intersection of strong epimorphisms is a strong_epimorphism. To show that m f is a monomorphism, consider P1, P2: P -> A in .M 1 with m f pi = M f P2. Since K is reflective in M1, we can restrict ourselves to P in K. The coequalizer c of pl and P2 in M' (or in K) yields a strong epimorphism c e f in K through which f factorizes-thus, c e f factorizes through e f , which implies that c is an isomorphism. h

K

56


The above factorization has the following "coherence" property: given f : A --+ X and P: A' --+ X in the category M 1 with A, A' E A, if f factor-

izes through f, then m f factorizes through m1'. In fact, given f = f h, let us form the pushout of e f and h in M' (or in K), h e f = if h. Since extremal and strong epimorphisms coincide in K (see 0.5), it is easy to see that if is a strong epimorphism in K. Now, f factorizes through if ; hence, e f, factorizes through if, and it follows easily that m f factorizes through mf,. (v) We will prove that K is a small-orthogonality class in .M', thereby verifying that K is locally presentable (Theorem 1.39). Let N be the (essentially small) collection of all reflections of A-generated objects of M' in K. We will prove that /C =)V1.

K C_ H' is obvious, thus, we just prove that each X E H' lies in K. Since M' is locally presentable, we know (from the necessity part of the present proof) that there exists a regular cardinal Al _> A such that each object of M1 is a A1-directed colimit of its A1-generated subobjects. Let D be such a diagram for the given object X, with a colimit cocone (D1 - X )iEI

Let ri : Di -* D; be a reflection of Di in K, then X E H' implies that di factorizes as di = d; ri for a unique d2: Di' -+ X. Since Di is A1generated in Ntl, and since both K and M' are closed under A1-direc-

ted colimits of monomorphisms in G, it follows that D; is A1-generated in K. By (ii) above we can assume D; E A. Then we have a factorization d'i = mi ei, for ei : D; - D=' as in (iv), and due to the above coherence property, the objects D;' form a A1-directed diagram of monomorphisms. This diagram obviously has a colimit (D®' '`* X)iEI. Since K is closed under A-directed colimits of monomorphisms in C, this proves that X lies in K. 0 1.71 Example of a category which satisfies the condition of the local generation theorem with A = w, yet, it is not locally finitely presentable. Let E be a one-sorted signature of countably many unary relation symbols ao, O 'l, .2, .... Let K be the full subcategory of Rel E of all structures satisfying the implication (011 = Qn

for all n > 1)

(3!x)vo(x).

Then (1) K is a reflective subcategory of Rel E: a reflection of a E-structure A outside of K is obtained by (a) adding a new element to (00)A in the case (vo)A = 0, or (b) merging all elements of (oo)A. Furthermore,

1.E. LOCALLY GENERATED CATEGORIES

57

K is closed under w1-directed colimits. Thus, K is locally w1-presentable by Theorem 1.39.

(2) K is also closed under directed unions in Rel E, thus, every finite structure in K is finitely generated in K. It follows easily that K satisfies the condition of Theorem 1.70 with A = w. (3)

K is not locally finitely presentable because the following E-structure A contained in K,

IAI={0,1}, oo={1}, is not finitely presentable in K. In fact, consider the following directed colimit of structures K,a in K: Ko is the structure with

IKoI =w,

and

and K is the quotient of K0 under the smallest equivalence merging 0, 1, ..., n. The w-chain of quotient maps K0 -+ K1 -> K2 ... in K

has a colimit (K k'-+

where

0 for all i < n. It is

obvious that A is not a directed colimit of finitely presentable objects of K.

1.72 Remarks (1) P. Gabriel and F. Ulmer introduce in [Gabriel, Ulmer 1971] the following concept of a locally generated category: a cocomplete category with a strong generator formed by A-generated objects and such that every A-generated object has only a set of strong quotients. They prove that a category is locally presentable if it is locally generated. It is an open problem whether the last condition (equivalent to co-wellpoweredness w.r.t. strong epimorphisms) can be deleted from their definition of locally generated categories.

(2) The difference between our local generation theorem and the result of Gabriel and Ulmer is that we pass from A-directed colimits to A-directed unions in both the places where they appear in the definition of locally

A-presentable category, whereas Gabriel and Ulmer do so only once (and they "pay" by the co-wellpoweredness condition). More precisely, a cocomplete category K is locally presentable if there exists a regular cardinal A such that (a) K has a set of A-presentable objects whose closure under A-direc-

ted colimits is all of K (by definition),


58 or

(b) K has a set of A-generated objects whose closure under A-directed colimits of monomorphisms is all of K (by our local generation theorem), or

(c) K has a set of A-generated objects whose closure under A-directed colimits is all of K, provided that K is co-wellpowered w.r.t. strong epimorphisms (by Gabriel and Ulmer).

EXERCISES

59

Exercises 1.a Directed Colimits (1) Prove that a colimit of a directed diagram D: (I, a2

(projections); (operation);

o: a2 -+ a o x id, id x o: a3 -+ a2

(here o x id and id x o are just notations for two morphisms).

The composition is defined freely subject to equations expressed by the commutativity of the following diagrams: a3

a3

/I a P1 P2

a

a2

a Pi a2 p2

(expressing the role of the two projections ql,2 and q2,3), a3

o x id'- ae

g1,2I

a3

o

3

g2,3I

a2

1PI

a2 0 ' a a

x id

idxo

2

a

a3

idxo

1P2

a2 0 -a (expressing the role of o x id and id x o), and

a

a2

a

HISTORICAL REMARKS

65 a3

id x of

°Xid--+ a 2 10

a2 0 )a (expressing the associativity). Further, let L contain two discrete diagrams: one of two copies of a and one of three copies of a. Finally, a assigns the following cones to them: a3

a2

\P2

y"INq3 q2

1.s Adjoint Functors (1) Prove that for each functor F: K -> C preserving limits and Adirected colimits, with K and C locally A-presentable, there exists a left adjoint C : C -+ K with G(Presa,C) C Presa K. (Hint: F is a right adjoint by Theorem 1.66, and any left adjoint of F preserves A-presentability of objects.) (2) Define a functor Q : lp,,- CAT -+ (ca- Cat)°P (see Remark 1.46) on objects by QK = (Presa K)°P and on morphisms F: K ->,C by choosing G as in (1) and denoting by QF the domain-codomain restriction of G°P. Verify that F is well-defined. (Hint: PresA K is skeletal.) (3) If A is a small category with split idempotents, prove that A-presentable objects of Conta A°P are precisely the representable functors. Conclude that QR = Id. (Hint: express a A-presentable object K as a A-directed colimit of hom-functors. Then K is a split subobject of one of them, and since A has split idempotents, K is then representable.) (4) Prove that RQ = Id. (Hint: by Theorem and Remark 1.46, every A-presentable category K is a A-free cocompletion of (Presa K)°1', i.e., K is canonically isomorphic to RQK.)

Historical Remarks The material of Chapter 1 is essentially contained in the lecture notes [Gabriel, Ulmer 1971]. We have found this text a continuous source of inspiration, in spite of the difficulties we had in discovering what it contains. The definition of locally A-presentable categories we have chosen as

66


basic seems to us the most fundamental both per se and also in connection with the A-accessible categories in Chapter 2, whereas Gabriel and Ulmer chose the condition of our Theorem 1.20 for the definition. The concepts of a presentable and a generated object have been introduced independently by [Artin, Grothendieck, Verdier 1972] and [Gabriel, Ulmer 1971]. Lemma 1.6 is, essentially, the proof in [Skornyakov 1964] of the famous

Iwamura' lemma, lc(1). The original source is [Iwamura 1944]. The idea of an iterative construction of orthogonal reflection is also due to Gabriel and Ulmer, and it was later developed by [Kelly 1980] from which our concrete form of the construction essentially stems. The reflectivity of orthogonal subcategories is discussed in [Freyd, Kelly 1972]. Limit sketches appeared first in the thesis [Lawvere 1963] where varieties of finitary algebras were shown to be sketchable by finite-product sketches; this was generalized to infinitary varieties and product sketches in [Linton 1966]. A full account of this is presented in Chapter 3 below. Limit sketches in general were introduced by [Ehresmann 1966] and studied by his school and, independently, by [Kennison 1968] and [Gabriel, Ulmer 1971]. The equivalence of directed and filtered colimits is well-known folklore; our treatment essentially follows [Artin, Grothendieck, Verdier 1972]. The example of Banach spaces as special totally convex spaces is due to [Pumplun, Rohrl 1984]. The local generation theorem is new. It has been published in [Adamek, Rosicky 1991], but the proof was much inspired by Gabriel and Ulmer.

Chapter 2

Accessible Categories We saw in Chapter 1 that locally presentable categories encompass a considerable collection of "everyday" categories. There are, however, other categories which are well-related to A-directed colimits, but are not (co)complete, e.g., the categories of fields, Hilbert spaces, linearly ordered sets, etc. These are the accessible categories we investigate in the present chapter. Accessible categories can be introduced in several equivalent ways: they are precisely

(1) the categories with A-directed colimits in which every object is a A-directed colimit of A-presentable objects of a certain set (Definition 2.1);

(2) the full subcategories of functor categories SetA closed under Adirected colimits and A-pure subobjects (Corollary 2.36);

(3) the free cocompletions of small categories w.r.t. A-directed colimits (Theorem 2.26); (4) the categories sketchable by a small sketch (Corollary 2.61); (5) the categories axiomatizable by basic theories in first-order logic (Theorem 5.35).

The relationship between locally presentable and accessible categories is that (a) a category is locally presentable iff it is accessible and complete (Corollary 2.47) and (b) a category is accessible iff it is a full, cone-reflective subcategory of a locally presentable category, closed under A-directed colimits (Theorem 2.53). Moreover, every accessible category has a full embedding into the category of graphs preserving A-directed colimits (Theorem 2.65). 67

CHAPTER 2. ACCESSIBLE CATEGORIES

68

The natural choice of morphisms between A-accessible categories is the A-accessible functors, i.e., functors preserving A-directed colimits. For example, each left or right adjoint between accessible categories is accessible (Proposition 2.23). In the last section we investigate properties of the 2category of A-accessible categories and A-accessible functors. We show, for example, that the Eilenberg-Moore category of a A-accessible monad is accessible (Theorem 2.78). In Chapter 1 we proved that locally presentable categories are precisely those sketchable by a limit sketch, i.e., the categories of set-valued functors preserving specified limits (in a small category). Here we prove that accessible categories are precisely those sketchable by a sketch, i.e., the categories of set-valued functors preserving specified limits and colimits (in a small

category); see Corollary 2.61. Our proof is simpler than any previously published proof and relies on the concept of a A-pure subobject (which is, roughly speaking, a A-directed colimit of split subobjects). The importance of A-pure subobjects lies in the fact that every A-accessible category has "enough" of them (Theorem 2.33). Consequently, a category is accessible if it is equivalent to a full subcategory of a functor-category SetA closed under A-directed colimits and A-pure subobjects (Corollary 2.36).

2.A Accessible Categories Definition and Examples Whereas locally finitely presentable categories generalize algebraic lattices, finitely accessible categories are a direct generalization of Scott's domains. The concept of an accessible category has developed from model theory

and category theory. We will explain the role that accessible and locally presentable categories play in model theory thoroughly in Chapter 5 below. In the present chapter we treat the concept of a A-accessible category as a natural generalization of a locally A-presentable category. We just require, instead of cocompleteness, the existence of A-directed colimits:

2.1 Definition. A category K is called A-accessible, where A is a regular cardinal, provided that (1) K has A-directed colimits,

(2) K has a set A of A-presentable objects such that every object is a A-directed colimit of objects from A.

2.2 Remarks (1) A category is called accessible if it is A-accessible for some regular cardinal A. (Then, as we show later, it is p-accessible for arbitrarily large

2.A. ACCESSIBLE CATEGORIES

69

regular cardinals p.) For the case A = w we speak of finitely accessible categories. (2) A A-accessible category has A-filtered colimits, see Remark 1.21.

(3) Every object of an accessible category is presentable (see Proposition 1.16).

(4) Every A-accessible category has, up to isomorphism, only a set of Apresentable objects. (The proof is analogous to Remark 1.9.) Therefore, we extend the notation of Remark 1.19 to accessible categories, and denote by Presa K a small, full subcategory of K formed by representatives of all A-presentable objects of K.

2.3 Examples (1) Every locally A-presentable category is A-accessible. The converse holds for cocomplete categories, thus, e.g., Top is not accessible (see Example 1.18(5)).

(2) Finitely accessible posets are precisely Scott domains, i.e., posets with directed joins in which each element is a directed join of finite elements. Scott domains which are not complete lattices are, obviously, not locally presentable. (3)

Every poset P of n elements is A-accessible for any regular cardinal A > n. (In fact, any A-directed diagram in P has the greatest element.) However, no large ordered class is accessible.

(4) The following poset P is not finitely accessible, although it has directed colimits and a dense set of finitely presentable objects. (This contrasts with the characterization of locally finitely presentable categories in

Theorem 1.11.) Let P be the set of all subsets of w which are either infinite or singleton, ordered by inclusion. The singleton sets are finitely

presentable, and they form a dense set in P. However, an infinite set is not a directed colimit of finitely presentable objects. (5)

The category Fld of fields and homomorphisms, which is far from being locally presentable (since morphisms are injective), is finitely accessi-

ble. It is evident that the category Fld has directed colimits, since it is closed under directed colimits in the category of commutative rings. Every field generated (in the usual algebraic sense) by finitely many el-


70

ements is, obviously, finitely presentable in Fld. Each field is a directed union (= directed colimit) of its finitely generated subfields. Finally, every finitely generated field is isomorphic to the field of quotients of some of the fields

Q[xl,...,xn],

(n = 1,2,3,...)

where Q is the field of rationals, and lip = Z/modp is the field of integers modulo a prime p. (6)

(7)

The category of sets and one-to-one functions is finitely accessible (and is not locally presentable). Analogously with other locally presentable categories K: the category of all K-objects and all K-monomorphisms is accessible, see Proposition 1.62 and Theorem 1.70.

For each regular cardinal A let Posa denote the category whose objects are A-directed posets and whose morphisms are order-embeddings (i.e.,

functions f : (X, does not have a colirnit in W). (9)

The category Hil of Hilbert spaces is wl-accessible when considered as a full subcategory of the (locally wi-presentable) category Ban (see Example 1.48). If di : Bi -> B, i E I, is an wl-directed colimit in Ban, then for any x E B there is i E I and y E Bi such that di(y) = x and Ilyll = JJxJJ. Since a Banach space is a Hilbert space if it satisfies the parallelogram law Ilx+y112+I1x-11112 = 211x112+211y1'2,

it immediately follows that Hil is closed in Ban under wl-directed colimits. Further, wl-presentable Hilbert spaces are either finite-dimensional or isomorphic to 12. Any infinite-dimensional Hilbert space is an wl-directed colimit of copies of 12.

2.A. ACCESSIBLE CATEGORIES

71

The category Hil is self-dual (see Example 1.65(2)), which contrasts with the fact that by Theorem 1.64 the dual of a non-trivial locally presentable category is never locally presentable. (10) Small categories (which, except for complete lattices, are not locally presentable) are often accessible, as we shall presently see. 2.4 Observation. Each accessible category has split idempotents, i.e., given f : A -* A with f f = f there exists a factorization f = i p where

In fact, each idempotent f : A - A defines a A-filtered diagram of one object A and two morphisms idA, f (for any A). A colimit of this diagram consists of an object B and a morphism p: A --> B; since f : A -p A forms a compatible cocone, we have a unique i : B ---> A with f = i p. The identity

p i = id is a consequence of p i p=p.

2.5 Remarks (1)

If a full, isomorphism-closed subcategory K of a category C has split idempotents, then K is closed in L under split subobjects.

In fact, if i : L -+ K is a split subobject of an object K from K, we choose p: K --+ L with p i = idL. Then f = i p is an idempotent which we can split in K (by io : Ko --r K and po : K KO). It is easy to see that p io : Ko -+ L is an isomorphism (with inverse po i).

(2) Every (small) category K has a (small) extension to a category k with split idempotents which is universal. That is, there is a full embedding

E : K -* K such that every functor from K to a category with split idempotents has an extension to K, unique up to natural isomorphism. See Exercise 2.b(2).

2.6 Proposition. Each small category with split idempotents is accessible. PROOF. Let K be a small category with split idempotents, and let A be a regular cardinal larger than the number of all K-morphisms. We are going to prove that for the cardinal successor A+ of A, the category K has A+directed colimits preserved by the Yoneda embedding Y : K --> Set C*'. It follows that K is A+-accessible, because every object of K is clearly A+presentable (since Y is a full embedding and each object YK is finitely presentable, hence, .A+-presentable in Set, K°P see Example 1.2(7)).


72

Thus, let D: (I, K, with A Apresentable, factorizes through some di, and given two such factorizations,

f = di p = di, p', there exists an upper bound j of i and i' such that

D(i _ j) p = D(i' -> j)

p'. Consequently, K is a canonical colimit of A-presentable objects (see Exercise 1.o), and the canonical diagram is A-filtered.

(iii) See Proposition 1.26.

Raising the Index of Accessibility We know that each locally A-presentable category is locally p-presentable for all regular cardinals p > A (see Remark 1.20). The corresponding result for A-accessible categories is not true, as we presently demonstrate. However, each A-accessible category will be proved to be p-accessible for arbitrarily large regular cardinals p.

74


2.9 Notation. For two cardinals /Q and A we denote

/3 A with f* bd = ba for all d E D°bi satisfies f f* bd = bd for each d, thus, f f* = id. Besides, f is a monomorphism because all A-presentable objects form a generator of K, and for each A-presentable object K, hom(K, f) is a monomorphism.

Thus, f* = f'1. (ii). Let (Dd -`-i C) be an arbitrary cocone with the above property, and let (Dd C*) be a colimit of D in K. The factorization map u: C* --> C (defined by cd = u ca) is an isomorphism by (i) above: for each A-presentable object K in K, hom(K, u) is, obviously, an isomorphism in Set. 2.22 Corollary. (Hom-functors are generic accessible functors.) Let K and,C be accessible categories. A functor F: K -> C is accessible iff for each

hom-functor hom(L, -): C --+ Set the composite hom(L, -) F: K

Set

is accessible.

PROOF. If F is accessible, then its composite with each of the (accessible, cf. Example 2.17(1)) hom-functors is accessible. Conversely, let F: K ,C have accessible composites with hom-functors. There exists a regular cardinal A such that K and L are A-accessible. We can find a regular cardinal A' D A such that hom(L, -) F is A'-accessible for each A-presentable object L of L. Since L is A-accessible, the hom-functors of A-presentable objects reflect A-directed (and thus, A'-directed) colimits (by Proposition 2.21(ii)), hence, F preserves A'-directed colimits.

2.23 Proposition. Every left or right adjoint between accessible categories is accessible.

PROOF. Let K and C be A-accessible categories. Every left adjoint

G: K

,C

preserves (A-directed) colimits, and hence, is A-accessible. For every right adjoint F: K ,C we choose a left adjoint G -I F; due to Theorem 2.19, we can assume that G preserves A-presentable objects. We shall prove that F

2.C. ACCESSIBLE CATEGORIES AS FREE COCOMPLETIONS

81

then preserves A-directed colimits. Let D : I --+ K be a A-directed diagram

in K, and let

u: colimFDi -> F(colimDi) be the induced morphism. To prove that u is an isomorphism, it is sufficient,

by Proposition 2.21, to show that each hom(L, -) with L A-presentable maps u to an isomorphism hom(L, u) : hom(L, colimFDi) --+ hom(L, F(colimDi)). In fact, hom(L, u) is the composite of the following isomorphisms:

hom(L, colimFDi)

2.C

colimhom(L, FDi) (L is A-presentable) colimhom(GL, Di) (G -i F) hom(GL, colimDi) (GL is A-presentable) hom(L, F(colimDi)) (G -1 F).

Accessible Categories as Free Cocompletions

In the present section we prove a representation theorem for A-accessible categories: they are just the free completions of small categories w.r.t. Adirected colimits. (We will later prove other equivalent conditions.) Recall from Theorem 1.46 that a category is locally A-presentable iff it is equivalent to a A-free cocompletion of a small category.

2.24. Recall from Remark 1.45 the concept of a Kan extension

F*: SetA°P -+ Set of a functor F: A --* Set. Lemma. For each functor F: A -+ Set, A small, the following conditions are equivalent: (i) F is a A-directed colimit of hom-functors in Set-4; (ii) F* preserves A-small limits;

(iii) F* preserves A-small limits of hom-functors (i.e., limits of A-small diagrams in SetA°P factorizing through Y : A --+ SetA*P)


82

PROOF. (i) . (ii). Observe that hom-functors have the property that a Kan extension can always be chosen as a hom-functor:

hom(A, -)* = hom(hom(-, A), -) This follows from the fact that for each functor S: SetA*p -> Set we have, by the Yoneda lemma, the following bijective correspondence:

hom(A, -) -* SY hom(hom(-, A), -) S Let F = colimH; be a A-directed colimit of hom-functors. Then F* _ colimH; (see Exercise l.p(3)) is a A-directed colimit of hom-functors, and we conclude that F* preserves A-small limits (since, by Proposition 1.59, they commute in Set with A-directed colimits). (ii) (iii) is clear. (i). Let F* preserve A-small limits of hom-functors. We will prove (iii) that the canonical diagram D: Y'(A°P) 1 F --* SetA

of F is A-filtered (where Y': A°P - SetA is the dual Yoneda embedding). It follows that F is a A-filtered (thus, also a A-directed) colimit of homfunctors. Let Do be a subcategory of Y'(A'P) 1 F of less than A morphisms, let hom(A;, -) --* F (i E I) be the objects of Do, and let ai E FAi be the corresponding points. Let Do : Do -* A°P be the composition of the domain restriction of D with the canonical isomorphism of Y'(A'P) with A°P. We

form a limit (H - hom(-,A;));EI of YDoP: DoP -> SetA*p, and by assumption we have a limit

(F*(H) F f', F*(hom(-,At)) = FA;);EI of F*YDoP in Set. Consequently, for the (compatible) collection of points

ai E FAi there exists a unique point b E F*(H) with a; = (F* f;)(b) for all i E I. Furthermore, F* preserves the canonical colimit of H w.r.t. homfunctors; thus, for b E F*(H) there exists a morphism f : hom(-, A) -+ H and a point a E FA with b = (F* f)(a). Consequently,

a;=F*(f; f)(a)

foralliEI.

Therefore, the object

hom(A, -) -> F of Y(A°P) I F

corresponding to a E FA is a codomain of a compatible cocone of the diagram Do in Y(A°P) 1 F.

2.C. ACCESSIBLE CATEGORIES AS FREE COCOMPLETIONS

83

2.25 Definition. Let A be a small category, and A a regular cardinal. A full embedding E: A -+ A* is called a free cocompletion of A with respect to A-directed colimits provided that A* has A-directed colimits and that for

each functor F : A --> B such that B has A-directed colimits there exists an extension F*: A* -. B preserving A-directed colimits, unique up to a natural isomorphism.

2.26 Representation Theorem. For each regular cardinal A and each category K the following conditions are equivalent: (i) K is A-accessible,

(ii) K is a free cocompletion of a small category A with respect to A-directed colimits, (iii) K is equivalent to the full subcategory of SetAOP formed by all A-directed colimits of hom-functors for some small category A,

(iv) K is equivalent to the category of all functors from SetA to Set preserving colimits and A-small limits (and of all natural transformations) for some small category A.

Remarks (1) In conditions (ii)-(iv) we can choose as A the subcategory Presa K. This generalizes the well-known fact that each Scott domain is a free completion of its finite elements. SetSetA

(2) The condition (iv) represents K as a full subcategory of , which is not a legitimate category. However, we will see in Theorem 2.58

that a natural restriction can be made as follows: there is a small subcategory L of SetA and a small collection of colimits in Sete such that K is equivalent to the category of all functors in Setz preserving the specified colimits and A-small limits. PROOF. (iv) ., (iii) by Lemma 2.24 and Proposition 1.45(i) applied to A°P. (iii) (i). Since K is closed under A-directed colimits in SetA°P, all homfunctors are A-presentable in K (since they are finitely presentable in SetA°D, see Example 1.2(7)).

(i) = (ii). Let F: Presa K -+ B be a functor, and let B have A-directed colimits. For each object K of K we choose a colimit of FDK in B, where


84

DK : Presa K 1 K -> Presa K is the canonical (A-filtered, see Proposition 2.8) diagram of K, with a colimit cocone

(FA

F*K)a: A-+K

The only condition put upon our choice is that if K lies in Presa K, we choose F*K = FK and a* = Fa. For each morphism f : Kl -> K2 in K we have a unique morphism

F* f : F*Kl -> F*K2

with

F* f a* _ (f a)*.

It is easy to verify that F*: K --+ B is a well-defined functor extending F. Since every object of Presa K is A-presentable, F* preserves A-directed colimits: whenever (Ki -+ K)iEI is a A-directed colimit of D in K, then each morphism a E Presa K 1 K factorizes (essentially uniquely) through (F*Ki F*k some morphism in Presa K I Ki, thus, a F K)iEI is a colimit of F*D in B. It is obvious that F* is unique up-to natural isomorphism. (ii) . (iii). It is our task to show that for each small category A, if K denotes the full subcategory of SetA*p formed by all A-directed colimits of hom-functors, then the Yoneda embedding Y : A --> K is a free cocompletion w.r.t. A-directed colimits. We know by (iii) (i) that K is A-accessible.

(a) Suppose that A has split idempotents (see Observation 2.4). It follows from Remark 2.5(1) that in SetA*p any split subobject of a hom-functor is naturally isomorphic to a hom-functor. Therefore A-presentable objects of K are, up to natural isomorphism, precisely the hom-functors, thus,

Presa K is equivalent to A. By the already proved implication (i) (ii) we conclude that K is a free cocompletion of A w.r.t. A-directed colimits. (b) If A is_ an arbitrary small category, we can find a small extension E : A -+ A with split idempotents which is universal in the sense of Remark 2.5(2). Then A and j have the same free cocompletion w.r.t. A-directed colimits (more precisely, if Eo : A -* B is such a cocompletion for j, then E0 E: A --+ B is such a cocompletion for A, and vice versa). Thus, (ii) for A implies (ii) for A, from which we know that (iii) follows. 0

2.D

Pure Subobjects

Pure subobjects are, roughly speaking, directed colimits of split subobjects. This is one of the central concepts of the theory of accessible categories. We will see, for example, that if a full subcategory of an accessible category is closed under A-directed colimits, then it is accessible if it is closed under

2.D. PURE SUBOBJECTS

85

A'-pure subobjects for some regular cardinal A. Although the concept of a pure subobject is rather abstract (subobjects are seldom pure), we are going to prove that every accessible category has "enough" pure subobjects.

Basic Properties of Pure Subobjects 2.27 Definition. A morphism f : A - B is said to be A-pure (A a regular cardinal) provided that in each commutative square

A' f B' A

B

with A' and B' A-presentable, u factorizes through f (i.e., u = u f for some U : B' --> A).

2.28 Examples (1) Split monomorphisms are A-pure for any A.

(2) In Set, A-pure morphisms are precisely the split monomorphisms (for any A).

(3) In Pos the embedding f : w -> w U {oo}, where oo is the largest element of w U loo}, is w-pure. In fact, given a commutative square as above

with A' and B' finite, let n E w be an upper bound of the image of u, then the map U: B'--+ w,

v(b)

U(b) = in

if v(b)

00

if v(b) = oo

is order-preserving, and u = u f. (4) In the category of complete semilattices, all non-constant morphisms are A-pure for any A. (Recall that non-trivial complete semilattices are not presentable, see Example 1.14(4).)

Remarks (1) A composite of A-pure morphisms is A-pure.

(2) If f- g is A-pure, then g is A-pure. (3) Every A-pure morphism is A'-pure for all A' > A.

86


2.29 Proposition. Every A-pure morphism in a A-accessible category is a monomorphism. PROOF. Let f : A -+ B be a A-pure morphism in a A-accessible category K. Since A-presentable objects form a generator Presa K of K, it is sufficient to prove that, given morphisms p, q : K -+ A with K A-presentable, then f p = f q implies p = q. Since A is a filtered colimit of its canonical diagram w.r.t. Presa K, there exists a morphism u : A' -+ A with A' A-presentable such that both p and q factorize through u: J

n

h

Since B is also a filtered colimit of its canonical diagram w.r.t. Presa K, there exists a morphism v : B -+ Bwith B A-presentable such that f u factorizes through v (say, f u = v f ), and then the equality

v (f p') =.f.p=f.q=r (7. q) implies that there is a morphism h : (B 1+ B) -+ (B' -+ B) in the canonical

diagram of B with h .(7.p') = h U. q'). Since f is A-pure, u factorizes

through f = h f , say, u = u . f'

.

Then

p=

q.

0

Remark. We thus speak about A-pure subobjects rather than A-pure morphisms. We are now going to show that A-pure subobjects in a locally A-presentable category K are just the A-directed colimits of split subobjects. The colimit relates here to the category K2 of K-morphisms (cf. Example 1.55):

2.30 Proposition. (A-pure subobjects are exactly the A-directed colimits of split subobjects.) (i) If K is a A-accessible category, then the full subcategory of K2 consisting of A-pure morphisms is closed under A-directed colimits and contains all split subobjects. (ii) If K is a locally A-presentable category, then every A-pure morphism in K is a A-directed colimit (in K2) of split monomorphisms.


87

PROOF. (i). Given a A-directed diagram in K2 with a colimit (a:,b.)

(fi

l AEI

if each fi is A-pure, then so is f. In fact, for each square as in Definition 2.27 there exists i E I with the following factorization: A'

B'

Ai - Bi

ul

V

B

A

since, obviously, both (Ai 4'+ A)iEI and (Bi -b '+ B)iEI are A-directed colimits in !C, and A' and B' are A-presentable. Then u' factorizes through f', thus, u also factorizes through f'. (ii) Let f : A -+ B be a A-pure morphism in a locally A-presentable category K. Following Example 1.55(2), we can express f as a A-directed colimit

(fi

(uuvv)

AEI with fi : Ai -> Bi such that Ai, Bi are both A-presentable

(i E I): Ai

A

ui

Bi

lvi

1

AfB

Since f is A-pure, ui factorizes through fi for each i E I; consequently, in the following pushout Ai

A

Bi

AyBi fi 7i is a split monomorphism (i E I). The induced morphisms bi,j : Bi , B i < j, constitute the A-directed diagram D = (f i

(idA,bi,i)

I f j) in 1C 2 . For


88

the unique v= : W, - B with f = v= .7i and vi = vi u= we have a A-directed colimit

f)iEI

ofDinK2. Corollary. In a locally A-presentable category, A-pure morphisms are precisely the A-directed colimits of split monomorphisms.

Remark. 2.30(ii) holds in every A-accessible category with pushouts (use Exercise 2.c). 2.31. We have seen above that A-pure morphisms in an accessible category are monomorphisms. It is an open problem whether they must be regular monomorphisms. This is true in any locally presentable category:

Proposition. Every A-pure morphism f : A --* B in a locally A-presentable category is a regular monomorphism.

Remark. We will prove that f is an equalizer of a pair B

B* such that

B* is a A-directed colimit of powers of B.

PROOF. In Proposition 2.30 we have found a A-directed collection of split monomorphisms fi : A --> Bi (i E I) whose colimit is f; more precisely, we have found a colimit (Bi b'+ B)iEI of a A-directed diagram D (of objects Bi

and morphisms bi,, for i < j) such that f = bi fi for each i E I and bi,j fi = fi for all i < j. Choose fi : Bi , A with fi fi = id. Let D* be the following A-directed diagram: D, is the power of B to

the set j i = {j E I I i < j}, for i E I, and d': D, --> Dj* is the canonical projection for each i < j. Let (d;: D= --> B*)iEI be a colimit of D*. The morphisms pi : Bi -+ Di whose jth components are

bi,i:Bi - Bi

(i<j)

form a natural transformation from D to D*, which defines a unique factorization p: B B*. Analogously, the morphisms qi: Bi -* Di whose jth components are fj fj bij yield a morphism q: B B*. We claim that f is an equalizer of p and q.

(1) p f = q f because for any i E I we see that pi fi = qi fi (since the components of the right-hand side are fj fj bij fi = f; fj fj _


89

f; = bi,j fi). Then

=di =di =q f. (2) To prove the universal property of f, it is sufficient to show that, for each A-presentable object H, each morphism h : H --f B merging p and q factorizes uniquely through f. We know that h factorizes through some bi, say, h = bi h'. Then

bi.h'

ph

=q h =q di (qi h')

which implies, since H is A-presentable, that di i pi h' = di j qi h' for some j > i. Thus, for h" = bi,3 h' we have pi h" = qj h": H -+ Dj* = Ilj A such that K is closed under A*-pure subobjects. This cardinal is obtained from the uniformization theorem 2.19: suppose that K is A*-accessible and that every A*-presentable object of K is A*-presentable in SetA. Then we shall prove that for each A*-pure morphism f : L --* K with K in K we have L E Kobi. Let us express L as a canonical colimit of A*-presentable objects (Li -+ L)iEI in Set A. Since K is A*-accessible, K is a A*-directed colimit of objects which are A*-presentable in K. Thus, for each i E I there exists a .*-presentable object Ki in K and a commutative square Li

Ki

LfK Since f is A*-pure in SetA, and since Ki is A*-presentable in SetA, we have ui : Ki --+ L with ui = ui fi. Now ui : Ki --> L is one of the colimit morphisms of the canonical colimit for L, thus, the canonical diagram has a cofinal subdiagram of objects in K. That subdiagram is A*-directed, see Exercise 1.o(3). Since L is a colimit of that subdiagram and K is closed under A*-directed colimits, this proves that L E K°bi.

O

2.33 Theorem. (Every A-accessible category has enough A-pure subobjects.) For each A-accessible category K there exist arbitrary large regular cardinals -y > A such that every subobject A B in K with A -y-presentable is contained in a A-pure subobject A -* B with A also y-presentable. Remark. We will prove a stronger result: every morphism f : A --> B with A -y-presentable factorizes through a A-pure morphism f: A-+ B with q -y-presentable.

PROOF. We know from Proposition 2.32 that K is equivalent to a full subcategory of SetA closed under p-directed colimits and p-pure subobjects (for some small category A and some regular cardinal p). We can assume that K actually is such a subcategory of SetA. Moreover, by uniformization theorem 2.19 (and Remark 2.28(3)), we can assume that every p-presentable object of K is p-presentable in SetA). Finally, we can assume that p is larger than the number of morphisms of A. (In fact, p can be chosen to be arbitrarily large in all these considerations.)


91

I. For each morphism f : A -* B in SetA we construct a factorization through a it-pure morphism f : A -* B in SetA. To this end, we define a chain f2 : A2 -* B, i < it, of subobjects of B by the following transfinite induction: First step: Ao = f (A) is the image of f and fo : AO -> B is the inclusion.

Isolated step: let fi: Ai --+ B be given. Consider a set of representatives of all spans (A2 A' 4 B') with A', B' p-presentable, such that f2 u factorizes through f'. For any such span let us choose v: B' -* B with f2 u = v f. Denote by fi+1: A;+1 --+ B the subobject which is the union of f2 with the images v(B) C B of all the chosen v's. This subobject contains fi (say fi = fi+1 ai,2+i) and has the following property: in each commutative square f'

A'

U

v

A2+1

f2+1

i

B

A

with A', B' p-presentable, the morphism a,,2+1 u factorizes through f'. (In fact, v factorizes through f2+i, and f2+1 is a monomorphism.)

Limit step: fi = U,B' u*

IV*

*

Ff

J

FAi

Fbj1,j 2

FBj1 --I

Fai

--> FBi2

Fb51

Fb?

11

FA Ff ) FB Since A' is A-presentable and F preserves the colimit expressing B above, it follows from the equation

Fb21 (Ff* u*) = Fbj1

(v*

f')

that there exists j2 > jl such that Fb11

2

- (F.f

u*) = Fbj1, 2 (v

.

f').

The morphism f is A-pure, and Ai and B32 are A-presentable objects. (bj1 j2 f*) it follows that there exists h: Bit --> A with f*.

Thus, from f ai = bj2

The morphism h* = F(h b21j2) v* then fulfils u = Fat . u*

F(h bj1,i2)

v*

=h*. f'

f'

which proves that F f is a A-pure morphism.

2.E

Properties of Accessible Categories

In the present section we will prove that if K is an accessible category, then all the functor categories KA and various comma-categories are accessible.


96

Furthermore, (co-)complete and accessible categories are precisely the locally presentable categories. We will also prove that a full subcategory of an accessible category is accessible if its embedding satisfies the solution-set condition. We postpone to Section 2.H the limit theorem which asserts that a lax limit of accessible categories is accessible, and some other properties of accessible categories (e.g., that algebras of an accessible monad form an accessible category).

2.39 Theorem. For each accessible category K, all functor-categories KA (A small) are accessible.

PROOF. By Proposition 2.32 we can assume that K is a full subcategory of some Set° (B small) closed under A-directed colimits and A-pure subobjects for some regular cardinal A. Then KA is a full subcategory of (Set g)A closed under A-directed colimits (since these are computed componentwise). For each object A of A the functor (Setn)A , Sets given by

CA(F) = F(A)

and

-tA('P) = SaA

is w-accessible. By Remark 2.19, there exists a regular cardinal p > A such that each 4tA preserves p-presentable objects. It follows from Proposition 2.38 that 4tA preserve p-pure morphisms, thus, KA is closed in (Set°)A under p-pure subobjects. Hence )CA is accessible by Corollary 2.36.

2.40 Remark. We know from Corollary 1.54 that )CA is locally A-presentable whenever K is locally A-presentable. The situation for accessible categories is different:

2.41 Example. The category Set, of infinite sets and mappings is wlaccessible. However, the uncountable product Sets _ fWl Sets is not wl-accessible: it has no wl-presentable object. To see this, consider, for each object K = (Xk)kEW, of Set", the following wl-chain Ki K5 (i < j < wl) in Set" : for each k < wl choose an infinite set Yk C Xk, and put Yk for k > i Ki = (X$,k)k<W, with Xi,k = Xk for k < i

and let Ki -+ K; be the morphism whose components are the inclusion maps. A colimit of this wi-chain is K together with component-wise inclusion maps ki: Ki -- K (i < wi). Since idK does not factorize through any ki, it follows that K is not wl-presentable.

2.E. PROPERTIES OF ACCESSIBLE CATEGORIES

97

2.42 Notation. Given functors F1: K1 -+ G and F2: IC2 -+ G, we denote by F1 I F2

the comma-category of all arrows F1K1 -f-+ F2K2 in G (with Ki E Kobe)

f

whose morphisms from F1K1 L F2K2 into F1K1 '

F2K2 are those

morphisms (k1, k2) of K1 X K2 for which the square F1K1

F2 K2 IF2k2

Flkll

F1 Ki

FZK2

commutes. Composition and identity morphisms are defined as in K1 X K2.

2.43 Theorem. The category F1 I F2 is accessible for arbitrary accessible functors Fi : Ki --+,C (i = 1, 2).

Remark. It will also be clear from the proof that the natural forgetful functors Pi : F1 1 F2 -* Ki are accessible for i = 1, 2. PROOF. By Remark 2.19 there exists a regular cardinal A such that (1) the categories K1i K2, and C are A-accessible, and (2) the functors F1 and F2 are A-accessible and preserve A-presentable

objects.

We will prove that F1 1 F2 is then a A-accessible category. Since F1, F2 preserve A-directed colimits, it is obvious that the category F1 1 F2 has A-directed colimits computed on the level of K1 X K2. Next we prove that each object F1K1 -ff+ F2K2 of F1 1 F2 is a A-directed colimit of objects F1A1 -°> F2A2 where Aj is A-presentable in K2 for j = 1, 2. As this implies that F,A,, is A-presentable in C, it is clear that these objects F1A1 °> F2A2

are A-presentable in F1 1 F2. Since they form, up to isomorphism, a set, this will conclude the proof. There exist a A-directed diagram D of A-presentable objects Di (and morphisms di,i,) in K1 with a colimit (Di k'+ K1)iEI, and a A-directed diagram D* of A-presentable objects DD (and morphisms dd;,) in K2 with

a colimit (Dj -+ K2) JET. We define a A-directed diagram D in F1 I F2 as


98

follows. The underlying poset of b is the set T of all arrows F1DE

7+

F2DD

in G (i E I, j E J) such that F2ki* f = f - Fik;. The ordering is given via the following square: F1Di

F2Dj*

Fid',''

IF'2,ul

I

F1 D,, -,

F2Dj*,

That is, 7 is below f in T if i < i' in I, j < j' in J, and the above square commutes. Let us prove that T is A-directed. Given a set To C_ T of less than A elements, there exists an upper bound io of all the corresponding i's in I, and an upper bound jo of all the corresponding j's in J. Besides, since FiDio is a A-presentable object of L and F2 preserves the colimit of D*, the morphism f Fik=o factorizes through F2kj* for some j E J-without loss of generality, we can assume that this holds for jo. Thus, we have a

f

morphism F1D=O for F2D.t*o in L:

F D; 1

Fid=,=o

I 111

F1D+ o

f

Flkio

'

F2 D

fr2,0 F2d* , ----------j-.

F2D i*°

----- > F2 D!, 7

IF2kj*,,

F2k

F1K1 f i F2K2 For each object F1D; .+ F2Dj* of To we know that F1D{ is a A-presentable object of L such that F2kjo - (fo Fads,:o) = F2kj*° (F2dj,1o

.

7).

Since F2 preserves the colimit of D*, there exists jo > jo in J such that F2dj*,jo

(fo Fid:,io) = F2D*(jo

jo) - (F2d*,,o .7).

Moreover, since J is A-directed and To has less than A elements, we can choose such a jo independent of the concrete object of To. It follows that the object Fzd.

FiDio

,jo

F2Dj*o


99

is an upper bound of To.

The diagram b assigns to each object F1D= 1+ F2Dj* the object itself and, to each pair of objects for which the square above commutes, it assigns the morphism (d1,1 , dd;,). The cocone of all

(F1Di LF2D,) -+(F1K1 f F2K2) is a colimit of D-this is obvious since F1 preserves the colimit of D, and F2 preserves the colimit of D*.

2.44 Corollary. For each accessible category K, every comma-category K 1 K and K J. K is accessible.

In fact, K I K = CK I IdK and K 1 K = Idj J. CK, where CK is the constant functor of value K.

2.45 Corollary. Every accessible functor satisfies the solution-set condition.

In fact, if F: K -+ G is accessible, then for each object L of ,C the comma-category F 1 CL, where CL : C -> C is the constant functor of value L, is accessible. Any set of generators of F J. CL is a solution set.

2.46 Remarks (1) We will see later that, under some set-theoretical assumptions, the above corollary can be reversed: given accessible categories K and C, any functor K -* G satisfying the solution-set condition is accessible. (See Theorem 6.30.)

(2) The above corollary is a natural generalization of the adjoint functor theorem (1.66). 2.47 Corollary. The following conditions on a category K are equivalent: (i) K is accessible and complete, (ii) K is accessible and cocomplete,

(iii) K is locally presentable. PROOF. Since (ii) (iii) (i) is clear (see Corollary 1.28), it remains to prove that (i) = (ii). Let K be A-accessible. Then the canonical functor E: K -> Set4°p, where A = Presa K, is A-accessible and preserves limits. By Corollary 2.45, E has a left adjoint and, consequently, K is cocomplete.


100

2.48 Reflection Theorem. Every accessibly embedded subcategory of a locally presentable category K closed in K under limits is reflective. PROOF. Let G be a full subcategory of a locally A-presentable category K

closed in K under limits and A-directed colimits. It follows from Remark 2.31 that L is closed in K under A-pure subobjects. Thus, by Corollary 2.36, £ is accessible. Hence, following Corollary 2.45, G is a reflective subcategory of K.

Corollary. Every full subcategory of a locally A-presentable category closed under limits and A-directed colimits is locally A-presentable. This follows from Theorems 2.48 and 1.39.

2.49 Theorem. Every accessible category with pushouts is co-wellpowered.

Remark. The following is, in particular, an elegant proof of the co-wellpoweredness of every locally presentable category (cf. Theorem 1.58). Whether or not each accessible category is co-wellpowered depends on set theory as we will see in Corollary 6.8 and Example A.19. PROOF. Given an accessible category K with pushouts and an object K of K, we will prove that the full subcategory EK of the comma-category K 1 K over all epimorphisms is accessible. Since EK is equivalent to a partially ordered class, this implies that EK is small (Example 2.3(3)). Let A be the following category

f

By Corollary 2.44 and Theorem 2.39, the comma-category C = (CK I KA),

where CK : A -+ K denotes the constant functor with the value K, is accessible. The objects of the category L are the commutative squares

K

AI

)AZ

A3

and morphisms are triples (a1, a2, a3) such that the following diagram


f

101

f'

id

N K

si

9'

Al

U

A12

a2

iA2 i

VI

A3 a3

al

A

All

u'

3'

commutes. The comma category K I K is isomorphic to the full subcategory Ll of G of the following squares

Kf 'A By Corollary 2.44, L, is accessible, and it is clearly accessibly embedded into L. Furthermore, the full subcategory G2 of L over all pushout squares is w-accessibly embedded into L (since pushouts commute with directed colimits), and it is accessible because, obviously, it is isomorphic to the comma-category (K, K) . K x K, and K x K is accessible by Theorem 2.39take for A the two-element antichain. By Corollary 2.37,,Cl fl,C2 is accessible. Since f : K -. A is an epimorphism if the corresponding square in L, is a pushout, L, fl L2 is isomorphic to EK, which concludes the proof. 2.50 Remark. In Section 2.H below we will study limits of accessible categories. We now just mention a special property which is needed in Section 2.F: if F : K -> G is an accessible functor, and if C1 is an accessible, accessibly embedded subcategory of.C, then F-'(G1) is an accessible, accessibly embedded subcategory of K. Here F-'(.C1) denotes a full subcategory of K, i.e., the pullback of F and the inclusion functor C1 --+ &

F'1(,Cl) Ti

+,Cl

1


102

The accessibility of F-'(L,) follows from the fact that there exists a regular cardinal A such that (i) F is A-accessible and preserves A-presentable objects, and (ii) C, is closed in C under A-directed colimits and Apure subobjects. By Proposition 2.38 the subcategory F-'(C,), which is, obviously, closed under A-directed colimits in K, is also closed under A-pure subobjects. Thus, it is accessible by Corollary 2.36.

Cone-reflective Subcategories We have seen in Corollary 2.45 that every accessible functor F : K --+ ,C satisfies the solution-set condition. In the case when F is the inclusion of K C C, this means that K is a cone-reflective subcategory:

2.51 Definition. Let A be a subcategory of a category K. By a conereflection of an object K of K is meant a cone (K a'* Ai)iEI of morphisms of K with Ai E A0'i (for i E I) such that for every morphism f : K -+ A,

A E A°bi, there exists i E I and a morphism f': Ai -* A in A with

f =f'.ai.

A subcategory is called cone-reflective in K provided that every object of K has a cone-reflection in the subcategory.

2.52 Examples (1) Every reflective subcategory is cone-reflective.

(2) The full subcategory A of Pos formed by posets with a greatest element is cone-reflective: a cone-reflection of a poset is obtained by adding a greatest element to it.

(3) The full subcategory A of the category Alg E, where E consists of one (one-sorted) unary operation, formed by all algebras with a cycle (i.e., algebras (X, a) such that an has a fixpoint for some natural number n > 1) is cone-reflective: a cone-reflection of an algebra K is formed by the cone (K K + Cn)nEw where C denotes a single n-cycle.

Remark. We have seen in Corollary 2.36 that for each accessible category K, the accessibility of accessibly embedded subcategories is characterized by closedness under pure subobjects. We now prove another result of

that sort. 2.53 Theorem. Let K be an accessible category, and let A be an accessibly embedded subcategory of K. Then A is accessible if it is cone-reflective in K.


103

PROOF. Necessity follows from Corollary 2.45. To prove sufficiency, let A be an accessibly embedded subcategory of an accessible category 1C. It follows from Proposition 2.8 and Example 1.41 that A is also an accessibly embedded subcategory of Rel E for some finitary signature E. Let

us prove that there exists a regular cardinal A such that (1) A is closed under A-directed colimits in ReIE and (2) every A-presentable object K of Ref E has a A-presentable cone-reflection in A, by which we mean a cone-reflection (K *4 A2)2Er such that each A1i i c I, is A-presentable in Re1E. Then A is A-accessible because for each object A of A the canonical diagram w.r.t. Pres,, A is clearly cofinal in the canonical diagram w.r.t. Presa Rel E. Since the latter diagram is A-filtered (because Rel E is locally finitely presentable), so is the former one, see Exercise l.o(3).

To prove the existence of such a A, first let E > card E be a regular cardinal such that A is closed in Rel E under 6-directed colimits. Since there is (up to isomorphism) only a set of A-presentable objects in ReIE, it is sufficient to find a regular cardinal A > 6 such that any morphism f : K A, A in A, factorizes through some A* in A which is A-presentable in Rel E. We know from Example 1.14(2) that for each regular cardinal A > card E, an object of ME is A-presentable if the power of its underlying set is smaller than A. We proceed by defining a chain A2 (i < 6) of cardinals by the following transfinite induction: A0 = 6;

A2+1 is the smallest regular cardinal such that every A2-presentable object of Rel E has a A2+1-presentable cone-reflection in A, and Al+1 > A2;

A2 = Vj A with

fi+1 ' ri = f ki+1

and

fi+1 ' ki,i+l = fi .

Since Ki+1 E C, there is a factorization

fi+1

i

s+1

r=+

* K"`

i+l

f!+ f A

with Ki+l E C. We define k:,i+1 = ri+1 ' ki,i+1 . Ki

Ks+1

ri+l = ri+1 ri+l : Ki+1 -> Ki+1. Limit step: Given a limit ordinal i < As, we form a colimit (K*

k'> Ki)j

of the previously defined chain. Since i < As and K7 E C for each j < i, it A follows that Ti E C (see Proposition 1.16). There is a unique 7,: Ki with fj* = f i k5,i for all j < i and we have a factorization

fi : Ki r`+ Ki f=* A

2.F. ACCESSIBLE CATEGORIES AND SKETCHES

105

with Ki E L. We define

ri =ri colimr,: Ki --* K. j 0 is a limit ordinal; moreover, cf /3 = w implies cf f03) = w. In fact, if cf /3 = w, then the f (/3n) are isolated ordinals with (f (/3n ), f (fl)) E a; by the definition of a this implies f (/3) = A + 1 or

cf /3 = w. The former cannot occur by (b), (c). If cf /3 > w, we have uncountably many ^y's with cf y = w and (y, f (fl)) E u (viz, y = f (a) for any a < /3 with cf a = w). Thus, cf f (/3) > w.

2.G. THE CATEGORY OF GRAPHS

113

(f) For each n < w, f (n) = n and f (a + n) = f (a) + n. In fact, f(2) is an isolated ordinal, f(2) # A + 1. Using the fact that a contains (0, 2), (0, 1) and (1, 2), we conclude that f(2) = 2. Consequently, f (0) = 0

and f (1) = 1. It remains to prove f (a + 1) = f (a) + 1 for each a. We know this if f (a + 1) is 2 or A + 1; if f (a + 1) # 2, A + 1, then since f (a + 1) is isolated, the unique y with (y, f (a + 1)) E v is the solution of f (a + 1) = y + 1. The statement follows from (f (a), f (a + 1)) E v. (g) f (fan) = f03)n for each limit ordinal f3 cofinal with w. This follows

from (e), (f): f(,an) = f(fn)+n. (h) f (a) = a for each a. Assume the contrary and choose the smallest a with f (a) # a. From (a), (b) we know that a < f (a) Re1 E'. In fact, let E' _ E + S where S is the set of sorts of E, ar(s) = 1 for s E S, and given o E E of arity (n,),ES, then a has arity > 3ES n, in V. For each o E E we have a canonical embedding

En

do: 11x,° --f $ES

tES(uXt)

ri(IHXt sES

tES

Define E : Rel E --* Rel E' on objects by

E(A) _ (II Xs, (dc(oA))oEE, (Xs),ES) \ sES

where A = ((X,),ES, (OA),EE), and on morphisms by E(f,) = USES .fs


114

(2) For every one-sorted signature E there exists a binary signature E' (i.e., with all arities equal to 2) and an accessible, full embedding

E: Re1E->Re1E'. In fact, let n be a cardinal larger or equal to ar o for all o E E, and let E' = E + n be a binary signature. For each o E E choose a one-to-one mapping h, = ar o -4 n. Define E : Rel E --> Re1 E' on objects by E(A) = (Xn, (aA),EE U (rA,:)iEn)

where A = (X, (oA),EE) and O'Al

X,

rA,i ={ (u, v) I u, v: n -+ X, v(j) = u(i) for each j E n

and on morphisms by E f = fn. It is easy to see that E is an embedding which preserves A-directed colimits for A > n (by Proposition 1.59). Let us verify that E is full. Let g : E(A) -> E(B) be a homomorphism. For the constant n-tuple [x] with value x we have ([x], [x]) E rA,i, thus (g [x], g [x]) E rB,i. Therefore g [x] is constant. Define f : IAA --+ IBI by g [x] = [f(x)]. Then g = fn because for each u E Xn and each i E n we have (u, [u(i)]) E rA,i; thus, (g(u), [ f (u(i))]) E rB,i and consequently g(u)(i) = f(i). To prove that f is a homomorphism in Rel E, consider any a E 0A, and choose u E Xn with u h, = a. Then (u, u) E o'' implies (f u, f u) E oB, thus f a =

f

(3) For any binary signature E there exists a finitely accessible, full embedding E: Rel E --> Gra. Let r be a rigid binary relation on the set E from Lemma 2.64. Define E: RelE --> Gra as follows: E(A) = (X + (X x X) + E + { 0, 1, 2, 3 }, PA) where X = JAI, BOA

U bA U ({O} X X) U (X X {1}) U

U({1} x E) U (E x {2}) U r U U (oA X {o}), aEE

y = { (0, 1), (1, 2), (2, 0), (0, 3), (3, 0) } and J

bA = { ((X I y)' x) }x,YEX U J (y' (x' y)) Ix,,EX'

2.H. LIMITS OF ACCESSIBLE CATEGORIES

115

For morphisms, E f = f + (f x f) + idE + id{ 0,1,2,3} It is easy to check that E is an embedding preserving directed colimits. Let us verify that E is full. Let g : E(A) -+ E(B) be a homomorphism. Since g preserves n-cycles for each n, and since E(A) has the following n-cycles (r is without cycles by Remark 2.64):

n = 1: none n = 2: (0, 3) and ((x, x), x)

n = 3: (0,1, 2), it is clear that g(0) = 0. Thus g(i) = i for i = 0, 1, 2, 3. From the path 0 --+ x -+ 1 we conclude that g(IAI) C JBI; let f : JAI -+ JBI be the restriction of g. From the path y -+ (x) y) --+ x we conclude g(x, y) = (f (x), f (y)); since the path 1 -+ o --+ 2 implies g(E) C E and since r is rigid, we have g(o) = o for each a E E. Consequently, g = E f . Finally, f preserves oA because (x, y) E oA implies ((x, y), o) E AA, thus ((f (x), f (y)), o) E PB, 0 i.e., (f (x), f(y)) E 0B.

2.66 Remark. The category Sgr of semigroups also has the universal property that every accessible category has an accessible, full embedding into it. This follows from the fact that Gra has a finitely accessible, full embedding into Sgr. This result, and a number of others of this kind, can be found in the monograph [Pultr, Trnkova 1980].

2.H

Limits of Accessible Categories

It is a remarkable property of accessible categories that a limit of a (small) diagram of accessible categories is accessible. This, however, does not hold for "ordinary" limits, but for lax limits, a concept from the theory of 2-categories which we explain in detail below. We do not assume any preliminary knowledge of the theory of 2-categories in our book. The reader can skip this section without breaking the continuity of the text. We denote by Cat the category of small categories and functors, and by CAT the quasicategory of all categories and functors. (See 0.13.) The sub-quasicategory of all accessible categories and all accessible functors is denoted by ACC. When talking about limits of accessible categories we mean limits (of

small diagrams) in ACC, The quasicategory CAT is well known to be complete, thus, limits can be always computed via products and equalizers. Products of accessible categories do not present problems, but equalizers (and limits in general) do:


116

2.67 Proposition. A product of accessible categories is accessible. More precisely, ACC is closed under products in CAT.

PROOF. Let Ki, i E I, be accessible categories. There exists a regular cardinal A such that Ki is A-accessible for each i E I, see Remark 2.14. Then the category fl EI Ki is A-accessible and the projections are A-accessible functors. In fact, REI Ki has A-directed colimits computed coordinatewise. Therefore, an object of [I Ki whose components are A-presentable is A-presentable. These objects form, up to isomorphism, a set, and every object REI Ki is a A-directed colimit of those A-presentable objects. 0 2.68 Example of a non-accessible equalizer of two accessible functors. Let K be a full subcategory of Set which is not accessible, and let

F: Set -+ Set

be a functor naturally isomorphic to Id such that FX = X if X is in K (for any set X). Then F, Id: Set --+ Set are finitely accessible functors whose equalizer is not accessible.

2.69 Definition. By the lax limit of a diagram D: D -* CAT is meant the following cone (d E Dobj)

Pd: LaxD -*Dd

in CAT: The objects of the category Lax D are the pairs ((Kd)dEDobi, (kf )fEDmor)

where each Kd is an object of the category Dd, and for each morphism f : d - d' in V we have a morphism kf : (Df)Kd -, Kd' in Dd' subject to the following conditions:

(i) if f = idd, then k f = idKd

(ii) iff = g h, then k f = kg (Dg)(kh). Morphisms of LaxD from ((Kd), (kf)) to ((Ks), (kf)) are families

k

(rd)dEDobi of morphisms rd: Kd --+ Kd in Dd such that the square (Df )Kd

(Df)rdI

Kd, Ird'

(Df )Ka k Ka f

2.H. LIMITS OF ACCESSIBLE CATEGORIES

117

commutes for each f : d --+ d' in D. The functor Pd : Lax D -+ Dd is the projection assigning the d-component to each object and each morphism.

2.70 Examples (1) Lax product and "ordinary" product coincide. (2) A lax equalizer of functors F1, F2: K --+,C is the category Lax Eq(F1, F2)

whose objects are pairs ((K, L), (kl, k2)) of morphisms ki : F;K L in .C and whose morphisms are all morphisms (r, r') of K x C such that the squares F1K

k'

LI

r'

FirIl

FiK'ki' commute (i = 1, 2), together with the obvious projection functors

Pl : Lax Eq(Fl, F2) -. K and P2: Lax Eq(Fl, F2) --* L. 2.71. In order to prove the limit theorem, which states that accessible categories are closed in CAT under lax limits, we introduce categories of inserters and equifiers. Given functors F1, F2: K --> C, the inserter category

Ins(Fi, F2) is the subcategory of the comma-category F1 1 F2 (see Notation 2.42) consisting of all objects F1K L F2K and all morphisms F1K

FlkI F1K'

f L--- F2K IF2k

F2K'

We denote by P: Ins(Fl, F2) -> K the natural forgetful functor.


118

Remark. There is an "inserted" natural transformation 0: F1 P --> F2P given by

0(1: F1K-.F2K) = f

2.72 Theorem. The category Ins(Fl, F2) is accessible for arbitrary accessible functors F1, F2: K --> G.

PROOF. There is a regular cardinal A such that F1 and F2 are A-accessible

functors. By Remark 2.19 and by Theorems 2.34 and 2.43 there exists a regular cardinal p > A such that (a) the categories K, L, Pure), K and F1 j F2 are p-accessible and

(b) the functors F1, F2: K ---f C, P1, P2 : F1

.

F2 -; K and the embedding

Pure), K -* K are p-accessible and preserve p-presentable objects.

Therefore, an object F1K1 4 F2K2 is p-presentable in F1 I F2 iff K1 and K2 are p-presentable in K. We are going to prove that Ins(Fl, F2) is a p-accessible category. Since it is, obviously, closed under p-directed colimits in F1 t F2, it is sufficient to prove that each object F1K -+ F2K of Ins(F1 i F2) is a /,t-filtered colimit of the (p-presentable) objects F1K -* F2K such that K is p-presentable in K. 1. First, consider a morphism in F1 J, F2:

F1 K1 f F2K2 Fig,

IF292

I

F1K 11j

h F2K

f

such that F1K1 1+ F2K2 is p-presentable in F1 1 F2. We will prove that it has a factorization F1K1 F191

I

F1K 11

f

F2K2

y

fr22 F2 T-1

F1tI

IF2t

F1K h F2K

2.11. LIMITS OF ACCESSIBLE CATEGORIES

119

with K a p-presentable object and t : K -- K a A-pure morphism in K. Since Purea K is p-accessible and the embedding PureA K -* K preserves p-presentable objects, there exists a p-directed diagram D of p-presentable objects and A-pure morphisms in K with a colimit (Di -+ K)iEI. The object K1 is p-presentable, thus, there is a morphism uo : K1 -> Di0 (io E I) with ki0 uo = g1. Since the objects K2 and F1Di,, are ppresentable, there exist io < it and morphisms fo: F1Di,, -> F2Di1 and vo : K2 -* Di1 such that kit . vo = 92

and

Since F2ki1 fo Fluo = F2ki1

h F1kio = F2ki1 . fo.

F2vo

f, we can assume without loss of

generality that

F2g2

f

F2K2

By transfinite induction we will construct morphisms fa : F1Dia -* F2Dia+1

for a < A (where ia. E I satisfies i« < ip for a < < A) such that a < implies that (D(i. -* ip), D(i«+1 i +1)) : f« - fp and

(ki, kia+1) : f. --> h are morphisms in F1 1 F2 (see the diagram above). ,


120

Isolated step: let fa: F1Dia -+ F2D;ai}1 be given; sentable, thus, there exist is+1 < is+2 and fa+l : FiD1. +1

is p-preF2Dgor}2 such

that

h Fiki.

+1 = F2kia+2 . fa+1

As above, we can assume without loss of generality that

fa+1 . F1D(i. - is+1) = F2D(ia+1

is+2)

f.-

Limit step: let fa: F1Dia --> F2D1 +1 be given for a < /3, where 0 < /3 is a limit cardinal. There exist ip _< ip+1 such that is < ip for any a < ,3, and fp : F1Di, -> F2Di,+, such that h F1kia = F2kia+1 ' f,6Since

F2kie+1'fp-D(iaF2kia+1

'

D(ia+1 -> ip+l) ' fa

for any a < /3, we can assume without loss of generality that for any a < /3

fp D(ia -> ip) = D(ia+1 --> ip+1) ' fa Since K is A-accessible, we can take a colimit K of Di., a < A. The

morphisms fa, a < A induce the morphism f : F1K , F2K such that f F1ka = F2ka+1 fa where ka: Di -* K is the colimit cocone. Then f is an object in Ins(Fi, F2) which is p-presentable in Ins(Fi, F2) (because

K is p-presentable in K by Proposition 1.16). Since the ki : Di. -+ K form a compatible cocone, there is a unique t : K --> K satisfying

for any a < A. Since, for any a < A

h'Fit'Fika =h'Fikio.

F2t'F2ka+1'fa =

we have

Therefore we obtain the factorization (91, 92) : f (sl'9

f -t' h

where g1 = ko uo and g2 = k1 vo. Moreover, t: K -* K is A-pure in K (by Remark 2.34).


121

F2K be in Ins(Fl, F2). Let Do be the comma-category II. Let F1K in Ins(Fi, F2) of h w.r.t. objects F1If 1+ F2K having K p-presentable, D the comma-category in F1 1 F2 of h w.r.t. p-presentable objects in F1 1 F2, and H : Do - D the inclusion. We will prove that Do is p-filtered and H cofinal, which will complete the proof.

The condition 0.11(a) follows by I. Moreover, if we take from I the factorization

g: f

(9''92

f t, h

of a morphism g belonging to Ins(Fi, F2) then gl = 92, i.e., (91, g2) belongs to Ins(Fi, F2) too. Indeed, since t gl = g = t 92 and t is A-pure, it follows by Proposition 2.29. Now, this property together with I implies that H is cofinal and Do A-filtered (see Exercise 1.o(3)).

2.73 Remark. It follows from the proof that the embedding

Ins(F1iF2)-*F1IF2 is accessible. Hence the natural forgetful functor P: Ins(Fl, F2) -> K is accessible (see Remark 2.43).

2.74 Notation. Let T : K - K be a functor. Denote by AIg T the category of T-algebras, i.e., arrows TK -+ K in K, and T-homomorphisms, i.e., morphisms f : K , K' such that the square

TK

commutes.

k

K'

2.75 Corollary. For each accessible functor T : K --> K the category Alg T is accessible.

In fact, A1gT = Ins(T, IdK). Remark. Moreover, if K is locally A-presentable and T is A-accessible then Alg T is locally A-presentable. In fact, the forgetful functor P : Alg T -* K has a left adjoint by Theorem 1.66, and free T-algebras over A-presenta-

ble K-objects are A-presentable in Aig T and form a strong generator. It suffices to apply Theorem 1.20.


122

2.76 Lemma. Let F1, F2: K -+ G be accessible functors. Then for each pair (p, 0: F1 -* F2 of natural transformations the equifier of cp, 'O, i.e., the full subcategory Eq(cp,V)) of K over all objects K with c'K = OK, is accessible.

PROOF. By Remark 2.19 there exists a regular cardinal A such that K and L are A-accessible categories, and the F; are A-accessible functors preserving A-presentable objects. It follows immediately that Eq(cp, 0) is closed under A-directed colimits. Moreover, since the F; preserve A-pure subobjects by Proposition 2.38, it is clear that Eq( F2

of natural transformations: the joint equifier, i.e., the full subcategory of K consisting of objects K with otK = V51K for all t E T, is accessible. (The proof is the same.)

2.77 Limit Theorem. A lax limit of accessible categories is accessible. More precisely, ACC is closed under lax limits in CAT.

PROOF. Let D: V -* ACC be a diagram, and let F1, F2:

11 dEDobi

Dd,

Dd' fJ f : d-+d'EDmor f idd

be the following functors: given f : d -> d' in Dm0'

f : idd, the f-

component of F1 is the projection Pd' of the first product to Dd', and the f-component of F2 is Df Pd. The inserter category Ins(Fi, F2) is accessible by Theorem 2.72. It is, obviously, isomorphic to the category Lax* D defined precisely as Lax D (see Definition 2.69) except that the condition (ii) is deleted. (In fact, the isomorphism from Ins(Fl, F2) into Lax* D "enriches" each object by defining k f = idKd for f = idd.) From Remark 2.73 and Proposition 2.67 we know that the projection functors Pd : Lax* D -* Dd are accessible. Every morphism f : d --* d' in D defines a natural transformation

f*: (D f) Pd - Pd,


123

which is the projection of each object to kf. For each composition d

h

d'

lg d"

in V consider the equifier of the natural transformations

f*, g* D(g)h*: (Df) . Pd -. Pd". The equifier is accessible (by Lemma 2.76) and consists of all objects in Lax* D which fulfil the condition (ii) for this instance of composition. By Remark 2.76 the joint equifier for all instances of compositions is also accessible. This joint equifier is isomorphic to Lax D. 2.78. As another application of the accessibility of equifiers we will prove that for each accessible monad, i.e., a monad T = (T, p,,j) over a category K

such that the functor T (and hence, the category K) is accessible, the Eilenberg-Moore category KT of T-algebras is also accessible.

Recall that a monad over a category K is a triple T = (T, p, 77) consist-

ing of a functor T : K - K and natural transformations p : T2 , T and 71: Id -* T such that the following diagrams

T3 Tp yT2 pT I

T2

IP

>T

T

T71

and

T

commute. We denote by K7 the Eilenberg-Moore category: the objects are T-algebras, i.e., arrows TK K satisfying (i) k '1K = idK and (ii) k Tk = k PK. Morphisms f : (K, k) -* (L,1), called homomorphisms, are K-morphisms f : K -> L such that f k = I T f . Theorem. The Eilenberg-Moore category of an accessible monad is accessible.

PROOF. We know that Alg T is an accessible category (Corollary 2.75). Let

P : Alg T -+ K be the (accessible) forgetful functor, and let 0: T U -> U be the natural transformation with SO(K k) = k. The Eilenberg-Moore category KT is the full subcategory of A1gT over T-algebras TK - K satisfying the above given conditions (i) and (ii). These conditions can be expressed by equifiers of the following natural transformations:

124


(i)

and (ii)
K

CD

I KA

Ea (SetB)A

H

IE

Seta

128


where B = (Presa 1C)°p, E: 1C -+ Set° is the canonical embedding, and H assigns to each G: A -> Set° the value limGD. Use Exercise 2.n and the argument of the proof of Theorem 2.60.

Historical Remarks The first concept closely related to that of accessible category was used by [Artin, Grothendieck, Verdier 1972]. They also defined A-accessible functors in much the same way as used here. Related concepts were investigated by [Banaschewski, Herrlich 1976], [Smyth 1978] and [Johnstone, Joyal 1982]. An important step was the introduction of locally multipresentable categories, see Chapter 4, by [Diers 1980a]. Categories axiomatizable in infinitary logic were characterized in terms of accessibility by [Rosicky 1981b]. The present definition of an accessible category is due to [Lair 1981], where the fundamental result that accessible categories are precisely those sketchable in the sense of [Ehresmann 1968] was proved. The name which C. Lair, used was "categories modelables". Accessible categories were rediscovered by M. Makkai and R. Pare, who gave them their present name and who later published a substantial treatise on the theory of accessible categories [Makkai, Pare 1989], whose ten-page introduction is a valuable source of further historical comments. Accessible categories were, independently, introduced in the thesis [Rosicky 1983] where the (unpublished) proofs of results announced in [Rosicky 1981b] are presented. Our treatment in sections 2.A-2.C closely follows [Makkai, Pare 1989]. The application of pure subobjects in 2.D is new. The concept stems from model theory (see Chapter 5), and its categorical formulation was presented by [Fakir 1975] in terms similar to our Definition 2.27. The fact that accessible categories have enough pure subobjects (Theorem 2.33) is new, but it could be derived from results on first-order logic such as Proposition 3.2.8 of [Makkai, Pare 1989]. The main result, that closedness under A-directed colimits and A-pure subobjects implies accessibility (Corollary 2.36), is new. Related concepts of purity were used by [Ulmer 1975] and [Bird 1984] in an investigation of constructions performed with locally presentable categories. The fact that a full subcategory of a locally A-presentable category, closed under limits and A-directed colimits, is reflective (Theorem 2.48) was proved for A = Ro by [Makkai, Pitts 1987] in a manner which cannot be generalized to arbitrary A; the general theorem was proved in an entirely different manner (using pure subobjects) in [Adamek, Rosicky 1989]. A nice example of the influence accessible categories have on the theory of locally presentable categories is the elegant proof that locally presentable categories are co-wellpowered (Theorem 2.49), due to [Makkai, Pare 1989].

HISTORICAL REMARKS

129

The characterization of accessible categories by means of cone-reflectivity (Theorem 2.53 and Corollary 2.54) was presented for categories with products in [Adamek, Rosicky 1993], and then generalized to all categories (using the same technique) by [Hu, Makkai 1994]. The results of section 2.G were proved by the "Prague School" in the 1960's, and our treatment closely follows that of [Pultr, Trnkova 1980]. The material of section 2.11 is from [Makkai, Pare 1989].

Chapter 3

Algebraic Categories The topic of the present chapter is varieties and quasivarieties, i.e., equational and implicational classes of algebras (with a set of many-sorted operations). Whereas locally presentable categories are characterized as categories of models of limit sketches, varieties can be characterized as categories of models of product sketches. These are limit sketches with discrete diagrams; in fact, to sketch a given variety of algebras, the Lawvere-Linton algebraic theory of that variety, considered as a product-sketch, can be used (Theorems 3.16 and 3.30). Moreover, varieties are precisely the accessibly monadic categories over many-sorted sets (Theorem 3.31). Quasivarieties can be abstractly characterized as precisely the locally presentable categories with a dense set of regular projectives; and varieties are then characterized as precisely the quasivarieties with effective equivalence relations (Theorem 3.33).

The name "presentable" stems from algebra: an algebra is a )-presentable object of a finitary variety if it can be presented by less than A generators and less than A equations in the usual algebraic sense [Theorem 3.12].

The chapter is concluded by a characterization of locally presentable categories which, although known as folklore, has never been published before: locally presentable categories are precisely the essentially algebraic categories, i.e., varieties of partial algebras in which the domain of definition of each partial operation is described by equations involving total operations only (Theorem 3.36).

Since all the results in the present chapter are quite analogous in the finitary case and in the general case, we present all the details for the (notationally simpler) finitary algebras, and then mention the general results more briefly. 131

CHAPTER 3. ALGEBRAIC CATEGORIES

132

3.A

Finitary Varieties

Algebras and Free Algebras We will show that varieties of finitary, many-sorted algebras are locally finitely presentable categories. More precisely: finitary varieties are precisely the categories sketchable by FP sketches, i.e., sketches in which the choice of limits is that of finite products. Let S be a set of sorts. By an S-sorted signature of finitary algebras is understood a set E (of operation symbols) together with an arity function assigning to each operation symbol o a finite word (Si, s2, ... , .s) in the alphabet S (determining the sorts of variables of a) and an element s E S (determining the sort of the result of o). Notation: 0':

X

In the case n = 0 we just write s. An algebra A of the signature E is an S-sorted set Al C= (A,),ES together with an operation

o'A: A,, xA,2 x... xA3. -' A, for each or E E of arity s1 x s2 x

x Sn ` s. In particular, nullary

operations o,: --+ s correspond to constants QA E A and unary operations o : s1 ` s2 to functions vA : A,, --> A,2.

A homomorphism from an algebra A to an algebra B (of the same signature E) is an S-sorted function f : Al I, I BI preserving the operations in the usual sense: given o : s1 X s in E, then X Sn h (oA(x1, ... , X.)) = O-B (fs, (xl), ... , fs, (xn))

for all xE E A,;, i = 1, ... , n. This yields the category Aig E of E-algebras and homomorphisms. The set of terms over an (S-sorted) set X of variables is defined in a standard manner, except that for each term r we must define a (value) sort

of r. Thus, the set TE(X)

of all terms over an S-sorted set X (of variables) is the smallest S-sorted set such that (a) each variable of sort s is a term of sort s, and

(b) for each operation o: s1 x s2 x . . . x sn --> s in E and each n-tuple of terms r = of sort si (i = 1, ... , n), we conclude that o,(r1...... n) is a term of sort s.

3.A. FINITARY VARIETIES

133

For reasons of clarity, we assume that the set E is disjoint with the set X,

of variables of sort s (for each s E S). Observe that terms o(rl...... ) are just formal expressions, thus, two terms o(rl...... ,,) and o'(rl', ... , r,,,) are

equal iffa=o',n=m, and We can consider TE(X) as an algebra: its underlying set is the S-sorted set of all terms, and its operations are given by (b) above.

3.1 Examples (1) Let E be the one-sorted signature of one binary symbol o. Terms over X = {x,y,z} are: x,

0'(X, Y),

o(o(x, y), z),

o(x, cr(y, z)) ,

etc.

(2) Let S = {ring, module} and let E be the signature of the following eight operations:

+: ring x ring -> ring 0: -+ ring -: ring -> ring x : ring x ring -> ring

+m : module x module - module Om : -+ module

-m : module - module : ring x module --> module

Let X = (Xring,Xmodule) be a 2-sorted set of variables, and suppose that we denote the variables in Xring by r, rl, etc., and those in by x, xl, etc. The following expressions are examples of terms:

-(0 x r),

rl + r2i

r,

x,

x1 + x2,

r.x,

-m(0.Om).....

(3) Sequential deterministic automata are algebras of three sorts:

S = { state, input, output } and of three operations: i:

-* state

n:

state x input - state

o: state --+ output

(initial state) (next state) (output)

Denote by al, 0'2.... variables of sort input. Here are some terms representing the reaction to /words (of input symbols) inside the automaton: i,

n(i, ol),

n(n(i, Q1), 02),

n(n(n(i, ol), 02), 03), .. .

and outside: o(i),

o(n(i, 01)),

o(n(n(i, or), 02)), ....


134

3.2 Proposition. The term-algebra TE(X) is a free E-algebra generated by X. That is, for each E-algebra A and each S-sorted function f : X - JAI there exists a unique extension off to a homomorphism f # : TE (X) -> A. PROOF. For each variable x of sort s we put f f (x) = f3 (x). For each term o(ri i ... , rn) of sort s we put

f%(o(r1...... n)) = oA(f (r1),..., f#(rn)). The unique f# defined by this rule is a homomorphism.

0

3.3 Corollary. The natural forgetful functor from Alg E to Sets is a right adjoint.

3.4 Remark: Properties of categories of algebras. The categories of finitary many-sorted algebras have the following properties:

(1) Alg E is complete, and the natural forgetful functor U : Alg E

Sets creates limits. This means that for each diagram D : V -> Alg E, given a limit (L -4 UDd)dEDobj of U D in Sets, there exists a unique algebra A with JAI = L and such that ld : A -+ Dd is a homomorphism for d E D°bJ; moreover, the latter cone is a limit of D. (In fact, the operations of A are defined coordinate-wise, i.e., by (Id)s 10-A (X1, ... , xn)] _ oDd((ld)sl(x1), ..., (ld)sn(xn)) for each d and each o : s1 x x sn -> s.) (2) Alg E has (regular epi, mono)-factorizations of morphisms: (i) Monomorphisms are precisely the homomorphisms which are injective

in each sort. (In fact, U is a faithful right adjoint, thus, it preserves and reflects monomorphisms.)

(ii) Every homomorphism bijective in each sort is an isomorphism. (In

fact, since f and f-1 f = id preserve the operations, so does f-1.) (iii) Given an algebra A, by a congruence - is meant a collection (--'s)sEs of equivalence relations ,...s on As such that each E-operation o: s1 x . . . X sn - s fulfils

0A(x1,...,xn) ^'s whenever

xi

yj for i = 1, ... , n.

For each homomorphism f : A , B the equivalences defined by x ..,s y iff f, (x) = fs(y) form a congruence on A, called the kernel congruence on A.


135

(iv) Let - be a congruence on an algebra A. The quotient algebra A/ is the algebra whose s-sort underlying set is (Al-), = A,/ i.e., the set of all equivalence classes [x] of elements x E A, (under the equivalence ',) and whose operations are given by CA/, l[xl], ... , [xn]) = [CA (X1, ... , xn)] .

The canonical homomorphism c: A -+ A/- assigning to each element its congruence class is surjective in each sort.

(v) For each homomorphism f : A -> B we have a factorization f = i c where c : A -+ Al- is the canonical homomorphism of the kernel equivalence of f, and i : A/ -+ B is the monomorphism defined by i3([x]) = f,(x).

(vi) Regular epimorphisms in Alg E are precisely the homomorphisms which are surjective in each sort, and each epimorphism represents the same quotient as the corresponding canonical homomorphism. In fact, let f : A -+ B be a homomorphism surjective in each sort. The kernel congruence - of f represents a subalgebra C of the product A x A. The two restricted projections ir1i 7r2: C , A have a coequalizer c: A -+ A/ and, since f is surjective, its canonical fac-

torization f = i c has is Al- -+ B bijective in each sort. Now use (ii).

(vii) Every epimorphism is regular in Alg E. See Exercise 3.b. (3) Alg E is welipowered [see (2.i)] and co-wellpowered [see (2.vii)].

(4) Alg E is cocomplete, and the forgetful functor creates directed colimits. In fact: (i) Coproducts are "free products". Given E-algebras At (t E T), the

coproducl is the quotient algebra of the free algebra Tr(jjtET IAtI) generated by the coproduct in Sets under the smallest congruence which merges the term a(x1 . . , xn) with the variable i .

x=CA,(XJI...,xn) for each a : s1 x ... x sn -+ s in E, each t E T, and each n-tuple

xiE(At),;,i=1,...,n.

(ii) A coequalizer of two homomorphisms f, g : A --+ B is given by the canonical homomorphism c: B -+ B/-r, where . is the smallest congruence on B with f,(a) ^,, g,(a) for every element a of A,.


136

(iii) Let D: (I, C, such that the resulting algebra A and the homomorphisms ci : Di -+ A form a colimit of D in AIg E.

(5) The free algebras TE(X), where X is a finite set of variables, are finitely presentable in AIg E, and they form a dense (essentially small) col-

lection. The former follows from the creation of directed colimits by U, and the latter from the fact that for each E-algebra A, in order to verify that a map f : JAI -- JBI is a homomorphism in A1g E, i.e., that fa (UA(al, ... , an)) = oB (f,, (al ), ... , f,,. (an )) (for o : sl x ... x sn -> s in E and for ai E A,), it is sufficient to show that f -h: TE ({ X-111'..., xn' }) -, B is a homomorphism, where h : TE ({ x1",..., x;,° }) A is the unique ho-

momorphism with h,,, (x'1) = al, ..., h,, (xn^) = an. Moreover, each of the free algebras is a regular projective, i.e., an object K such that for every regular epimorphism e : A' --* A and each mor-

phism f : K - A there exists a morphism f': K -* A' with f = e f'. In fact, for every homomorphism e : A' -> A surjective in each sort and each homomorphism f : TE(X) -> A we can clearly choose a many-sorted function e: JAI -- IA') with e e = idIA1, and we can extend the function g = e` f .?7x: X -* JA'1 to a homomorphism g# : TE(X) -* A. The equality e g# = f then follows from the fact that

(6) Alg E is a locally finitely presentable category. This follows from (4) and (5), see Theorem 1.20.

(7) The forgetful functor U: AlgE -> Sets creates absolute coequalizers. That is, given homomorphisms f, g : A -> B in AIg E and given a coequalizer UA

Uff

Ug

UB

X


137

in Sets which is absolute (i.e., preserved by every functor F: Sets --> ,C), then there exists a unique E-algebra C such that UC = X and c : B -* C is a homomorphism; moreover, c is a coequalizer of f and g in Alg E. In fact, for each operation o: sl x x s -+ s in E we have

cs'O'B'(f.

X ... X fs,.) =csfs'O'A = Cs

9s 'O'A

= C3 . OB ' (gs1 x ... X gs,.)

The functor Sets --+ Set given by (X3)$ES `-4 Xs1 x ... x Xsn preserves the coequalizer of f and g (since it is absolute); thus, there exists a unique map with cs oB = oC (C31 x ... X csn)'

This equips the S-sorted set X with operations ac (o E E) such that c : B -* C is a homomorphism. It is easy to verify that c is a coequalizer of f and g in Alg E (see Exercise 3.b(2)). (8) Alg E has effective equivalence relations, i.e., every equivalence relation is a kernel pair of some morphism. Recall that a kernel pair of a morphism f : K -+ K' in a category is a pair el, e2: E --+ K such that the square

E el K e21

if

K is a pullback. Recall that a relation on an object K is a subobject of K x K (usually represented by a pair el, e2: E --+ K of morphisms such that the morphism

(el, e2) : E -* K x K is a monomorphism). We call el, e2: E --> K an equivalence relation provided that it is

(i) reflexive, i.e., the diagonal of K x K is contained in the subobject represented by (el,e2), (ii) symmetric, i.e., the monomorphisms (el, e2) and (e2, el) represent the same subobject, and

(iii) transitive, i.e., when we form the pullback of e2 and el:


138

E\e1

E / K

e2

el

K

2

K

then the subobject represented by (el e' l, e2 e2) is contained in that represented by (ei,e2). To verify that Alg E has effective equivalence relations, it is sufficient to show that every equivalence relation in Alg E is a kernel pair of its coequalizer. This follows from the fact that U : Alg E -- Sets creates coequalizers of equivalence relations (because they are absolute, see Exercise 3.h), and that Sets has effective equivalence relations.

Equational Presentation We now assume that a "standard" many-sorted set V of variables has been chosen such that for each sort s the set V$ is denumerable. An equation is a pair (T1,T2) of terms in TE(V) of the same sort; notation: Ti = T2.

A E-algebra A satisfies this equation if for each interpretation of the standard variables, i.e., each S-sorted function f : V , JAI, we have f f (Ti) = f#(T2). An equational class or variety of E-algebras is a class C for which there exists a set E C TE(V) x TE(V) of equations such that a E-algebra lies in C if it satisfies all equations in E. The pair (E, E) is called an equational presentation of the class C, and Calgebras are also called (E, E)-algebras. Each equational class is considered as a full subcategory Alg(E, E) of Alg E.

3.5 Examples (1) Abelian groups form an equational class of one-sorted algebras of signature {+, -, 01 with arities s x s -- s, s --> s, and s, respectively,


139

presented by the following equations:

(x+y)+z=x+(y+z) x+y=y+x x+0=x X + (-x) = 0. (2) Modules over arbitrary rings form an equational class of two-sorted algebras of the signature of Example 3.1(2) presented by the following set of equations

E=ErUE.UE0

where E, are the four equations of (1), Em are the corresponding four equations for +m, Om, and -m, and Eo are the following equations

rx(r xr")=(rxr')xr" rx(r'+r")=(rxr)+(rxr") (r' + r") x r = (r' x r) + (r" x r) r x (x +m x') _ (r x) +m (r x')

(r+r') x= (3)

x)

For each small category A, the functor-category SetA is isomorphic to an equational class of unary algebras (besides being a small-orthogonality class of binary relations, see Example 1.41). The objects of A play the role of sorts and the morphisms the role of operation symbols: Let E = Amor. and S=Aobi each morphism

f: s- s'

has arity, as indicated, s -+ s'. Let

E=E'comUEid be the following set of equations: Eco,,, consists of all the equations

h(x) = f(g(x))

where h = f g : s -+ s' in Amor and x is a variable of sort s, and Eid consists of all the equations id,(x) = x


140

for s E A°bi, where x is a variable of sort s. Then each (E, E)-algebra A

defines a functor A: A -> Set by A(s) = A, (s E Aobj) and A(f) = fa (for f in E = A-or). In this way we obtain an isomorphism

(--): Alg(E, E) -> SetA defined on morphisms by r" = (r, ), EAob; .

3.6 Remark: Properties of Varieties. Every variety Alg(E, E) of Ealgebras has the following properties. (1)

Alg(E, E) is closed under subalgebras in Alg E: if an algebra A satisfies all E-equations, then so do all subalgebras B C A because terms are computed in B as in A.

(2) Alg(E, E) is closed under products in Alg E (because terms are computed in a product coordinate-wise).

(3) Alg(E, E) is a (regular epi)-reflective subcategory of Alg E. This follows from (1) and (2) since Alg E is a complete, co-wellpowered category with (regular epi, mono)-factorizations (see Remark 3.4). (4) For each many-sorted set X of variables there exists the smallest congruence - on the term-algebra TE(X) for which the quotient algebra TE,E(X)

=TE(X)/^'

lies in Alg(E, E). Moreover, TE E(X) is a free (E, E)-algebra with the universal map 9: X ---> Tr,E (X) assigning to each variable x its congruence class [x]. (That is, for each (E, E)-algebra A and each many-sorted function f : X --+ CAI there exists a unique E-homomor-

phism f# : TT,E(X) - A with f = f# 9.) All this follows from (3): consider the reflection of TE(X) in Alg(E, E). (5)

Alg(E, E) is closed under quotients (that is, under homomorphic images) in Alg E. In fact, given (E, E)-algebra A and a E-homomor-

phism h: A -> B which is surjective in each sort, then there is a many-sorted function k: CBI -> Al J(not a homomorphism, in gen-

eral) with h k = idB. It follows that each equation r = r' satisfied by A is also satisfied by B: given a function f : V --p BI, we have

thus, if (k

f)#(r) = (k f)#(r'), then f#(r) = f#(r').


141

(6) Alg(E, E) is closed under directed colimits in Alg E. In fact, let (A2 -3 A)2EI be a directed colimit of algebras A2 satisfying an equa-

tion r = T' in the variables x1i ... , x,,. For each many-sorted function f : {xl,... , xfz) - JAI there exists a factorization f = hi g for some i E I, and then f# = hi g#. Thus, g#(T) = A# (,r') implies f# (T) = f#(T') (7) Alg(E, E) has effective equivalence relations. This follows from Remark 3.4(8): from the closedness under (finite) products and subobjects it follows that Alg(E, E) is closed under equivalence relations in Alg E, and from the closedness under quotient algebras it follows that Alg(E, E) is closed under coequalizers of equivalence relations in Alg E (which are created by the forgetful functor).

(8) Alg(E, E) has a dense subcategory consisting of finitely presentable regular projectives. The argument is analogous to that of Remark 3.4(5), it is just necessary to substitute TE(X) by TE,E(X). 3.7 Corollary. Every variety of finitary algebras is a locally finitely presentable category. In fact, this follows from Theorem 1.11, since every variety is cocomplete (see Remarks 3.4(4) and 3.6(3)) and has a dense set of finitely presentable objects (by Remark 3.6(8)). 0

3.8 Example of a locally finitely presentable category which is not equivalent to a many-sorted variety: the category Pos of posets does not have effective equivalence relations. In fact, let K = (X, io. (Since the di,5 are homomorphisms, such tuples form a sub algebra of fl EI Di.) Then the following is an epimorphism

in Alg E:

f : B ---r C,

fs (xi)iEI = [xi0] whenever xi = (dio,i)s(xia) for all i > io.

Let us consider all finite subsets Z of the standard set V of variables (i.e., many-sorted subsets of a finite power), and let cZ : TE(Z) -+ TE(Z)/be the canonical map forming a reflection of the free E-algebra in A. We claim that A is presented by the set E of all equations r = r' where Z C V is finite and (r, r') lies in the kernel of cz. Observe that a E-algebra satisfies all equations in E if it is orthogonal to each cz. Thus, we need to prove that A = { cZ I Z C_ V, Z finite }1. By Theorem 1.38, A is the orthogonality class of all reflection maps of finitely presentable E-algebras. Thus, it is sufficient to prove that orthogonality to each cz implies orthogonality to the reflection map of any finitely presentable algebra A. Let Z C JAI be

a finite set of generators, and let k: TE(Z) - A be the corresponding epimorphism. Let us form a pushout of k and cZ in Alg E: TE(Z) cZ I TE(Z)/

kI

A

1k'

F

A'

Since k' is an epimorphism, and A is closed under quotients, we have A'

in A-thus, r is a reflection of A in A. It is obvious that a E-algebra orthogonal to cz is orthogonal to r.

O

3.10 Presentation and Generation of Algebras. We are now going to prove that an algebra in Alg E is (1) A-generated (see Definition 1.67) if it has less than A generators in the usual algebraic sense, and

(2) A-presentable if it can be presented by less than A generators and less than A equations in the usual algebraic sense.


143

(These results explain the terminology used throughout our book.)

Recall that for each algebra A in E- Alg, a (many-sorted) subset X of JAI is said to generate the algebra A provided that every subalgebra of A containing X is equal to A. This is, obviously, equivalent to A's being isomorphic to a quotient algebra of the free algebra TE(X) w.r.t. the extension of the inclusion X IAI to a homomorphism TT(X) --> A. We say that A has less than A generators provided that it can be generated by a subset X of power smaller than A. By a presentation of a E-algebra A is meant a set X of (many-sorted) variables, not necessarily standard, and a set ri = i' (i E I) of equations in TT (X) such that A is isomorphic to the quotient of TE (X) modulo the smallest congruence - with ri i' for each i E I. We say that A is presentable by less than A generators and less than A equations provided that we can choose a presentation with X and I both of power smaller than A. More in general, for each variety V of E-algebras a presentation of an

algebra A in the variety V is a set X of variables and a set of equations ri = ri in TT(X) such that A is isomorphic to the algebra TE(X)/- for the smallest congruence - such that TT(X)/- lies in V and ri - ri for each i. 3.11 Proposition. For each regular cardinal A, an algebra is a A-generated object in Alg E if it has less than A generators. PROOF. I. If A is A-generated, consider the diagram D whose objects are all subalgebras of A on less than A generators, and whose morphisms are the inclusion morphisms B -+ B' for B C B' in D. Then A is a canonical colimit of D, and hom(A, -) preserves that colimit. Thus, idA factorizes through B --* A for some algebra B in D; it follows that B = A.

II. If A has less than A generators, it is obvious that hom(A, -) preserves A-directed unions (= colimits) of subalgebras. 0 3.12 Theorem. For each regular cardinal A, an algebra is a A-presentable object of Aig E if it is presentable by less than A generators and less than A equations.

PROOF. I. Let A be an algebra presentable in Aig E by less than A generators and less than A equations. We can assume that A = TE(X)/- where #X < A and - is the smallest congruence containing a set E of equations

with card E < A. Denote by ko : X -* A the domain restriction of the canonical homomorphism k: Tn(X) -* A. Let D be a A-directed diagram in Aig E with a colimit (Di -c-'+ C)iEI. Given a homomorphism f : A -* C, there exists io E I with f(JAI) C cio(IDi.1): this follows from the fact that,


144

since I is directed, Co = UiEI ci(IDi1) is a sub algebra of C, hence Co is equal

to C. From the A-directedness of I we get io with f(k(X)) C cia(IDi.1). Thus, there is a mapping h : X --* JDio I such that cio h = f ko, i.e., cio h# = f k for the homomorphism h#: TE(X) -* Dio extending h. For each (u, v) E E we have cio h#(u) = cio h#(v), and since I is directed, it follows that there exists i > io such that the homomorphism dio,i : Dio ---+ Di of D merges h#(u) and h#(v). Since card E < A, there exists it > io such that dio,i, merges (h#)2(E), thus, it merges Then f factorizes through ci, : there exists a mapping f: IAI --> JDi,I (h#)2(,-).

with f k = dia,i, h#, and this is a homomorphism f: A -; Di, (see Exercise 3.b(2)) with f = ci, f'. To show that the factorization is essentially unique, let f": A -> Di, be a homomorphism with f = ci, f". For each x E X since (ci,), (f, (x)) = (ci,)., (f"(x)) it follows that there exists i > it with di,,i (f, (x)) = di,,i (f,'(x)) . Since #X < A, there exists i2 > it with di,,i2 . f ko = di,,i2 f" ko. This implies di,,i2 . f' k = di,,i2 f" k, and since k is an epimorphism, we conclude that di,,i2 f _ d%,%2 f

II. Let A be a A-presentable object of Alg E. By Proposition 3.11 the algebra A has less than A generators. Let X C JAI be a set of generators of power less than A, and let k: TE(X) -* A be the canonical homomorphism. Let ker k denote the many-sorted kernel set of k:

(ker k), = { (t, t') I t, t' E TE(X), and k,(t) = k,(t') }.

Consider the set {Ei

I

i E I} of all many-sorted subsets Ei g ker k of

power less than A. We have a A-directed diagram D' whose objects are the

quotient algebras Di' = TE(X)/-i, where -i is the congruence generated by Ei, and for Ei C Ej the D'-connecting morphism is the homomorphism kij: D; -+ Dj' defined via the quotient maps ki : TE (X) --> D; by

k%a'k%=ki It is obvious that a colimit of D' is formed by the canonical homomorphisms E: D= A (i E I) defined by k1 ki = k. Di'

k%,.i

A

Since A is A-presentable and the diagram D' is A-directed, idA factorizes through some kg, say

k'u=idA.


145

Next we use the fact, established in I above, that each Di' is A-presentable: since D' is a A-directed diagram, and since k; merges idDy with u k; (we

have k; u k; = ki), there exists j > i with

kij We claim that kj' is an isomorphism with the inverse ki,; u. In fact,

k j ' -(kia u) =k; u=idA, and to prove (ki,; u) kj' = idD' , we use the fact that ki j is an epimorphism:

[(kia'u)-ki]

-kij=kia.u.k:=kia.

Thus, A = Dj' has a presentation by less than A generators and less than A equations.

3.13 Corollary. In every variety V of finitary algebras, A-presentable objects are precisely the algebras presentable by less than A generators and less than A equations in V (for each regular cardinal A). For A > card E these are precisely the algebras whose underlying set has power less than A. PROOF. Every variety is locally finitely presentable by Corollary 3.7. V is closed under directed colimits in Alg E, and every E-algebra A has a reflection A -+ Al- in V for some congruence -, see Remark 3.6. If an algebra A of V is presentable by less than A generators and less than A equations in V, then it is a reflection of a E-algebra Ao presentable by less than A generators and less than A equations in Alg E. Since A0 is A-presentable in Alg E, it follows that A is A-presentable in V. Conversely, let A be a A-presentable

object of V. The proof that A can be presented by less than A generators and less than A equations is analogous to the above proof in the case V = A1g E. Remark. For each A-presentable algebra A of V there exists a coequalizer in V

TE,E(I)

TE,E(X) -A

with X and I of power less than A. In fact, let us assume that A = TE(X)/L for a set X of generators of power less than A and for a congruence - gen-

erated by less than A equations ri = ,', i E I. We can consider I as an Ssorted set: the sort of i E I is the sort of ri. Denote bye : TE (X) -+ TE,E (X )

the canonical homomorphism. Then the maps f, f: I -* ITE,E(X)l given by f, (i) = e, (r,) and fs (i) = e, (T;) have extensions to a pair of homomorphisms f #, (f')#: TE,E(I) --> TE,E(X) whose coequalizer is the canonical homomorphism TE,E (X) --r A.


146

Algebraic Theories With each variety V of many-sorted algebras we associate a category Th V, called the theory of V-algebras, such that V is equivalent to the category

of all set functors on Th V preserving finite products. This presentation of V via its theory has the fundamental property of being independent of signatures and equations. Recall that we assume that a standard set of variables is given; in each sort s the variables are denoted by xi, x2, x3, .... We have shown that the forgetful functor of V has a left adjoint F: Sets -> V (see Remark 3.6(4)).

3.14 Definition. Let V be a variety of finitary S-sorted algebras. The theory of V-algebras, denoted by Th V, is the dual category of the category of all V-free algebras F{ xi...... xn^ } for all n-tuples of sorts s1, ... , s E S (n = 0, 1, 2, ... ), as a full subcategory of (AIg E)°P.

3.15 Remarks (1) In the one-sorted case the objects of Th V are just F{xl,... , xn}. Thus, they can be identified with the natural numbers n. Given two objects n and m, morphisms from n to m in Th V are the homomorphisms f : F{ xl, ... , x,,, } --> F{ xl, ... , xn }. Since f is fully determined by its values f (xi) E F{ xl, ... , xn }, we can describe hom(n, m) as the set of all m-tuples of terms in the variables Si, ... , xn.

(2) In the many-sorted case, the objects F{ xi', ... , x ,' } of Th V can be identified with the expressions sl x x sn . Morphisms from sl x . . . x s to tl x . . x t,n are m-tuples (Tl, ... , where Ti is an element of sort ti of F{ xi', ... , x;,° }. For a detailed description see Exercise 3.i. (3) The category Th V has finite products: the free algebra F0 is a terminal

object, and every object F{ xi', ... , x;,° } is a product of the F{ x;' } (i = 1, ... , n) with the ith projection given by the homomorphism F{ x7 } - F{ xi', ... , x;,^ } mapping x,' to itself.

3.16 FP sketches. Recall from Definition 1.49 that a limit sketch .' = (A, L, o) consists of a small category A, a choice L of diagrams in A, and a function a assigning a cone v(D) to each diagram D in L. If all diagrams in L are finite and discrete, then .' is called a finite-product sketch, briefly an FP sketch. Thus, a model of an FP sketch is a set-functor turning specified cones to product cones. In particular, each small category A with finite products can be consid-

ered as an FP sketch with L consisting of all finite, discrete diagrams to which o, assigns product-cones. Then the category ModFP A of models of


147

that sketch is the full subcategory of SetA over functors preserving finite products. Theorem. Each finitary variety V is equivalent to the category of models of its theory, considered as an FP sketch, i.e., V

ModFP Th V.

PROOF. Since V is a locally finitely presentable category (Corollary 3.7), it is equivalent to Mod .9 for the limit sketch .9 given by all finite limits in

the category (Presu, V)°P-see Theorem and Remark 1.46. We are going to show that Mod .9 is equivalent to Mod S° for the FP sketch 9 on the category Th V. For each model H : (Pres,,, V)°P -; Set of So the domain restriction to Th V is, obviously, a model of Y. Thus, we have a natural functor

G: Mod.? , Mod.?,

H H H/ThV

(assigning to each morphism the domain-codomain restriction). This functor is full and faithful-to verify this, recall from Remark 3.13 that every

object of Presu, V can be expressed by a coequalizer (in Alg E) of two homomorphisms between objects of Th V. To prove that G is an equivalence (0.12), it remains to verify that each model H : Th V -+ Set of .? is naturally isomorphic to a domain-restriction of a model of We are going to find an algebra A such that H is naturally isomorphic to the domain restriction of horn(-, A) (and the latter functor, restricted to (Pres,, V)°P, is a model of .7, of course). Let V = Alg(E, E) be an equational presentation. For each set X of variables the free V-algebra FX is a quotient of the term-algebra TEX. Thus, given a term r in (TEX) we have the unique E-homomorphism [r] : F{xi} --> FX assigning to xi the congruence class of r in (FX)s. Consequently, for each term r in (TE{xi',..., xn^})s we get a morphism [r] : F{xil, ... , xn^} -+ F{xi}

in Th V.

Observe that the cone

(F{x11,..., xn°} [x`J `1L

F{xi'})n

i_1

is a product in ThV (Remark 3.15(3)). Since H: Th V -- Set preserves finite products, we can assume that n

HF{31 xl , ... , xn I=

HF{x } S;

S

i-1


148

is the cartesian product with projections H[x7'] in Set. Define a E-algebra A as follows. The underlying set of sort s is A, = HF{x'1}

(s E S).

For each operation symbol o : sl x . . . x sn --> s put oA = H[o(xa1l,...,xsnf1) : A,1 x ... x A,n -> A,.

In order to show that A lies in V, we will prove that for each term r in TE{xi', ... , xn^ } of sort s the value of the map H[r] at a given n-tuple (a1i ... , an) in A,1 x . . . x A,,, (or, equivalently, at each many-sorted map a: {xi',...,xn^} .-> JAI) is the value

H[r](a) = a#(r) x;,°} -> A assigns to r. It then which the homomorphism a*: TE{x'i follows that, whenever all V-algebras satisfy an equation r = r' in variables xi', ..., xn°, then A satisfies it too, because [r] = [r']. Since E is finitary, we conclude that A satisfies all equations in E, thus, A E V. We prove the rule H[r](a) = a* (-r) by induction on the complexity of r: (a) If r = x1', then H[r] is the ith projection (by our hypothesis on H),

and a# (xi') = a, too. (b) If r = o(rl...... ,n) and the rule holds for each of the terms rk E (T{x'1', ..., x,")1 k (k = 1, ... , m), we prove the rule for r. In fact, the morphisms [-rk] : F{x",..., xkk} -, F{xlk} yield a morphism m

([rl],...,[rm]): F{xi`}:_1

flF{xlk} =F{xi',...,xm}. k=1

Composing it with [o(x=' )] in Th V, we get [r]. Thus, by our hypothesis on H we have H[r] = H [o(x;'] (H[rl],... , H[rk]), therefore

H[r](a) = H[o(x;')] (H[rl](a),..., H[rm](a)) (definition of 0A) = QA(H[rl](a), ... , H[rm](a))

(induction)

= QA(a#(rl), ... a# (rm))

(a# is a homomorphism)

= a# (LrA(r1, ... , rm))

= a# (r).

Let us verify that H is naturally isomorphic to the domain restriction of hom(-, A). Since both functors preserve finite products, and every object


149

of Th(V) is a finite product of the objects F{xi}, s E S, it is sufficient to observe that hom(F{xi}, A) = A, = HF{xi}.

3.17 Remark: FP sketches define finitary varieties. We have seen that each variety can be described by an FP sketch. Conversely, we will now show that for each FP sketch 9 = (A, L, o) the category Mod. of models is a variety. Let us extend the signature of Example 3.5(3) to E* by adding, for each

o-cone c = (s - si )i=1,.., n, an operation c of arity C : S1 X

X Sn --> S,

and by expanding the equations E to a set E* by the equations 6(7r1(x), ... , ir,,(x)) = x

for each i=1,...,n

'riMX

for each o-cone c = (s L+ si) (including the case n = 0, where the first line reads c = x, and the second line is empty).

The variety Alg(E*, E*) is equivalent to the category of models of (A, L, o). In fact, each algebra A defines a functor F : A --> Set as in 3.5(3)

(Fs = A F f = fA) which preserves L-products because the above equations guarantee that the function A, -* A,1 x .

x A,,, with components F7ri

is a bijection (whose inverse is cA). That is, (Fs F"` Fsi) is a product in Set. Conversely, for any functor F : A -> Set preserving the specified products we define an algebra A in Alg(E*, E*) as follows: A, = Fs, hA = Fh

for h E Amor and given c = (s - si) in L, then cA : Fs -> Fs1 x ... x Fs is the canonical bijection.

3.18 Finitary Monads. Another way of describing varieties of algebras independently of signatures and equations is via monads (see 2.78). A monad T is called finitary provided that T preserves directed colimits. A category isomorphic to )CT for a finitary monad T is said to be finitary monadic over K.

Theorem. Varieties of S-sorted finitary algebras are precisely the finitary monadic categories over Sets.

PROOF. I. For each variety Alg(E, E) of finitary S-sorted algebras the forgetful functor

U : Alg(E, E) has the following properties:

Sets


150

(1) U has a left adjoint F (see Remark 3.6(4)), (2) U creates absolute coequalizers (see Remarks 3.4(7) and 3.6(5)), (3) U preserves directed colimits (see Remarks 3.4(4) and 3.6(6)). By Beck's theorem (see e.g. [MacLane 1971]), conditions (1) and (2) guarantee that Alg(E, E) is isomorphic to (Sets )T for the monad T generated

by the adjoint situation F -4 U. Condition (3) guarantees that T = UF preserves directed colimits (since F preserves colimits), thus, T is finitary.

II. Each finitary monadic category (Sets)T is isomorphic to a variety of S-sorted algebras. In fact, let E be the following signature: for each word sl x x s in S and each element s E S the E-operations U: S1 x

X S16 --> S

are precisely the elements of sort s in the free algebra over the variables xi...... xn" i.e., 01 E (T{xi',... , xn^}),. For every T-algebra (TA -+ A) we define a E-algebra A* on the underlying set of A as follows: given

0-: s1 x...xs,,-+sin E,the value Of crA.:A,, x...xA,,, --r A, inanntuple (al, ... , an) E A,, x

x A,, is defined by forming the corresponding

S-sorted map

a: {xl',...,xn^}-JAI and putting 0-A(al, ... , an) _ (Ta),(cr)

This yields a functor

G: (SetS)T - A1gE

with G f = f on morphisms. This functor is an embedding, since T is finitary: each T-algebra is a canonical directed colimit of the free T-algebras T{xi',... , x ,' }, and free T-algebras are free E-algebras. Thus, the category of T-algebras is isomorphic to the image of G. It is sufficient to verify that im G is a variety. Since G obviously preserves products and subalgebras, imG is closed under products and subalgebras. It remains to prove that imG is closed under quotients. Let A be a E-algebra in imG, and let h: A --- B be a surjective homomorphism. The kernel pair

P1, p2: Aa - A

of h (formed by the projections of the subalgebra A0 of A x A where (x1, x2) E (A0), iff h,(xl) = h,(x2)) lies in imG, since G preserves products

3.B. FINITARY QUASIVARIETIES

151

and subalgebras. Let U : Alg E --p Sets be the natural forgetful functor. Then UPI

UAo

UA -p UB

Up2

is an absolute coequalizer in Sets (see Exercise 3.h). The forgetful functor U G of the monadic category (Sets)T creates absolute coequalizers. Consequently, B lies in imG.

3.B

Finitary Quasivarieties

Finitary quasivarieties are classes of many-sorted algebras that can be described by implications (rather than equations). An implication is a formula of the form A(Ti=T2')

(T-T')

where Ti = T' and r = T' are equations. There is some flexibility in the concept of implication, related to restriction on the size of the set of premises; in the present section we assume that there are only finitely many premises Ti = i , but we will return to this question later. Thus, given a many-sorted signature E, by an implication we understand a formula (*)

(T1

=TT)A...A(TT =Tn)

(r=r)

where Ti = i' and T = T' are equations. A E-algebra A is said to satisfy the implication provided that for each interpretation of the standard variables f : V - JAI such that f#(r) = f # ( 7i') holds for i = 1, ..., n it follows

that f#(T) = f#(T'). 3.19 Remark. A finitary quasivariety of algebras is a class V of E-algebras for which there exists a set of implications (*) such that a E-algebra lies in V iff it satisfies each of the given implications. Every quasivariety V has the following properties. (1) V is closed under subalgebras and products in E (i.e., it is an SP-class, or, equivalently, a full, epireflective subcategory, of Alg E).

(2) V is closed under directed colimits in Alg E. (3) V has free algebras. (4) V has a dense subcategory of finitely presentable regular projectives.


152

The proof is analogous to Remark 3.6.

3.20 Examples (1) Every variety is a quasivariety.

(2) Torsion-free Abelian groups form a quasivariety presented by the equations for Abelian groups (Example 3.5(1)) together with the following countable set of implications:

x+x=0 x+x+x=0

(3)

x=0 x=0

Permutation automata are precisely the finite sequential automata (see Example 3.1(3)) satisfying the implication

. q=4

n(q,o)=n(9,a)

(i.e., each input a leads to an injective next-state function n(-, a)). (4) The category of graphs can be presented as a quasivariety of 2-sorted

unary algebras. The two sorts are "edges" and "vertices", and the operations assign to each edge its source and target, respectively. That is, we put S = { vertex, edge }

and

E = { s, t },

where s, t both have the arity edge-vertex. The quasivariety A presented by the implication (s(x) = t(x)) A (s(y) = t(y)) = x = y

for standard variables x, y of sort edge is equivalent to Gra. In fact, let E : Gra -> A be the functor assigning to each graph K = (X, a) the algebra A, where JAI = (X, a) SA(u, V) = U

tA(U, v) = v

for each edge (u, v) E a. To every graph homomorphism f : K -> K' we assign the E-homomorphism E f = (f, fo), where fo is the domain-

codomain restriction of the map f x f. It is obvious that E is a

3.B. FINITARY QUASIVARIETIES

153

full embedding. To show that this is an equivalence of categories (see 0.12), consider an algebra A in A, and denote by K the graph K = (IAlvertex, a), where the relation a consists of all pairs (sA(e), tA(e)) with e E IAledge. Then A is, obviously, isomorphic to EK.

(5) More generally, for each relational signature E the category Rel E is equivalent to a quasivariety of unary algebras.

3.21 Remarks (1) Unlike Gra and ReIE, the category of posets is not equivalent to a quasivariety: Pos does not have a dense set of regular projectives. In fact, the only regular projectives are the discretely ordered sets. To verify this, consider the following regular epimorphism in Pos:

e

0H0, 1-1,

a

a; F-+ a.

For each poset (X, mor with x y defined if d(x) = c(y).

A partial algebra A of signature E is an S-sorted set JAI = (A,),Es together with partial functions vA : fief A,; - A, for all operation symbols o,: 11iEI Si -> s in E. We denote by Palg E the category of all partial algebras of signature E and all homomorphisms, where a homomorphism from A to B is an S-sorted function f : JAI -+ IBI such that, for each o: fiElsi --> s in E, whenever aA(ai) is defined, then 7B(f,i(ai)) is defined and is equal to f,(rA(ai)). The concepts of terms and equations are introduced exactly as in the

theory of total algebras above. We say that r = r' is an equation in standard variables xj (E Vs,) for j E J if both r and r' are terms in TE({xj }jEJ). A partial algebra A satisfies that equation in the elements aj (E A,,) provided that both r and r' are defined in A under the substitution xj aj and that they give the same result.

3.34 Definition (1) An essentially algebraic theory is a quadruple

r = (E, E, Et, Def)

consisting of a many-sorted signature E of algebras, a set E of Eequations, a set Et C E of "total" operation symbols, and a function Def assigning to each operation symbol o : fliEI si , s in E - Et a set

Def(a) of Et-equations in the standard variables xi E V; (i E I). (2) We say that the theory r is A-ary, for a regular cardinal A, provided that E is.\-ary, each of the equations of E and Def(o) uses less than \ standard variables, and each Def(o) contains less than A equations. (3) By a model of an essentially algebraic theory t we mean a partial Ealgebra A such that (a) A satisfies all equations of E, (b) for each v E Et, the operation cA is everywhere defined,

3.D. ESSENTIALLY ALGEBRAIC CATEGORIES

163

(c) for each a E E - Et with a : H3 EJ Si -* s and any aj E A,, (j E J) we have that 0A(aj) is defined if A satisfies all equations of Def(o) in the elements aj.

The category of all models and homomorphisms is denoted by Mod t. A category is called essentially algebraic if it is equivalent to Mod r for some essentially algebraic theory F.

3.35 Examples (1) Every equational theory is essentially algebraic: we simply put E = Et.

(2) The implicational theory of graphs (Example 3.20(4)) can be turned into an essentially algebraic theory by introducing a new operation Cr E E - Et whose only role is to translate the implication

(s(x) = t(x)) A (s(y) = t(y))

x=y

into a definability statement. Thus, we put S = { vertex, edge }, E_{s,t,o} with s,t: edge -> vertex,

Et={s,t},

a : edge x edge - edge, Def(er) = { s(x) = t(s), s(y) = t(y) }, E _ { o(x, y) = x, o-(x, y) = y } .

A partial E-algebra is a model of this essentially algebraic theory IF if its total operations s, t satisfy the above implication. Thus, Mod t is equivalent to Gra. (3) More generally, every theory given by implications can be easily translated to an essentially algebraic theory: for each implication AiEI(ri =

t') = r = r' we add a new operation symbol c E E - Et depending on all the variables which appear in all ri ,', and then we put

Def(er)={ri= i'iEI), while extending E by the equations a = r and c = r'. (4) The following is an essentially algebraic theory of posets. We add three operations to the theory of graphs in (2): a total operation 6: vertex -+ edge


164

expressing the reflexivity (in each poset 6(x) = (x, x)), a partial operation r (which will translate the implication of antisymmetry (t(x) = s(y)) A (t(y) = s(x)) =* x = y

into a definability condition):

r: edge x edge -* edge Def(r) = { t(x) = s(y), t(y) = s(x) }

and an operation o (expressing the transitivity, i.e., in each poset with edges x = (u, v) and y = (v, w) we have o(x, y) = z for z = (u, w)): o : edge x edge --f edge

Def(o) = { t(x) = s(y) } . We extend E by the following equations: t6(x) = x,

s6(x) = x

r(x, y) = x,

r(x, y) = y

so(x,y) = s(x),

to(x,y) = t(y)

(reflexivity)

(antisymmetry) (transitivity).

The resulting essentially algebraic theory IF has the property that Pos is equivalent to Mod F.

Remark. Since Pos is not equivalent to a quasivariety of algebras (see Remark 3.20), the last example demonstrates that essentially algebraic theories have a wider expressive power than quasivarieties.

3.36 Theorem. A category is locally presentable if it is essentially algebraic.

Remark. In more detail, a category is locally A-presentable if it is equivalent to the category of models of a A-ary essentially algebraic theory. (Thus, locally finitely presentable categories can always be described by an essen-

tially algebraic theory with all arities finite, and with the sets Def(o) all finite.)

PROOF. I. Let r be an A-ary essentially algebraic theory, then we will prove that Mod r is locally A-presentable. It is sufficient to prove that the category Palg E is locally A-presentable, and that Mod r is closed under limits and A-directed colimits in Palg E. It then follows that Mod r is a reflective subcategory of Palg E (by the reflection theorem, 2.48) and

3.D. ESSENTIALLY ALGEBRAIC CATEGORIES

165

the closedness under A-directed colimits guarantees that Mod t is locally A-presentable, see Theorem 1.39. To prove that Palg E is locally A-presentable, we consider it as a subcategory of the category Rel E* of relational structures of the signature E* = E where for each operation symbol o: fliEI si --+ s in E we define

the arity of the (relation) symbol o in E* to be (fiEI si) x s. Then we have a full embedding E : Palg E - Rel E* assigning to each partial Ealgebra A the relational structure EA on the same underlying set, and with each partial operation oA: RE, A,; -+ A, interpreted by its graph, i.e., by the corresponding subset of (fl EI A,;) x A,. The image of this full embedding is a full subcategory of Rel E* characterized by the property that for each symbol o E E* of arity (fl EI si) x s the following holds: if oA contains ((a=)iEi, a) and ((aj)=EI, b), then a = b. It is easy to verify that this full subcategory is closed in Rel E* under limits and A-directed colimits. Since Rel E* is locally A-presentable (see Example 1.18), it follows that Palg E is locally A-presentable by Theorem 1.39.

The subcategory Mod r is closed under limits in Palg E because an equation (in E or in Def(a)) holds in the limit of a diagram of partial algebras Ai if the corresponding equations hold in each Ai. Analogously, a partial operation is defined in an I-tuple of the limit if the corresponding partial operations are defined in the corresponding I-tuples in each Ai. Closedness of Mod t under A-directed colimits follows from the fact that all the conditions (a), (b), (c) defining a model of IF (see Definition 3.34) involve in each instance less than A variables.

II. Let K be a locally A-presentable category. Then K is equivalent to a full, reflective subcategory of SetA (A a small category), closed under A-directed colimits, see Theorem 1.46. Without loss of generality, let us assume that K is actually a full, reflective subcategory of Set A, closed under Adirected colimits. Recall that SetA is isomorphic to a variety of unary algebras (Example 3.5(3)).

Thus, we can consider K as a full, reflective subcategory of the category Alg E of unary algebras of signature E, closed under A-directed colimits. Moreover, by Theorem and Remark 1.39, we have a set M _ { A; m-'+ A; }iEI of morphisms of Alg E such that

K=Ntl, each Ai is A-presentable in Alg E, and mi : Ai -+ A= is a reflection of Ai in K. Let us choose a presentation of Ai by ai < A variables, say, variables yi,k of sort si,k for i E I, k < a;


166

and by ,Qi < .\ equations (see Theorem 3.12). Denote by pi : {yi,k} -> JAI the interpretation of variables used by the presentation. We now define an essentially algebraic theory F = (E*, E, E, Def)

for which we will prove that K Mod F. The set E of total operation symbols is the above unary signature E. This we extend to E* by choosing, for each i E I and a E IA I, an operation

hi a : J Si,k

S.

k P are different because the elements of the two copies of B \ A are never merged. Thus, for each epimorphism e : A --* B, the subalgebra e(A) of B is not proper.) (2) Prove that for each epimorphism e : A --> B and each E-algebra C, given an S-sorted function f : J BI JCJ such that f e : A -> C is a homomorphism, then f : B --* C is also a homomorphism.

3.c Regular projectives. Find a dense subcategory formed by regular projectives, if possible, in the categories Sets, Vec, Grp, Gra, Aut, CPO.

3.d Finitely Generated and Finitely Presentable Algebras (1) Prove that a vector space in Vec is finitely generated if its dimension is finite. Conclude that finitely presentable and finitely generated vector spaces coincide.

(2) Find a finitely generated semigroup which is not finitely presentable.

(Hint: consider generators x, y, z and equations xy" = xz" for

n= 1,2,3,....)


170

3.e FP sketches and Varieties (1) Construct an FP sketch describing the variety of semigroups. (2) Construct an FP sketch describing deterministic sequential automata.

(3) Which variety of algebras is described by the FP sketch 9 _ (A, L, a), where A is the four-element Boolean algebra { 0, a, a,1 }, L consists of the two discrete diagrams 0 and { a, a }, Q(O) is the empty cone with domain 1, and v({ a, a}) is the cone a +- 0 --> a'?

3.f Quasivarieties (1) Let E be the one-sorted signature of a unary operation a and nullary operations ro, r1, r2, .... Is the class of all E-algebras A satisfying

((r2n)A = (r2n+1)A

for all n E w)

vA(x) = x

a (finitary) quasivariety? Is it an epireflective subcategory of Alg E? (2) Can the quasivariety of torsion-free groups be described by finitely many implications?

(3) For each many-sorted relational signature E find a quasivariety of algebras equivalent to Rel E. (Hint: see Example 3.20(4)).

3.g Effective Equivalence Relations (1) Verify that Set has effective equivalence relations. (2) Prove that if K is a category with effective equivalence relations, then so is K-4 for each small category A. (3) Verify that Pos does not have effective equivalence relations.

3.h Absolute Coequalizers (1) Verify that in every category a coequalizer A

3 B `> C 9

A and c : C , B with is absolute whenever there exist morphisms f : B f f = idB, c - c = idc, and g f = c - c. (Hint: show that the given equations together with c f = c g imply that c is a coequalizer of f and g. Every functor "preserves" such equations.)

(2) Prove that in Sets every coequalizer of an equivalence relation is absolute.

HISTORICAL REMARKS

171

(3) Conclude from (1) and (2) that the forgetful functor U: Alg E --> Sets creates coequalizers of equivalence relations.

3.i Theory of V-algebras (1) For each variety V of one-sorted finitary algebras prove that the theory Th V (see Definition 3.14) is isomorphic to the category whose objects are natural numbers, and whose morphisms from n to m are all mtuples (r1, ... , rm) of terms in the variables x1,. .. , x (i.e., m-tuples of elements of F{ xl, ... , with composition defined by substitution as follows. Given morphisms k

(T,,...,*n)

n

m

their composite is

(r1...... ,)

-

(.1....,Qm) _ (r1(0-1,...,em)),...,rn(.71,...,em))

where r{ H rq (al, ... , um) denotes the substitution of x2 by o, for i =

1, ..., M. (2) In an analogous way, describe Th V for a variety of S-sorted finitary

algebras as the category whose objects are expressions Si x ... X Sn and morphisms from sl x x sm to t1 x ... x tm are m-tuples (r1, ... , rm) where ri is an element of sort t; in F{ xil, ... , xn' }.

Historical Remarks The father of universal algebra is Garret Birkhoff who introduced (onesorted) algebras as sets endowed with operations, and who, inter alia, characterized equational classes of algebras as HSP-classes (Theorem 3.9) in [Birkhoff 1935]. The many-sorted approach, inspired by early computer science, was first formalized in [Birkhoff, Lipson 1970].

Categories of algebras presented by equations and implications were studied e.g. by [Mal'cev 1958], [Isbell 1964], [Linton 1966], and [Felscher

1968]. The characterization of quasivarieties in Theorem 3.22 is due to [Mal'cev 1956]. The abstract characterization of quasivarieties (Theorems 3.24 and 3.33) is due to [Isbell 1964], an abstract characterization of varieties was presented in [Lawvere 1963].

172


An entirely new view of algebraic categories was presented by F. W. Lawvere in his dissertation [Lawvere 1963]: an algebra is a finite-productpreserving functor, and a homomorphism is a natural transformation (Theorem 3.16). A generalization of Lawvere's algebraic theories to infinitary algebras is due to [Linton 1966]. Monads as a means of describing algebraic categories were first used by [Eilenberg, Moore 1965], and a characterization of monadic categories over Set was presented in [Linton 1966]. The idea of essentially algebraic category stems from [Freyd 1972], and it was studied in [Adamek, Herrlich, Rosicky 1988] and [Adamek, Herrlich, Tholen 1989]. The characterization in Theorem 3.36 has not been, to our knowledge, published before. Further information about varieties of partial algebras can be obtained in [Reichel 1984].

Chapter 4

Injectivity Classes In the present chapter we study two special classes of accessible categories: weakly locally presentable categories, which are the accessible categories with products, and locally multipresentable categories, which are the accessible categories with connected limits. Recall that locally presentable categories can be characterized by orthogonality (in the sense that they are just the small-orthogonality classes in categories SetA, see Theorem 1.46). We will show that weakly locally presentable categories can be characterized by injectivity (Theorem 4.11), and locally multipresentable categories by cone-orthogonality (Theorem 4.30). And, while locally presentable categories are precisely the categories sketchable by limit-sketches, weakly locally presentable categories are sketchable by limit-epi sketches (i.e., sketches whose models are set-valued functors preserving certain limits and certain epimorphisms) (Theorem 4.13), and locally multipresentable categories are sketchable by limit-coproduct sketches (i.e., models preserve certain limits and certain products) (Theorem 4.32). Orthogonality w.r.t. a morphism can be generalized to orthogonality (or injectivity) w.r.t. a cone as follows: an object K is orthogonal if every morphism from the domain of the cone to K has a unique factorization through a unique member of the cone. (And K is injective if every morphism from

the domain of the cone to K has a factorization through some member.) Accessible categories are then described as precisely the categories of objects injective w.r.t. a set of cones in some locally presentable category (Corollary 4.18). (This is closely related to the characterization of accessible categories as cone-reflective subcategories of the categories SetA, see Corollary 2.54.) Locally multipresentable categories are described as precisely the categories of objects orthogonal w.r.t. a set of cones in a locally presentable category (Theorem 4.30). 173

CHAPTER 4. INJECTIVITY CLASSES

174

4.A Weakly Locally Presentable Categories Injectivity The following concept "relieves" that of orthogonality by omitting the uniqueness requirement:

4.1 Definition (1) An object K is said to be injective with respect to a morphism

m: A, A' provided that for each morphism f : A , K there exists a morphism P: A' -> K such that the triangle A

M

A'

Zfl K commutes.

(2) For each class M of morphisms in a category K we denote by M- Inj the full subcategory of K of all objects injective w.r.t. each morphism

in M. Conversely, a full subcategory of K is called a (small-) injectivity class provided that it has the form M- Inj for a (small) collection M of morphisms in K. Observe that in the case where m : A --> A' is a monomorphism, we can say that injectivity is the possibility of extending any morphism from the

subobject A of A' to the whole A.

4.2 Examples (1) A poset P is injective w.r.t M = strong monomorphisms in the category Pos iff P is a complete lattice. In fact, we have seen in Example 1.33(5) that each poset P is a strong subobject of the complete lattice I(P) of ideals of P. If P is injective with respect to the embedding P I(P), then this embedding is a split monomorphism, and given h: I(P) --> P with h{x E P I x < p} = p, then each subset M C P has a join, viz., h(M), where M is the smallest ideal containing M.

4.A. WEAKLY LOCALLY PRESENTABLE CATEGORIES

175

Conversely, if P is a complete lattice, then for every strong monomorphism m : A -+ A' we can extend each order-preserving function

f:A,P to A' by the following rule

f'(a)=V{f(x)I xEA, m(x) Y, if Y is M-injective, then X is M-injective. Let m : A --> A' be a morphism in M. In (i), given a morphism f : A REI X, each component of f can be extended to fi along m (since X2 is M-injective), and the morphism f': A' -+ f2EI Xi whose components are fi extends f along m. In (ii), given a morphism f : A , X, there exists

P: A' -* Y with f m = u f (since Y is M-injective) and we just compose f with any morphism v such that v u = idx.


176

4.4 Remarks (1) In a category with pushouts each orthogonality class is an injectivity class. In fact, for each morphism m : A -+ A' we can form the pushout of m and m, and then factorize idA,, idA': A

M .A'

A'

Given a class M of morphisms, put M* = M U {m* I m E M}. Then

M1 = M*- Inj . (2) In a category with pushouts we thus have the following hierarchy of conditions on full, isomorphism-closed subcategories: small-orthogonality class 4 orthogonality class 4

small-injectivity class

4 injectivity class

4

closed under limits = closed under products and split subobjects

4.5 Definition. Let A be a subcategory of K. By a weak reflection of a K-object K in A is meant a morphism r : K -> K* with K* in A, such that every morphism f : K -+ A with A in A factorizes (not necessarily uniquely) through r. If each K-object has a weak reflection, then A is said to be weakly reflective.

Remarks (1) The following implications reflective

weakly reflective

cone-reflective

are obvious.

(2) For subcategories of a category with products we have

cone-reflective and closed under products = weakly reflective. (In fact, for each cone-reflection (K -r 4 Ai)iEI the morphism (ri) : K -> rjiEi Ai is a weak reflection.)

4.A. WEAKLY LOCALLY PRESENTABLE CATEGORIES (3)

177

Any weakly reflective subcategory A of K closed under split subobjects is an injectivity class, therefore it is closed under products. This is not true for weakly reflective subcategories in general (see Exercise 4.c(3)).

(4) For locally presentable categories we will later see that, under certain set-theoretical assumptions, every injectivity class is weakly reflective, and every class closed under products and split subobjects is an injectivity class, see Theorem 6.26.

4.6 Examples (1) Posets with a greatest element form a weakly reflective, full subcategory

of Pos.

(2) The full subcategory of Pos formed by complete lattices (cf. Example 4.2(1)) is weakly reflective in Pos. MacNeille completions (see Exercise 4.c(2)) are weak reflections. This subcategory is not reflective. Analogously, complete Boolean algebras form a full, weakly reflective subcategory in the category of distributive lattices. (3)

Divisible Abelian groups form a weakly reflective subcategory in Ab. Divisible hulls are weak reflections. Again, this subcategory is not reflective.

(4) The cone-reflective subcategory of unary algebras with a cycle (see Example 2.52(3)) is not weakly reflective in the category of unary algebras (because it is not closed under products).

Small-injectivity Classes We know that, for each locally presentable category K, the small-orthogonality classes in K are precisely the reflective, accessibly embedded subcategories (see Theorem 1.39 and Remark 1.35). In the present section we will prove that the small-injectivity classes in K are precisely the weakly reflective, accessibly embedded subcategories. 4.7 Proposition. Every small-injectivity class in an accessible category is an accessibly embedded, accessible subcategory. PROOF. Let M be a set of morphisms in an accessible category K. There exists a regular cardinal A such that K is A-accessible and each domain and codomain of an M-morphism is a A-presentable object.

178


1. M- Inj is closed under A-directed colimits: This is quite analogous to kt) the proof in Proposition 1.35. If (Kt K)tET is a A-directed colimit of M-injective objects Kt, then for every m: A -+ A' in M, each morphism f : A -* K has a factorization f = kt fl for some fl : A , Kt and, choosing

fl : A' -> Kt with fl = fl m, we get f = kt fl with f = f m. Thus K is M-injective.

II. M-Inj is accessible: By Corollary 2.36 it is sufficient to observe that M- Inj is clearly closed under A-pure subobjects.

Example. The category of complete lattices and order-preserving maps is not accessibly embedded into Pos, thus, it does not form a small-injectivity class (although it is an injectivity class, see Example 4.2(1)).

4.8 Theorem. Let A be a full subcategory of a locally presentable category K. Then the following conditions are equivalent: (i) A is a small-injectivity class in K;

(ii) A is accessible, accessibly embedded, and closed under products in K;

(iii) A is weakly reflective and accessibly embedded in K. PROOF. (i) = (ii) follows from Propositions 4.7 and 4.3. (ii) . (i). Let A fulfil (ii). There is a regular cardinal A such that K and A are A-accessible categories and the inclusion of A into K preserves A-directed

colimits and A-presentable objects (see Theorem 2.19). Let K be a Apresentable object of K and let (ft: K -+ A;)jEI be a solution set for K (see Corollary 2.45). The morphism

I=(ft):K--.lA; iEI

is, clearly, a weak reflection of K in A. There is a factorization off through some r: K --* K*, where K* is A-presentable in A. It is evident that r is a weak reflection of K in A. We have verified that every A-presentable object K in K has a weak reflection rK : K K* with K* A-presentable in A. We will now prove

that A= {rK I K E PresAIC }-Inj. In fact, each object of A is, obviously, {rK}-injective. Conversely, let A be an {rK}-injective object of K. Since K is locally A-presentable, A is a


179

colimit of its canonical diagram D : V -* K w.r.t. Presa K, and D is Afiltered, see Proposition 1.22. Next, let D* be the canonical diagram of A w.r.t. { K* I K E Presa K }. Then D* is a subdiagram of D (since K* is Apresentable in K for each K E Pres),K) which is cofinal (see 0.11) because for each D-object K -f-L A the rK-injectivity of A implies that there is a

D*-object K* -L A with f = f* rK (i.e., such that rK: f -* f* is a D-morphism). It follows that D* is A-filtered (see Exercise 1.o(3)), and that A is a colimit of D*. Therefore, A belongs to A because A is closed under A-filtered colimits in K (see Remark 1.21). (ii)

(iii). This follows from Corollary 2.45 and Remark 4.5(2).

(ii). This follows from Theorem 2.53 and Remark 4.5(3), see also Observation 2.4. 0 (iii)

Remark. The above proof of the equivalence of conditions (i) and (ii) is independent of Theorem 2.53.

Weakly Locally Presentable Categories Just as initiality generalizes orthogonality, we will now generalize cocompleteness and local presentability. Recall that a category is locally presentable if it is accessible and cocomplete.

4.9 Definition (1) By a weak colimii of a diagram D is meant a compatible cocone of D through which every compatible cocone of D factorizes (not necessarily uniquely). A category is weakly cocomplete if every diagram in it has a weak colimit. (2) A category is called weakly locally A-presentable provided that it is Aaccessible and weakly cocomplete.

Remark. Any weakly reflective, full subcategory A of a cocomplete category K is a weakly cocomplete: a weak colimit of a diagram D : D --> A is given by a weak reflection of a colimit of D in K.

4.10 Examples (1) The category of non-empty sets and mappings is weakly locally finitely presentable: it has colimits of all non-empty diagrams, and any object is weakly initial.


180

(2) The category of divisible Abelian groups is weakly locally finitely presentable.

(3) The category of CPO's with bottom and continuous (not necessarily strict) maps is weakly locally wl-presentable.

4.11 Characterization Theorem. The following conditions on a category K are equivalent: (i) K is weakly locally presentable;

(ii) K is an accessible category with products; (iii) K is equivalent to a weakly reflective, accessibly embedded subcategory of SetA for some small category A;

(iv) K is equivalent to a small-injectivity class in a locally presentable category.

PROOF. (i) = (iii). Let K be a A-accessible category with weak colimits and put A = Presa K. Then the canonical functor E: K -> SetA°P is full, faithful, and A-accessible (see Propositions 2.8 and 1.26). Each object of SetA°P is a colimit of objects from E(K) (because a set-valued functor is

a colimit of hom-functors). We will prove that if a diagram D: D E(K) has a colimit (Dd 2-+ C)dEDOb; in then C has a weak reflection SetAoP,

in E(K). In fact, let (Dd be a weak colimit of D in E(K). There exists r : C -> C with r cd = Td for all d E DIN. Then r is a weak reflection of C in E(K), since any morphism f : C -+ A with A in E(K) defines a compatible cocone (Dd f-`dam A) dED° b. of D in _ E(K). The unique

f : C - A with f Td = f cd for all d E D°bi fulfils T r = f because f cdforalldEDobi

f

(iii) = (iv) follows from Theorem 4.8. (iv)

. (ii) follows from Propositions 4.7 and 4.3.

4.A. WEAKLY LOCALLY PRESENTABLE CATEGORIES (ii)

181

(i). Let K be a A-accessible category with products. Then K is equiv-

alent to a full subcategory E(K) of SetA°P closed under products and )directed colimits, where A = Presa K (cf. Propositions 2.8 and 1.26). Hence E(K) is weakly reflective in SetAOP (by Theorem 4.8), and thus it has weak colimits (see Remark 4.9). Remark. The above theorem cannot be specialized to a given A (in contrast to Theorem 1.39). In fact, there are full, weakly reflective subcategories of locally finitely presentable categories which are closed under directed colimits, but are not finitely accessible. Consider the following embedding of graphs:

m

a

a

b

Then {m}- Inj is the full subcategory of Gra over all graphs such that any vertex is a source of an arrow. This subcategory is weakly reflective and closed under directed colimits in Gra (see Theorem 4.8) but is not finitely accessible (see Remark 2.59).

4.12 Definition. By a limit-epi sketch is meant a sketch 9 _ (A, L, C, o) such that each C-diagram is a span A

e

B

e

B

B

to which a assigns the cocone A

el

IMB

B B.

B

Thus, models of a limit-epi sketch are functors turning specified cones to limits, and specified morphisms to epimorphisms.

Remark. The sketches in Examples 2.57(1) and (3) are limit-epi sketches. 4.13 Theorem. A category is weakly locally presentable iffit is sketchable by a limit-epi sketch.


182

PROOF. I. Let K be a A-accessible category with weak colimits and put B = Presa K. We first prove that any A-small diagram D : V -> B has a weak colimit in B. Let (Dd -+ K)dEDobj be a weak colimit in K. Since K is A-accessible, K is a A-directed colimit of a diagram D* of objects Di E B k7

and morphisms di j: Di D , say colimD* = (D; =* K)iEI. For each d E D°bi, since Dd is A-presentable, the morphism kd factorizes through kb (d) for some i(d) E I. The number of objects d of V is less than A, and I is A-directed, thus, there is an upper bound i E I of all i(d)'s. Then each kd factorizes as

for some hd:Dd->Di.

kd=kL hd

Given a morphism b : d1 , d2 in D, the equation k; (hd2 Db) = kd2 Db = kdl = k4 hdl

implies, since Dd1 is A-presentable, that there exists i(b) > i such that di i(6) hd2 D6 = d; i(6) hd1. Again, there is an upper bound j E I of all i(6)'s, and then the morphisms Td

form a compatible cocone of D: for each b: d1 --> d2 in D we have hd2 d=(5),5 . d=,i(a) . hd2 . D(6) = d1(5),.i

di,i(6) hd.

= di,; hdl = hd1.

The cocone (Dd Dj*) is a weak colimit of D because the given weak colimit (Dd -f K) factorizes through it: we have kd = ki hd for each d. The object Dj* is A-presentable.

We are going to present a limit-epi sketch 9* for K by modifying the sketch 9' of Remark 2.58. In that sketch .So = (A, L, C, v), every object of A of the form A(D), where D E L, is the domain of the limit cone o(D). Put v(D) = (A(D) aD,d+ Dd)dEDobj, where D: V -* Y(B°P) is a A-small diagram. As proved above, every A-small diagram in B has a weak colimit in B-thus, every A-small diagram in Y(B°p) = B°" has a


183

weak limit in Y(B°P). Let us choose a weak limit (B(D) b°') Dd)dED°bj of D with B(D) in Y(B°P). Denote by

eD : B(D) - A(D) the unique factorization (defined by aD,d eD = bD,d). The sketch Y* is obtained from .9 by substituting each of the canonical diagrams of A(D) (i.e., the diagrams from C) by the span A(D) E°- B(D) -* A(D). That is, .So* is the limit-epi sketch

9* = (A,L,C*,o*) where C* is the set of all spans A(D) f`D- B(D) A(D) for D E L, and a* (D) = o(D) for each D E L. To conclude the proof, we show that .9 and 9* have the same models:

Mod 9* = Mod.?

(-_ K, see Remark 2.58).

Let G : A -> Set be a model of Y. For each D E L we know that G preserves the canonical colimit of A(D) w.r.t Y(B°P). Thus, to prove that GeD is an epimorphism, it is sufficient to prove that each morphism c: YC -+ A(D) with C in B°P factorizes through eD. To this end, factorize the following compatible cone (YC 'D"*') Dd)dED°bj of D through the above weak limit: there exists c: YC --> B(D) with

bD,d c = aD,d c

for each d E D°bj

Then aD,d c = aD,d eD c for all d E D°bi, which implies c = eD c. Conversely, let G : A -* Set be a model of 9* . By Lemma 2.24, to prove that G is a model of V it is sufficient to verify that its domain restriction F to Y(B°P) = BOP is a A-directed colimit of hom-functors. This can be done

analogously to the proof of Lemma 2.24: we make use of the fact that GeD is an epimorphism and (in the last part of the proof) we choose, for the given a E G(A(D)), an element b E G(B(D)) with GeD(b) = a. II. Let V = (A, L, C, o-) be a limit-epi sketch. The category ModY is accessible (by Theorem 2.60) and has products (in fact it is closed under products in SetA) because any product of epimorphisms in Set is an epimorphism. Hence, Mod. is locally weakly presentable by Theorem 4.11.

0

Remark. Again, the theorem cannot be stated for a fixed A (in contrast to Remark 1.52): the sketch from Remark 2.59 is a limit-epi sketch with finite limit-diagrams whose category of models is not weakly locally finitely presentable.


184

4.B A Characterization of Accessible Categories In the present section we prove that accessible categories are precisely the small-cone-injectivity classes of locally presentable categories.

4.14. Recall that a cone is a set of morphisms with a common domain. The set can be empty (in which case the cone is given by the object which is considered as the domain).

Definition (1) An object K is said to be injective with respect to a cone (A " Ai)iEI provided that for each morphism f : A --+ K there exists i E I and a

morphism f': Ai -+ K with f = f'. mi. (2) For each class M of cones in a category IC we denote by M- Inj the full subcategory of K of all objects injective w.r.t. each cone in M. Conversely, a full subcategory of IC is called a (small-) cone-injectivity class provided that it has form M- Inj for a (small) collection of cones in K.

Remark. The injectivity of K w.r.t. the empty cone A means that there is no morphism A K. Observe that every cone-injectivity class is closed under split subobjects.

4.15 Examples (1) Every injectivity class is a cone-injectivity class.

(2) An Abelian group is a torsion group iff it is injective w.r.t. the following cone:

7zP

(formed by the trivial homomorphism and the canonical quotient maps of the additive group 7L of integers to Z p, where p is prime). Thus, torsion groups form a small-cone-injectivity class in Ab.

4.B. A CHARACTERIZATION OF ACCESSIBLE CATEGORIES 185 (3)

Analogously, fields form a small-cone-injectivity class in the category Rng of commutative unitary rings. Here we use the ring 7L[x] of integer polynomials, which we first factor canonically onto Zp[x] (p a prime), and then we take the field Zp[x]* of fractions over it, thus getting a canonical homomorphism hp : 7L[x] -* Z [x]*. Moreover, we denote by ho : 7L[x] , Q[x]* the canonical embedding into the field of fractions over rational polynomials.

A ring has inverses for all non-zero elements iff it is injective to the following cone: Q[x]*

h2

Z2[x]*

Z3[x]*

4 [x] *

The condition 0 0 1 for fields is expressed by injectivity to the empty cone with the domain {0 }.

(4) Linearly ordered sets form a small-cone-injectivity class in Pos, presented by the following single cone: b

ab

a

(5) Connected graphs (i.e., non-empty graphs in which each pair of nodes can be connected by a directed path) form a small-cone-injectivity class


186

in Gra, presented by the unique morphism from the empty graph to any discrete non-empty graph, and by the following cone:

0

e2

a

b

a

-a -> b

a

b

4.16 Proposition. Every small-cone-injectivity class in an accessible category is an accessible, accessibly embedded subcategory. The proof is quite analogous to that of Proposition 4.7.

4.17 Theorem. For each locally presentable category K the small-coneinjectivity classes in K are precisely the accessible, accessibly embedded subcategories of K. PROOF. We know that every small-cone-injectivity class in K is accessible and accessibly embedded by Proposition 4.16. Conversely, assume that A is an accessible, accessibly embedded subcategory of a locally presentable category K. There is a regular cardinal A such that K and A are A-accessible categories, and the inclusion of A into K preserves )-directed colimits and A-presentable objects (see Theorem 2.19). Let K be a A-presentable object

in K, and (fi : K - Ai)iE, be a solution set for K (see Corollary 2.45). For any i E I there is a factorization of fi through some rK,i : K -> K; where Ki is A-presentable in A. Denote the cone (rK,i)iEI by rK. Then

A = {rK I K E PresAK}-Inj. The proof is the same as in Theorem 4.8. 4.18 Corollary. A category is accessible iffit is equivalent to a small-coneinjectivity class in SetA for some small category A.

This follows from Theorem 4.17 and the fact that each A-accessible category K is equivalent to the image of the canonical functor into SetA (Proposition 2.8) for A = (Presa K)°P.

4.C. LOCALLY MULTIPRESENTABLE CATEGORIES

4.C

187

Locally Multipresentable Categories

The concept of orthogonality has been generalized above in two steps: first

to injectivity (by giving up the uniqueness) and then to cone-injectivity (by working with cones rather than single arrows). In the present section we take the second step alone: we consider orthogonality to a cone, and we characterize the small-cone-orthogonality classes of locally presentable categories as precisely the accessible categories with connected limits, or equivalently, the accessible categories with multicolimits. (This corresponds nicely to the characterization theorem 4.11.) Accessible categories with multicolimits are called locally multipresentable categories.

Cone-orthogonality Classes 4.19 Definition (1) An object K is said to be orthogonal to a cone (A Wit, At)tET provided

that for each morphism f : A -i K there exists a unique t E T such that f factorizes through mt and that the factorization is unique. That is, there exists a unique morphism f': At -+ K such that the triangle A

mt At

If K commutes.

(2) For each class M of cones in a category K we denote by M1 the full subcategory of K of all objects orthogonal to each cone in M. Conversely, a full subcategory of IC is called a (small-) cone-orthogonality class provided that it has the form M1 for a (small) collection M of cones in K.

4.20 Examples (1) Every orthogonality class is, of course, a cone-orthogonality class. (2) Fields and torsion Abelian groups are small-cone-orthogonality classes, see Examples 4.15.

(3) Let Pos© denote the category of strict partially ordered sets (i.e., sets with an antireflexive, antisymmetric, transitive relation) and strict


188

order-preserving maps. The class of linearly ordered sets is a smallcone-orthogonality class presented by the following single cone: b

a

a b

a b

0 In contrast, linearly ordered sets do not form a cone-orthogonality class in Pos, as we can see from the following:

4.21 Remark. Let K be a category with a terminal object 1. Every (small-) cone-orthogonality class of K containing 1 is a (small-) orthogonality class.

In fact, let M be a collection of cones with 1 E M1. For each cone (A -'-n--'+ At)tET the unique morphism A -> 1 factorizes through a unique t, thus, T contains just one element.

4.22. We know that orthogonality classes are closed under limits (Observation 1.34) and injectivity classes are closed under products (Proposition 4.3). We will now show that cone-orthogonality classes are closed under connected limits, i.e., limits of diagrams D: V - K such that V is a connected category (which means that D is non-empty and is not a coproduct of two non-empty categories).

Proposition. Every cone-orthogonality class is closed under connected limits.

PROOF. Let M be a class of cones in a category K, and let D : D - K be a lconnected diagram with Dd in M1 for each d E D°bi. If (L Dd)dEDae; is a limit of D, we will prove that L lies in Ml. Choose a cone (A - At) tET

in M. For each morphism f : A - L and each d E Dobi there exists a unique t(d) E T such that ld f factorizes (uniquely) through mt(d). We will prove that t(d) is independent of d. Since V is a connected category, it is sufficient to prove that for each morphism t : d -* d' in D we have


189

t(d) = t(d'). In fact, Id, f factorizes both through t(d') and through t(d) (since ld' f = DS (ld f)), and since Dd' is orthogonal to the given cone, we conclude that t(d) = t(d'). Put t = t(d) for any d. Then for each d E Dobi we have a unique fd : At -+ Dd with ld f = fd' mt. The cone (fd) is compatible: given b: d -+ d' in D then Id' f factorizes through mt both

via fd, and via DS fd, thus, f, , = D8 f'd. Let f': At - L be the unique morphism with fd' = Id f for all d E DON. Then, obviously, f = f mt. The uniqueness of t and f is easy to verify. 4.23 Proposition. In a category with pushouts each (small-) cone-orthogonality class is a (small-) cone-injectivity class.

Remark. This is analogous to Remark 4.4(1).

PROOF. For each cone (A -* At)tET we can form the pushouts of m, and mt for all pairs s, t E T:

AAA, mt1 At

p

JP2

A; ,t

and for s = t we have a unique factorization

Given a class M of cones, let M* be the class of cones extending M by adding to each cone (A - A2)tET (i) the empty cones with domains A; t (for all s, t E T, s # t), (ii) the morphisms A* t - At (for all t E T), considered as singleton cones.

Then

.Ml = M*-Inj.


190

Locally Multipresentable Categories 4.24 Definition (1) A functor F: K --> L is called a right multiadjoint provided that for each object L of £ there exists a cone (L -+ FKi)iEI, with each Ki in K, such that for every arrow L -k + FK with K in K there exists a unique i E I for which a commutative triangle

Lki

FKi f : Ki -+ K a K-morphism

FK exists, and moreover, f is also unique. (2) A subcategory K of a category L is called multireflective provided that the inclusion functor K -+,C is a right multiadjoint.

4.25 Examples (1) Linearly ordered sets are multireflective in the category Pos© of strictly ordered posets. Given a connected strictly ordered poset (X, b' D d

Dd)dEDobj

for i E I

with Bi(D) in Y(B°P) for each i E I. For each i E I denote by ei,D : Bi (D)

A(D)

the unique factorization (defined by aD,d ei,D = bi,D,d) The sketch 9* is obtained from Y by substituting each of the canonical diagrams of A(D) by the discrete diagram { Bi (D) }iEI That is, 9* is the limit-coproduct sketch

.2*=(A,L,C*,v*) where C* consists of the discrete diagrams { Bi(D) }jEI for D E L with Q* ({ Bi(D) }) = (Bi(D) "DD A(D)) iEI and

v* (D) = o(D)

for all D E L.

To conclude the proof, we show that 9 and 9* have the same models:

Mod9* = Mod.

(;z: K, see Remark 2.58).

This follows from the observation that for each D E L the discrete catei,D eory ofBiDA(D), i E I, is a full, cofinal subcategory of the g ( ) comma-category Y(B°P) 1 A(D). In fact, if i # j, then ei,D does not factorize through ej,D (by the definition of multilimit). Given an arrow a YC -c+ A(D) with C in B°P, the compatible cone (YC Dd) of D factorizes through bi,D,d for a unique i E I, i.e., we have c: YC Bi(D) (for each d E D°bi). This implies aD,dei,D.c = aD,de with bi,D,d'e =


195

for each d E D°bi, i.e., c = e=,D c; thus c is the unique morphism from c to e;,D in the comma-category Y(B°p) 1 A(D). Consequently, a functor preserves the canonical colimit of A(D) w.r.t. Y(B°P) if it preserves the e. n coproduct (B=(D) A(D))=EI'

II. Let 9 = (S, L, C, v) be a limit-coproduct sketch.

The category Mod .9 is accessible (by Theorem 2.60) and accessibly embedded to SetA (in fact, closed under A-directed colimits for any regular cardinal A such that all diagrams in D are A-small). Moreover, because connected limits commute with coproducts in Set (see Exercise 4.f), Mod S° is closed in SetA under connected limits. Therefore Mod .9 is locally multipresentable by Theorem 4.30.


196

Exercises 4.a Connected Topological Spaces. Prove that connected spaces do not form an injectivity class in Top (although they are closed under products and split subobjects). (Hint: for any cardinal A there are arbitrarily large connected spaces having all subspaces of cardinality A discrete. Consequently, the two-point discrete space belongs to any injectivity class in Top containing all connected spaces.)

4.b Injective Modules. Let R be a unitary ring. Prove that an Rmodule A is injective w.r.t. the class M of all monomorphisms iff for any right ideal U in R and any R-module homomorphism f : U --> A there is an extension off to a R-module homomorphism f : R -+ A (Baer's criterion). Conclude that M- Inj is a small-injectivity class.

4.c Weakly Reflective Subcategories (1) Show that all graphs containing a loop form a small-injectivity class in Gra closed under split subobjects. Describe all weak reflections of the graph with one vertex and no edge.

(2) For each poset P and each set A C P put ApEPIP< a for each a E A } and A+ = I p E P I p > a for each aEA } . The MacNeille completion of P is the poset P* of all cuts, i.e., all A C P with A = A+-, ordered by inclusion. Verify that the map r : P P* defined by r(p) = {p}- is a weak reflection of P in the category of complete lattices and order-preserving functions.

(3) Show that all sets with at least two elements form a weakly reflective full subcategory of Set which is neither closed under split subobjects nor under products. (4) Show that if A is a weakly reflective subcategory of B and B is a weakly reflective subcategory of C, then A is weakly reflective in C.

4.d Injective Hulls (1) Let M be a collection of morphisms in a category K. A morphism m: A -+ A' in M is called M-essential provided that for each morphism f : A' -* A" we have that

fm E M implies f EM. By an M-injective hull of an object K is meant an M-essential morphism from K into an M-injective object.

EXERCISES

197

Show that an M-injective hull is essentially unique and that it forms a weak reflection in the full subcategory of M-injective objects. That is, if p: K -* P is an M-injective hull of K, then

(a) for each M-injective hull p': K P there exists an isomorphism

i:P--*P' with p' = ip, and

(b) for each morphism f : K --> A such that A is M-injective there exists

a morphism f': P , A with f = f p. (2) A full subcategory A of a category K is called stably weakly reflec-

tive if for any K in K there is a weak reflection r: K -> K* to A such that

fr=r

f is an isomorphism

for any morphism f : K* -f K*. Prove that the following statements are equivalent for any full, isomorphism-closed subcategory A of a category K: (a) A is an injectivity class with injective hulls; (b) A is stably weakly reflective.

(3) Consider the following morphism m in Gra:

0--4

0

1

2

Show that { m }- Inj is not stably weakly reflective in Gra.

4.e Projectivity. The dual concept of injectivity is that of projectivity: an object is said to be projective w.r.t. a morphism m: A -> A' provided that for each morphism f : K -+ A' there exists a morphism f': K A

such that m f = f. (1) Prove that, for any unitary ring R, an R-module is projective w.r.t. M = monomorphisms if it is a split subobject of a free module. Conclude that an Abelian group is projective if it is free. (2) Let

0->A->B--*C-0 be an exact sequence of Abelian groups. Prove that an Abelian group G is projective w.r.t. e if Ext(G, A) = 0.

198


4.f Connected Limits. Prove that connected limits commute with coproducts in Set.

4.g Locally Multipresentable Categories. Prove the following generalization of Proposition 1.16 to multilimits: For any A-small diagram D: V --+ K consisting of A-presentable objects, its multicolimit (Dd ka ` K1)dEDobi, i E I, has all objects K`, i E I, A-presentable.

4.h A Limit-Coproduct Sketch for Fields. Find a limit-coproduct sketch whose category of models is equivalent to Fld. (Hint: proceed in the spirit of Example 1.50(2), adding to the objects a and a2 two objects t (terminal) and u (units). The only colimit diagram is the discrete diagram with objects u and t, and the corresponding cocone has codomain a.)

Historical Remarks Injectivity w.r.t. morphisms is a classical algebraic concept (injective modules, injective groups, etc., see e.g. [Fuchs 1970]). The categorical concept of an injectivity class was introduced by [Maranda 1964]. Limit-epi sketches are introduced by [Lair 1987]. Theorems 4.11 and 4.13 are from [Adamek, Rosicky 1994a], a different, but closely related, result to Theorem 4.13 can be found in [Lair 1987]. Cone-injectivity was introduced by [John 1977] (in the dual form), and

it was studied in the spirit of our treatment by H. Andreka, I. Nemeti, and I. Sain in the late 1970's. See [Andreka, Nemeti 1979a] and [Nemeti, Sain 1982]; in the latter paper the authors attribute the idea of using cones to E. Nelson. The equivalence of sketchable categories and small-coneinjectivity classes in locally presentable categories was proved by [Guitart, Lair 1980]. Our treatment (Corollary 4.18) follows [Adamek, Rosicky 1993]. Locally multipresentable categories are due to J. Diers who introduced

the concept and studied its properties in [Diers 1980a,b]; Theorem 4.32 is due to [Guitart, Lair 1980]. Further generalizations and the relationship to sketches can be found in [Ageron 1992].

Chapter 5

Categories of Models In this chapter we study the axiomatization of classes of structures in (finitary and infinitary) first-order logic. We will show that locally presentable categories are precisely the categories of models of limit theories, and accessible categories are precisely the categories of models of basic theories. There is a substantial difference between those two cases: in the locally presentable case we can specialize to a given cardinal A, and we see that locally A-presentable categories are precisely the categories of models of limit theories in the A-ary logic LA,. Nothing like that is possible in the case of accessible categories: we will see that a finitely accessible category need not have an axiomatization in the finitary logic L,,,, and that a basic theory in L,, need not have a finitely accessible category of models.

5.A

Finitary Logic

Formulas, Models, and Satisfaction Up to now, we have worked with many-sorted operations and many-sorted relations separately (see Example 1.2(4) and Chapter 3). We will now combine them: by a signature we will mean a set E of operation and relation symbols of prescribed arities. A E-structure A is, then, a many-sorted set together with appropriate operations and relations, and a homomorphism is an S-sorted function preserving the given operations and relations. Although infinitary logic is needed for general locally presentable categories, we start with the classical finitary (but many-sorted) first-order logic.

5.1. E-structures and homomorphisms. Let S be a set (of sorts). A finitary S-sorted signature is a set E = Eope U Erel with Eope and Erei dis199

CHAPTER 5. CATEGORIES OF MODELS

200

joint. The elements of Eope are called operation symbols, and f o r each operation symbol ' r E Eope an arity s i x . . x sn - > s is given (s, s1i ... , sn E S);

notation: T : s1 x . . . X sn --> s. The elements of Erel are called relation symbols, and for each relation symbol o- E Erel an arity s1 x ... X sn is given

(si, ... , sn E S). For example, the (one-sorted) signature EOG of ordered Abelian groups has (EoG)ope = {+, -, 01 of arities s x s --> s, s -+ s and s, respectively, and (EOG)rel = {G} of arity s x s. By a E-structure A is meant an S-sorted set JAI = (A$)$ES together with operations

QA:A31 x...xAIn -'As for all o : s1 X

X Sn -> $ in EOpef and relations

CACA31 x XA$ for all a in Erel of arity s1 x . . . x sn. A homomorphism from a structure A

to a structure B is an S-sorted map f : JAI --j JBI satisfying fs (UA(a1, ... , an )) = aB (fs 1(a1), ... , fs. (an ))

for each operation o,: Si x ... X sn -+ s, and (fsl x ... X fsn)(UA) C QB

for each relation o- of arity s1 x x sn. The category of E-structures and homomorphisms is denoted by StrE.

Remark. The category Str E has a number of properties analogous to the corresponding category Alg Eope of algebras. Let us denote by

(-)° : Str E -* Alg Eope

and

1-1: StrE --> Sets

the natural forgetful functors. (1) StrE has (regular epi, mono)- and (epi, regular mono)-factorizations of morphisms. Given a E-structure A, by a substructure of A is meant a E-structure B with IBI C JAI such that for each relation symbol Q of arity s1 x x sn we have UB = vA fl (B3, x ... x If A is a substructure of B, then the inclusion A B is a regular monomorphism (and every regular monomorphism in StrE has such a representation). For each homomorphism f : A --> B consider the (regular epi, regular mono)-factorization f°----- - B0 A°

\l C

5.A. FINITARY LOGIC

201

in Alg Eope. Then an (epi, regular mono)-factorization of f is obtained by defining the relations of C to form a substructure of B, i.e., ac = (m,, x x m,,,)-1(CB) for each relation symbol a of arity s1 x .. x s,,. A (regular epi, mono)-factorization is obtained by defining ac = (e,, x ... x e,n)(CA) for each relation symbol a of arity Si x . . . x s,,.

(2) StrE is complete. A limit of a diagram D in StrE is obtained by forming a limit A° = lim D° of the corresponding diagram D° of algebras in Alg Eope7 and defining, for each a E Erel, the relation CA as the largest relation on A° for which the limit-maps are E-homomorphisms.

For example, a product A = rLEIAi in StrE is obtained from the product HIEI A° of the corresponding algebras (with operations defined coordinate-wise) by defining CA, for a E Ere of arity s1 x . . x s, to consist of precisely those n-tuples ((ai )iEI, ... , (as )iEI)) such that for each i E I

we have (aL,...,a;)ECA. (3) Str E is cocomplete. A colimit of a diagram D in Str E is obtained by forming a colimit A° = colimD° of the corresponding diagram of Eope algebras, and defining, for each a E ECe1, the relation CA as the smallest relation on A for which the colimit-maps are E-homomorphisms. For example, if D: (I, Str E is a directed diagram, then a colimit is formed on the level of sets: if (IDil - . C)iEJ is a colimit in Sets,

then we define operations on C in a unique manner that makes each ci a E-homomorphism (see Remark 3.4(4)), and for each o E Erel of arity s1 x ... x s,,, we put ac = U:EI(Ci)ai x ... X (ci)8 (oDi) (4) A E-structure A is A-presentable in StrE if (a) the algebra A° obtained from A by forgetting the relations is A-presentable (i.e., can be presented by less than A generators and less than A equations, see Theorem 3.12) and

(b) A has finitely many edges, i.e., card U CA < no. oEErei

(5) Str E is a locally finitely presentable category. In fact, for each Estructure A we have, by Remark 3.4(6), a directed diagram D of finitely presentable algebras in Alg Eope whose colimit is the algebra A°. Consider the (obviously directed) diagram D* of all finitely presentable E-structures B

with B° an object of D; then A is a colimit of D*.


202

(6) The forgetful functor I -I : StrE -* Sets is a right adjoint because it is a composite of the forgetful functor of Alg Eope (which is a right adjoint

by Remark 3.4) and the functor (-)° (which has a left adjoint assigning to each Eope algebra A the E-structure with oA = 0 for each relational symbol o). For a left adjoint F : Sets -+ Str E of we call FX the free E-structure generated by the S-sorted set X. 5.2 Formulas. As in Chapter 3, we assume that a set V of standard variables is given, with V, denumerable for each sort s. Terms are defined as in 3.A, using operations only, that is (a) each variable of sort s is a term of sort s, (b) for each operation o : sl x ... x s --+ s and each n-tuple of terms Ti of sort si (i = 1, ... , n), the expression o(Tl...... 11) is a term of sort s. The set Var(T) of (free) variables of a term T is defined by Var(x) = x Var(o(Ti ... r )) = Var(rj) U .. U Var(Tn).

By an atomic formula is meant either an equation Tl = T2 (i.e., a pair of terms), or an expression o(Ti...... n) where o is a relation symbol of arity sl x . . . x sn and Ti is a term of sort si (i = 1, ... , n). For example, x + y = y - x and x < y are atomic formulas in EoG Formulas are built up from the atomic formulas co, V).... by the wellknown logical operators: -IV cp

Eli

cp A i/,

cp V'0

(Vx)cp, (3x)cp

(negation of cp) (ca implies 0) (conjunction of cp and (disjunction of cp and

(universal and existential quantifiers using a variable x E V).

The free variables of a formula cp are those variables which appear nonquantified in cp. That is, the set Var cp of free variables is defined by the following induction:

Var(ri = T2) = Var ri U Var r2

Varo(T1i...,Tn) = Var T1 U...UVarr Var(-'cp) = Var cp Var(cp = Vi) = Var(cp n 0) = Var(cp V 0) = Var cp U Var O Var(Vx)cp = Var(3x)cp = Var cp - {x}.

5.A. FINITARY LOGIC

203

Example: Var(x + y) = {x, y} and Var((dx)(3y)(x + y) = 0) = 0. We write i, j > V.

Since (I, i} for i E I. Define, as above, D a = fu Di. and A = fJu Di, A* _ flu Ds . Further define fl, f2: A --+ A* by fl ([xi]iEI) = [[xi, xi, xi, ... ]] iEI'

f2([xi]iEI) = [[Dij(xi)i<j]iEI. Then A contains the equalizer of fl, f2i which is the substructure E of A over all [xi]iEI for which there exists U E U such that

i E U ==* Ucontains Uiaef

EI

i,Di,j(xi)=xj

The morphism

h: C -+ E,

h([x,i]) = [Di,j(x)] j>i

is one-to-one. In fact, given (x, i), (x', i') with [Dj,i(x)]j>i = [Dj,i,(x')]j>i,, there exists U E U such that for each j E U fl (Ti) fl (Ti') (E U) we have

Di,j(x) = Di,,j(x'), thus, [x, i] = [x', i']. The morphism h is also surjective: for each [xi]iEi E E we choose U as above, then given any io E U we have h([xio, io]) = [xi]iEI; in fact, Ui,, E U and for each j E Uio we know that xj = Dio,j(xio). To see that h is an isomorphism, it remains to show that for each n-ary relation symbol v E E the pre-image of QE under h" is contained in ac. Any n-tuple ([X11, ... , [4 11) in vE can be chosen so that (x=, ... , xi) E OD; for each i E I (see Remark 5.1(3)). For every t = 1, ... , n we have Ut E U corresponding to [x;] E E, and when choosing io E nt 1 Ut, we get h([x;o, io]) = [xf]iEI fort = 1, ... , n. Since C is a quotient of [J1EIDi and (xso,...,x o) E °D;0, we conclude that [xi1

n io] o, io],... , [xia

Et.

5.21 Corollary. Every axiomatizable class of E-structures in LW closed under finite limits in Str E is a locally finitely presentable category (as a full subcategory of Str E). In fact, this follows from Theorems 5.20 and 2.48; the subcategory is reflective in StrE. 5.22 Example of a finitely accessible category which is not axiomatizable

in the logic L,,. Let A be the full subcategory of Alg E where E is the

5.A. FINITARY LOGIC

219

one-sorted signature of operations a (unary) and a (nullary), given by the condition o"(a) = vn-1(a) for some n > 2. A is finitely accessible. However, A is not axiomatizable in L,,, because it is closed under finite limits

in Alg E, but is not closed under infinite products. A stronger property of A is shown in Exercise 5g.

5.23 Theorem. Directed colimits in axiomatizable categories are "standard", i.e., the inclusion ModT -* StrE preserves directed colimits for each theory T in L,,. PROOF. Let D : I -* Mod T be a directed diagram with colimits

(Dj-`.A)jEI

inModT

(Di a`* A )=EI

in Str E.

and We will first prove that the homomorphism h : A -+ A defined by h ai = ai (i E I) is w-pure, and then we will conclude that h is an isomorphism. Denote by E the extension of E by a nullary operation symbol ca

of sort s

for each element a of A of sort s. We consider A as a E-structure. We can characterize w-pure E-homomorphisms f : A -+ B into models B of T via an extension of T: let T be the theory in signature E whose axioms are

(1) all axioms of T, (2) all positive-primitive sentences (Definition 5.14) of signature E which are satisfied by A,

(3) all negated positive-primitive sentences of signature 'E which are satisfied by A.

Let B be a model of T. We obtain (i) a model Bo of T by forgetting the constants ca and (ii) a E-homomorphism f : A -+ Bo assigning to each a in IA! the interpretation of the constant ca in B. By Proposition 5.15, the homomorphism f is w-pure. Conversely, given a model Bo of T and an w-pure E-homomorphism f : A - Bo, we obtain a model B of T by interpreting each ca as the value of f. To prove that h is an w-pure homomorphism, it is sufficient to show that the theory T has a model. In fact, we then get an w-pure homomorphism

f : A - B0 for Bo in Mod T. The cone f a; is compatible with D, thus, it factorizes as f ai = f ai (i E I) for some f: A --> Bo. It follows that


220

f = f h. The purity of f thus implies the purity of h, see Remark 2.28. We are going to use the compactness theorem, 5.19: let T' C T be finite. We will prove that there exists io E I such that Dio is a model of T' for a "suitable" definition of the constants ca. We proceed by induction on the size of T': the case ' = 0 is clear, and for the induction step it is sufficient to prove that for each sentence cp of one of the above types (1)-(3) there exists io E D such that each Dio is a model of cp. The case E T is clear. Let us turn to the case that cp is a positive-primitive sentence W = (J x1 ... xn)(b1 A ... A bk) satisfied by A, where the Vij(x1, ... , xn) are atomic formulas (j = 1, ..., k). Each V)j is either an equation a = ,3 for two terms of signature E, or Oj = a(a1i ... , an) for a relation symbol a E E and for terms a1, ... , an of

signature E. A term a of signature E can be "translated" into a term a* of signature E by choosing, for each occurrence of ca in a, a variable xa of the same sort (with xa # x1 i ... , xn and xa # xb if a # b) and by changing all occurrences of ca to xa. We denote by Oj the corresponding formula of signature E: if V)j is a = /3, then 1/ij* is a* = /3*; if V)j is a(a1,... , an), then Oj is a(ai, ... , an). Suppose that all elements of 1Al for which we substituted ca for xa in 01, ... , Ok are a1, ... , am, and denote (p* _ (3 xal

... xam)(3 x1 ... 'xm)( l A ... A 0k ).

Since A, considered as a E-structure, satisfies cp, it is clear that as a Estructure it satisfies p*. Using the fact that A is a directed colimit of D, we will show that there exists io E I such that Dio satisfies o* . It is then obvious that Dio can be turned into a model of
i* such that the connecting

5.B. INFINITARY LOGIC

221

map Di. - Di+ merges all the different interpretations of x to one interpretation (for any u = 1, ... , n). Thus, Di0 is a model of So. The case where cp is a negated positive-primitive formula is quite analogous. Consequently, by the compactness theorem, T has a model, which proves

that h is an w-pure morphism. To prove that h is an isomorphism, we proceed analogously to the proof of Proposition 2.31. The sets I i = { i E I I

i < j } are pairwise non-disjoint for i E I. Consequently, there exists an ultrafilter U on the set I which contains each of them. Let us form the ultrapower

A*=II Ai

Ai=A

of the object A. Then we can find homomorphisms

p,q:A--+ A* such that h is an equalizer of p and q. (This is completely analogous to the proof of Proposition 2.31.) Since A* is a model of T by Proposition 5.18, the equations

p = q. Thus, h is an isomorphism.

5.B

Infinitary Logic

We have seen above that locally finitely presentable categories exactly correspond to finitary limit theories. It is no surprise that for locally A-presentable categories infinitary logic is needed. But then, again, there is an exact correspondence between locally A-presentable categories and A-ary limit theories. In case of accessible categories, we have seen that finitary logic is insufficient. In infinitary logic we introduce the concept of a basic theory and prove an exact correspondence between accessible categories and basic theories.

Formulas, Models, and Satisfaction 5.24 Infinitary Signature. As in the finitary case, we work with a mixed S-sorted signature E, where S is an (arbitrary) set of sorts. That is, E is a disjoint union of the set Eope of operation symbols a each having an arity Q: 11 si -* s iEI

(Si, s E S)


222

and the set Erel of relation symbols o of arities r(;EI si (si E S). Analogously to the finitary case we work with category Str E of E-structures and homomorphisms. The signature E is called A-ary if A is a regular cardinal with card I < A for all the index sets I in the arities of operation and relation symbols of E. We assume that a standard many-sorted set V of variables is given with card V3 = A for each sort s. Terms are defined analogously to the finitary case: each term is given together with its sort, and

(a) every variable in V$ is a term of sort s, (b) every expression o(ri)iEI where o: HZEI si -+ s in an operation symbol and ri is a term of sort si (for each i E I) is a term of sort s. Atomic formulas are, analogously to the finitary case, either equations rl = r2 between terms, or expressions o(ri)iEI where o is a relation symbol

of arity REI si, and ri is a term of sort si (i E I). Formulas are built up from the atomic formulas by applying the following logical operators finitely many times: -' JAI from the index set I to the underlying set of A (with a(i) of sort si), then we simply write AJ:= go[a].

The logic La is the collection of all formulas of La with the above satisfaction relation J. The notion of sentence is defined precisely as in the finitary logic L,. When no cardinal restrictions are considered, we work with the logic here the standard variables form a proper class in each sort, and the formulas and satisfaction form a union of the logics LA for all regular cardinals A. Analogously to L, , a set T of sentences in La (or Lam) is called a theory, and the class of all models of T (i.e., all E-structures satisfying every sentence in T) is denoted by Mod T. We do not consider large theories T (even


225

in Lam); thus every theory in L,, is actually a theory in La for some A. Observe that in the logic L,,o (which has no size restriction on conjunction nor quantification) we always have Mod T = Mod (p for cp = AT. A class of E-structures is called axiomatizable in La (or L,) if it has the form Mod T

for some theory T in La (or L,, resp.).

5.27 Examples (1)

Finite sets are axiomatizable in the logic Lw, (but not in L"). In fact, consider the following sentence in the trivial signature (one sort, no symbol):

(V xo, xl, x2, ...) V (xi = xj)i,j Ew

i0j

More generally: all sets of cardinality less than A form an axiomatizable class in LA+ (where A+ is the cardinal successor of A).

(2) Torsion groups are axiomatizable in L,,,: extend the theory of Abelian groups (see Example 3.1) by the following sentence: (Yx) V (xn = 1).

nEw (3)

Analogously to the finitary case, every infinitary quasivariety is axiomatizable in L by a universal Horn theory, which is a theory of sentences (Vx)(A2 E J cpj #- cp) with cp and cpj (j E J) atomic formulas.

More precisely: a class of structures is closed under products, subobjects, and A-directed colimits if it is axiomatizable by a universal Horn theory in LA,. The proof is quite analogous to that of Theorem 5.12. (4) o -complete semilattices are axiomatizable in L,,,: extend the theory of partially ordered sets (see Example 5.4(2)) by the following sentence: (*)

(VX) (3y) [(

xA

x < Y) A (Vz)

\ \ xA x < z/

(y< z)/ J

where X is countable.

(5) Well-ordered sets are axiomatizable in Lw,: extend the theory of linearly ordered sets (Example 5.4(3)) by the sentence

(Ylxi}iEw) (, A ((xi+1 < xi) A-'(xi = xi+1)) iEw

//


226

Limit Theories and Locally Presentable Categories 5.28 Presentation and Orthogonality Formulas. Analogously to the finitary case (see Remark 5.1(4)), for each A-ary signature E a E-structure A is A-presentable if the underlying algebra A° is A-presentable and card CA < A. We can assume, without loss of generality, that A° is the quotient algebra of the free algebra generated by less than A standard variables xi (i E I) modulo a congruence generated by less than A equations

Tj(xi)iEI = j(xi)iEI for j E J. The relations of A have less than A edges, so we can list them as ok(ok,l)lEIk for k E K (where card K < A). We define a sentence in LA as follows: (IrA)

A (Tj (xi) = Tj (xi)) A A Qk (Pk,l)LElk . kEK jEJ

Models of IrA form the comma-category

A I Str E, in the same sense as explained in detail in L,y (see Remark 5.5).

Let h : A -+ B be a homomorphism in Str E with both A and B Apresentable. Analogously to Remark 5.6 we choose presentations of A (in variables {xi I i E I}) and of B (in variables {yj I j E J}); for each i E I we choose a term of whose congruence class in B is the image of the congruence class of xi under h, and we get the following formula of LA:

(id)I1rA(xi)

(j!)(TB(Yj)AA(x: =Pi(Yj)))] iEI

Models of Irh are just the E-structures orthogonal to h; this is completely analogous to the case A = w in Remark 5.6. 5.29 Definition. By a limit theory in LA is meant a theory each sentence of which has the form (Vx) (o(x)

(3!y) '(x, y))

where p(x) and 1(x, y) are conjunctions of atomic formulas.

5.30 Characterization Theorem. Let A be a regular cardinal. Locally A-presentable categories are precisely the categories equivalent to categories of models of limit theories in LA.

The proof is analogous to that of Theorem 5.9.


227

Basic Theories and Accessible Categories 5.31 Definition. A formula is called (1) positive-primitive if it has the form (3y) b(x, y), where '(x, y) is a conjunction of atomic formulas, (2) positive-existential if it is a disjunction of positive-primitive formulas,

(3) basic if it has the form (V x) (`p(x)

'0(x))

where Y with X infinite. This category is wi-accessible, but not finitely accessible.

5.41 Downward Lowenheim-Skolem Theorem. For each A-ary signature E there exists a regular cardinal) > A such that every E-structure A has the following property: each subset of JAI of power less than a is contained in a A-elementary substructure of A of power less than . .

PROOF. Let A be a E-structure. For each subset X of JAI we construct a A-elementary substructure X,, of A, and then we will show that there exists a regular cardinal A such that if X has power less than A, then X), also has power less than a. We define a chain Xi (i < A) of many-sorted subsets of JAI by the following transfinite induction:

First step: Xo = X. Isolated step: given Xi, consider the set of all triples (cp, Z, a) where cp is a formula of LA,

Z is a subset of Var y, and

a: Z --> JAI is a mapping for which there exists an extension a: Vary -+ JAI with A 1 y[-a].


233

For each such triple we choose an extension a : Var iO -> JA with sp[a], and we denote by im a the many-sorted image of a. Then A we define

Xi+1 = X2 U U ima (W,Z,a)

(where (

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