Algebra of Proofs (Studies in Logic and the Foundations of Mathematics)

ALGEBRA OF PROOFS

M. E. SZABO Concordia University, Montreal

1978

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Library of Congress Cataloging in Publication Data

Szabo, M . E. Algebra of proofs. Studies in logic and the foundations of mathematics; v. 88) Bibliography: p. Includes indexes. 1. Proof theory. 2. Categories (Mathematics) 3. Combinatory logic. I. Title. 11. Series. QA9.54284 51 1'.3 ISBN 0-7204-2286-8

77-706

PRINTED IN THE NETHERLANDS

To Isabel and Julianne

PREFACE

In this monograph, we study the algebraic properties of the proof theory of intuitionist first-order logic in a categorical setting. Our work is based on the confluence of ideas and techniques from proof theory, category theory, and combinatory logic, and this book is addressed to specialists in all three areas. Proof theorists will find that categories give rise to a non-trivial semantics for proof theory in which the concept of the equivalence of proofs can be investigated from a mathematical point of view. Categorists, on the other hand, will find that proof theory provides a suitable syntax in which commutative diagrams can be characterized and classified effectively. Workers in combinatory logic, finally, may derive new insights from the study of the algebraic invariance properties of their techniques established in the course of our presentation. We have divided the material into thirteen short chapters in which we explore systematically the algebraic properties of the usual operations of logic. In each case, we begin by constructing a category of a certain type as an algebraic model for the class of proofs being studied. We then prove a completeness theorem to the effect that the arrows of the constructed category can be represented by derivations in a Gentzen system with cut elimination. In the propositional cases, we use the algorithmic nature of the cut elimination process as a basis for an effective description of the arrows of the constructed categories and develop decision procedures, in the form of Church-Rosser theorems, for the commutativity of the finite diagrams of these categories. As corollaries, we obtain solutions of the word problems for the associated free-object functors. We then show that quantifiers fit smoothly into the calculus of adjoints, and describe the topos-theoretical setting in which the proof theory of intuitionist first-order logic possesses a natural semantics. The chosen functorial characterization of quantifiers necessitates a brief sortie into the world of infinitary logic for the construction of suitably complete categories. vii

viii

PREFACE

This study is self-contained. All globally relevant definitions have been collected together in Chapter 1. More specialized concepts are introduced when needed. We eschew the use of subscripts, and have tried to make the notation as simple and readable as possible. We have also attempted to strike a happy balance between formal definitions and informal proofs. In each case, we have tried to present the essential ideas by avoiding an excessive level of generality. The end of the proof of a theorem or of the statement of a result whose proof is omitted is marked with the symbol 0. Cross references to an item in the text are given in terms of the last section number preceding the item, and references to an item in the Bibliography are given by name of author and year of publication. Moreover, the deductive systems required for Chapters 2-12 are summarized in Appendices A and B for easy reference, and the statements of the cut elimination and normalization algorithms on which most of our results depend are relegated to Appendices C and D in order to avoid unnecessary duplications. The investigation of the basic connections between proof theory, category theory, and cornbinatory logic was pioneered by Joachim Lambek. In many places, our own work relies heavily on the conceptual insights reflected in his articles on these and related subjects. We gladly acknowledge this debt of gratitude. We are also grateful to Saunders MacLane for reading critically a preliminary version of one of our papers. This has led to major changes and improvements in our work. Furthermore, our thanks go to the logicians and categorists of Montreal, Oxford, and Tlibingen who have listened patiently to successive presentations of our ideas as they developed. Additional thanks are due to the National Research Council of Canada for their continuous support of our research, and to the Committee for the Advancement of Scholarly Activities of Concordia University for their assistance of the technical production of this book in the form of a publication loan. Finally, we thank Dr. E. Fredriksson and the North-Holland Publishing Company for their encouragement of our work and Dr. Th. van den Heuvel for his care and skilful editing of our manuscript. M. E. Szabo

CHAPTER 1

INTRODUCTION

The formal proofs of deductive systems may be studied as mathematical objects in their own right, and in this book we report the results of an algebraic investigation of the intuitionist proof theories of the usual operations of first-order logic and of the relationships between them. The conceptual basis for our programme is the realization that the Lindenbaum-Tarski algebras of formulas may be viewed as categories, and that the formal proofs of the associated deductive systems determine structured categories as their canonical algebras which are of the same type as the Lindenbaum-Tarski algebras of the formulas of the underlying languages. In the cases of interest, the algebras of formulas and the corresponding algebras of formal proofs are linked by Gentzen’s theorem which asserts that provable formulas code their own proofs. This theorem and reducibility relations with the Church-Rosser property are our principal syntactic tools. We study twelve separate theories of varying linguistic complexity and deductive strength. They divide into two main types: the monoidal type, in which we investigate systems based on the common algebraic properties of conjunction and disjunction, and the Cartesian type, in which conjunction and disjunction have their proper meanings. The finer subdivisions arise from the symmetry and distributivity properties of conjunction and disjunction, and from the absence or presence of implication, distinguished constants, quantifiers, and infinitary operations. The following table (p. 2) displays the propositional theories treated in Chapters 2-12. A check mark in a square indicates that the symbol in the left-hand column of the row determined by the square occurs in the language of the theory mentioned in the top row of the column determined by the square. The symbol I denotes, ambiguously, the symbols T (true) and 1 (false), and the symbol XI denotes, also ambiguously, the symbols A 1

2

INTRODUCTION

[1.0

(and) and v (or).The intended meaning of j is i f . . . then, and that of if. The abbreviations in the top row stand for monoidal, symmetric monoidal, Cartesian, bicartesian, distributive bicartesian, monoidal closed, symmetric monoidal closed, Cartesian closed, biCartesian closed, residuated, and monoidal biclosed, respectively. The common feature of the theories in the above table is the fact that the commutative diagrams of their generic models are effectively classified by the deducibility relations on their associated deductive systems. Hence quantifiers do not appear in this catalogue since the notion of derivability for formulas with quantifiers is not decidable. Negation is absent since, in keeping with intuitionist practice, we do not take this operation as primitive. Whenever required, we can define it in terms of and 1. Our reason for limiting ourselves to intuitionist theories, finally, is algebraic. In Chapter 10, we show that Boolean algebras do not generalize to non-equivalent categories. Since the Lindenbaum-Tarski algebra of classical logic is Boolean, the algebra of classical proofs therefore reduces isomorphically to its motivating algebra of formulas. The ultimate source of inspiration for our work is Gentzen’s doctoral

+ only

+

I .O]

PRELIMINARIES

3

dissertation. In view of its singular influence on all aspects of proof theory, we give it prominence by listing it here in the Introduction separately from the other entries in the Bibliography: GERHARDGENTZEN [ 19351 Untersuchungen iiber das logische Schliessen, Mathematische Zeitschrift 39, 176-210, 405431.

An English version of this paper may be found in SZABO[1969], and all references to Gentzen’s work are given in terms of their English equivalents in that volume. An effective proof of Gentzen’s theorem that provable formulas code their own proofs requires a sequent calculus as an auxiliary structure and is based on the distinction between structural and operational rules of inference. It turns out that even at the categorical level, this type of structure and the distinction between two kinds of rules of inference provides the right conceptual setting for a variety of difficult combinatorial problems. For this reason we give a full axiomatic description of these structures in this Introduction. They will be called sequential categories and appear as background structures in each chapter below. However, we omit the individual verifications that the particular sequential categories required satisfy the stated axioms since such verifications are tedious and always routine. Attempts to apply sequential methods to category theory have their beginnings in LAMBEK[ 19691. The use of reducibility relations with the Church-Rosser property in this monograph replaces the earlier use of combinatorial invariants such [1969] and as the scope (LAMBEK[1968]) and the generality (LAMBEK SZABO[ 1974a1) of a proof to classify equivalence classes of derivations. In the mathematically interesting cases, these concepts fail to capture the essential complexity of the equivalences involved, and their successful application requires a drastic normalization of the derivations to which they apply. Carrying out these preliminary calculations is tantamount to proving a Church-Rosser theorem, except for certain trivial uniqueness requirements made unnecessary by these invariants. Hence we have abandoned this approach. A similar failure to classify the arrows of symmetric monoidal closed categories (KELLYand MACLANE [1971]) by means of simple external criteria such as graphs shows up the difficulties inherent in any attempt to give context-free descriptions of structurally subtle mathematical properties.

4

INTRODUCTION

[1.1

Although there is a reasonable consensus among mathematicians that a proof is the equivalence class of its representations, the question of the equivalence of proofs is far more controversial. KREISEL[1971] and PRAWITZ[1971] have made a healthy start by tackling this difficult problem from a philosophical point of view. This monograph represents a corresponding mathematical beginning. It is clear from the work of MANN [1973] that the two approaches interact and in some cases lead to analogous results. In this section, we assemble the necessary facts from category theory and logic to make this monograph accessible to both logicians and categorists. 1.1. Categorical preliminaries

The basic notions of category theory are category, functor, and natural transformation. We motivate their definition by an example. 1.2.2. EXAMPLE. Let P = (P, s P ) be a pre-ordered set, i.e., a set P together with a reflexive, transitive relation S P on P, and let S P X P S p be the set of all ordered pairs ((y, z ) , (x, y)), with (y, z ) and (x, y) E S p . The pre-ordered set P then determines three functions, which we shall call domain, codomain, and composition, and which we abbreviate as dom, cod, and comp, respectively. They are defined as follows: (1) dom : S P+P is defined by the equations dom((x, y)) = x. (2) cod : S P + P is defined by the equations cod((x, y)) = y. (3) comp : S P X P S P +P is defined by the equations comP((Y7 z ) , (x, Y), = (x, 2).

The system C=(P,SP, dom, cod, comp) is the most elementary example of a category. The elements of P are called the objects of C, and we write ObC in place of P. The elements of S P are called the arrows of C, and we write Arc in place of S P . In this notation, S P X P SP = Arc XOW Arc is the set of all pairs (f,g ) of arrows of C with the property that domu) = cod(g). These data satisfy the following axioms:

(C 1) For every A E ObC there exists a 1(A) E A r c such that

1.11

5

CATEGORICAL P R E L I M I N A R I E S

( I ) dom( 1(A))= cod( 1(A)) = A. (2) If dom(f) = A, then comp(f, 1(A))= f. (3) If cod(f) = B, then comp( l(B), f ) = f. (C2) The following equations hold f o r all (f. g ) E A r c ( 1 ) dom(comp(f, 8))= dom(g). (2) cod(comp(f, g ) ) = cod(f).

XObC

Arc:

(C3) The following equations hold for all (f, g, h ) E A r c X o b C A r c X O b C A r c : comp(f, COmp(g, h ) ) = comp(comp(f, g ) . h ) . 1.1.2. DEFINITION. A category is a system C = ( O b C , A r c , dom, cod, comp) consisting of a class ObC, a class A r c , and three functions dom : A r c + ObC, cod : A r c + ObC, comp : A r c XObC A r c + A r c satisfying Axioms (Cl), (C2), and ((23) above, together with the following set-theoretical condition:

(C4) The class C(A, B ) of all f E A r c with dom(f) = A and cod(f) = B is a set, f o r all A , B E ObC. 1.1.3. REMARK.Here and below, the words class and set are used in the sense of Godel-Bernays set theory. 1.1.4. NOTATION. If the category C is clear from the context, we often write [A, B] in place of C(A, B), and write f : A + B in place of f E C(A, B). Moreover, fg and f . g often abbreviate comp(f, g). 1.1.5. DEFINITION. A category C is small if ObC is a set, and large if ObC is a proper class. 1.1.6. DEFINITION. A category C is discrete if C ( A , B ) = 0 for A # B and C(A, A) = {l(A)} otherwise. 1.1.7. DEFINITION. A category C is simple if C(A, B ) = 0 or C(A, B) = { * } for all A, B E ObC. 1.1.8. DEFINITION. An arrowgram is a system A= (A, S A , dom, cod, comp) determined by a transitive, but not necessarily reflexive set (A, IA).

6

INTRODUCTION

[1.1

Let P be the pre-ordered set introduced earlier, and let Q = (Q, SQ)be another pre-ordered set. Then a homomorphism F : P + Q is simply a monotone function, i.e., a function F : P + Q with the property that (x, y ) E PI implies (F(x), F ( y ) )E IQ. We can therefore think of F as a pair (F,, Fa) of functions F, : P + Q and F a : (P --* S Q satisfying the following axioms :

(F4) Fa(l(A)) = I(Fo(A)) for all A E ObC. Thus we are led to the following definition of a homomorphism of categories, known as a functor: 1.1.9. DEFINITION. A functor F : C + D is a pair (F, : ObC+ObD, : Arc + ArD) of functions satisfying Axioms (Fl), (F2), (F3), and (F4) above. F a

1.1.10. DEFINITION. A functor F : C + D is faithful if f# g in C implies that Fcf)# F ( g ) in D for all f, g E Arc. F is also called an embedding.

1.1.11. DEFINITION. A functor F : C + D is full if for every f E D(F(A), F(B)) there exists a g E C(A, B) such that f = F(g), for all A, B E ObC. 1.1.12. DEFINITION. A functor F : C + D is constant if there exists a B EObD with the property that F ( A ) = B and Fcf)= 1(B) for all A E ObC and all f E Arc. We denote F by Const B.

1.1.13. DEFINITION. A diagram in a category C (relative to an arrowgram A) is a pair of functions (F, : ObA+ ObC, Fa: ArA+ Arc) satisfying Axioms (Fl) and (F2) of a functor. 1.1.14. DEFINITION. A commutative diagram in a category C (relative to an arrowgram A) is a diagram in C satisfying Axiom (F3) of a functor.

1.11

CATEGORICAL PRELIMINARIES

7

It will be represented pictorially in the usual way. The set hom(P, Q) of homomorphisms of pre-ordered sets can itself be made into a pre-ordered set by defining f s g iff f ( x ) s Q g ( x ) for all x E P. In an analogous way, the class Funct(C, D) of functors from a category C to a category D can be made into a category, provided that it exists as a class in Godel-Bernays set theory. If C is small, its existence is assured. The arrows of this category are families of arrows of D. 1.1.24. DEFINITION. A natural transformation from a functor F to a functor G is a class v = {v(A) : F ( A ) + G(A) E ArD I A E ObC} whose elements satisfy the equations comp(G(f), v(A)) = comp(v(B), Fcf)) for all f : A + B E Arc. The elements of v are called the components of v. The above equations assert that all diagrams in D of the form

F(A)

"(A)+

G(A)

commute. In order to make Funct(C, D) into a category, we define the composite of two natural transformations v : F + G and p : G + H as follows: comp(p, v ) = {comp(p(A), v(A)) : F ( A ) + H ( A ) E ArD I A E ObC}. This makes the class { l(F(A)) : F ( A ) + F ( A )E ArD I A E ObC} the identity transformation 1(F) of F. Any pre-ordered set P determines another pre-ordered set Popcalled the opposite of P. The set P O p = ( P , Z P ) is defined by the condition that (x, y ) E Z P iff ( y , x) E S P . Analogously, every category C determines an opposite category Cop. 2.2.25. DEFINITION. The opposite category of a category C is the category Cop obtained from C by interchanging the values of the functions dom and cod of C,' and by defining compcw(g, f ) = compc(f, g ) for all f,g E Arc.

1.1.16. DEFINITION. A contravariant functor F : C + D is a functor

F : Cop+D, or a functor F : C +DOP.

8

INTRODUCTION

[1.1

By Ens we mean the category whose objects are the Godel-Bernays sets, and whose arrows-are functions between such sets. Functors with values in Ens will be called set-valued functors. Any A E ObC gives rise to a set-valued functor C(A, -) : C +Ens, and to a set-valued functor C(-, A) : CoP+Ens as follows: (1) Let f : B + C E A r c , and define C(A, f ) : C(A, B)+C(A, C) by the equation C(A, f)(g) = comp(f, g). (2) Let f : C + B E A r c , and define C(f, A ) : C(B, A ) + C ( C ,A) by the equation Ccf, A)(h) = comp(h, f). The functors C(A, -) and C(-, A) are known as hom functors. Let C and D be two categories. Then we can form a new category C x D as follows: Ob(C x D) = ObC x ObD, and Ar(C x D) = A r c x ArD, with composition defined by comp((h, k ) , (f,8 ) ) = (comp(k f), comp(k, g ) ) . 1.1.17. DEFINITION.A bifunctor is a functor whose domain is of the form C x D.

The functor C(-, -) : Copx C +Ens obtained in the obvious way from the hom functors above is a bifunctor. If F : C + D and G : D + E are functors, we can define their composite comp(G, F) : C + E as follows: and

comp(G, F)(A) = G ( F ( A ) )for all A E ObC, comp(G, F ) ( f )= G ( F c f ) )for all f E Arc.

By Cat we mean the category whose objects are all small categories, and whose arrows are all functors between such categories, with composition as defined above. This makes the assignments F ( A ) = A and Fcf) = f for all A E ObC and all f E A r c the identity functor 1(C) on C. 1.1.18. DEFINITION.An arrow f : A + B E A r C is an isomorphism if there exists an arrow g : B + A E A r c such that comp(f, g ) = 1(B) and comp(g, f) = 1W). 1.1.19. DEFINITION.If F, G : C + D are two functors, a natural transformation v : F + G is a natural isomorphism if there exists a natural transformation p : G + F such that comp(p, v) = 1(F) and comp(v, p ) =

1.11

CATEGORICAL PRELIMINARIES

9

Two objects A, B E ObC are isomorphic if there exists an isomorphism between them, and two functors F, G : C + D are isomorphic if there exists a natural isomorphism between them. We write A = B and F = G, respectively. Two categories C and D are equivalent if there exist functors F : C + D and G : D + C with the property that comp(G, F) = 1(C) and comp(F, G ) = l(D). The following lemma will be useful for proving objects isomorphic: I . 1.20. YONEDALEMMA.If F : C + Ens is a functor and A E ObC, then there exists a bijection Y : F ( A ) .+Nat(C(A, -), F ) between F ( A ) and the set Nat(C(A, -1, F) of natural transformations from C(A, -) to F.

PROOF.Let x E F ( A ) , define the function Y(x)(B) by the equation Y(x)(B)(f) = F(f)(x), and let Y(x) = { Y(x)(B) : C(A, B ) + F ( B )E ArEns 1 B E ObC}.

for all B, C E ObC and all g : B + C . On the other hand, let u E Nat(C(A, -), F). Then

1.1.21. COROLLARY. If u : C(A, -)+C(B, -) is a natural isomorphism, then u(A)( I(A)) : B + A is an isomorphism. Similarly for C(-, A)

C(-, B). 0

10

[1.1

INTRODUCTION

Next we define one of the most central notions required in this monograph: 1.1.22. DEFINITION. Two functors F : D + C and G : C + D are adjoint if there exists a natural isomorphism a(-,-) : D(-, G(-))4C(F(-), -) of , are the set-valued bifunctors bifunctors, where D(-, G(-)) and C ( F ( - ) -) o n D o P x C determined by F and G. The isomorphism a is called an adjunction.

The adjunction (Y determines two natural transformations 17 and respectively called the unit and counit of the adjunction:

E,

17 = {q(B): B + G ( F ( B ) )E ArD I B E ObD}, E = {€(A) : F(G(A))+

A E A r c I A E ObC},

where q(B)= a-'(l(F(B))) and €(A) = a(l(G(A))). The naturality of 77 is a consequence of the commutativity of the following diagram: C(F(A), F(A))

D(A, G

C ( F ( A ) ,FUN

J.

C(F(B), F ( B ) )

4

D(B, G ( F ( B ) ) ) .

The commutativity of a similar diagram establishes the naturality of E . The transformations 17 and E relate the values of (Y and a-' to the composition laws of the categories C and D as follows:

PROOF.By the naturality of a, the diagram

1.11

CATEGORICAL PRELIMINARIES

D(G(B), G ( B ) ) IN.G ( B ) )

i

D(A, G ( B ) )

a(G(B).B),

I1

C(F(G(B)), B )

I

U F W . 8)

>

C(F(A), B )

commutes. Hence a ( A , B)(f) = comp(a(G(B), B)(l(G(B)), F(f))= comp(E(B), F(f)) by the definition of E. A similar commutative diagram based on the naturality of a - ’establishes the second equation. 0 The following is an easy example of adjoint functors: 1.1.24. EXAMPLE. Let prOrd be the category whose objects are preordered sets and whose arrows are order-preserving functions. Then there exists a pair F : Ens+prOrd and U : prOrd+Ens of functors defined on objects as follows: F ( S ) = (S, %), with 5 s = {(x, x) 1 x E S}, i.e., F ( S ) is the free pre-ordered set generated by S, and U ( ( P ,I P )=)P, i.e., the set of elements of the pre-ordered set (P,IP). F and U are functorial and adjoint in the sense of Definition 1.1.22. The free monoid construction yields another example of adjoint functors:

1.1.25. EXAMPLE. Let Mon be the category whose objects are monoids and whose arrows are homomorphisms of monoids. Then there exists a pair M : Ens+Mon and U : Mon+Ens of functors defined on objects as follows: M ( S ) is the monoid consisting of the set of all finite sequences of elements of S, with concatenation as binary operation. U((A, X, I ) ) = A, i.e., the set of elements of the monoid ( A , x, 1). M and U are functorial and adjoint in the sense of Definition 1.1.22. The object M ( S ) is called the free monoid generated by S. The functors U in the previous examples are called forgetful functors. Their left adjoints F and M are called the free pre-order functor and the free monoid functor, respectively. They are two examples of many familiar free object functors F : Ens+C, left adjoint to forgetful functors U : C +Ens. An important combinatorial problem for free C-objects is the solution of the word problem for these objects, i.e., the problem of constructing

12

INTRODUCTION

[I.!

an algorithm for deciding the equality relation on these objects. Since free C-objects are determined, up to isomorphism, by a free C-object functor F, we can think of the word problem for free C-objects as the word problem for the free C-object functor F. By analogy, we define the word problem f o r a left adjoint functor F : Cat +C to a forgetful functor U : C + Cat as the problem of finding an algorithm which decides the equality of arrows of the objects E ( X ) relative to the equality of arrows of the objects X. Part of our work in Chapters 2-12 below consists of solving word problems for free object functors on Cat by proof-theoretical means. It is clear from the work of Gentzen (cf. SZABO[1969]) that the appropriate setting for certain syntactic problems of logic are systems with generalized transitivity. It has turned out that this need for generalization derives from the nature of the problems being tackled and the effectiveness of the methods of solution being employed. Hence it is not surprising that related problems in category theory are solved most easily by working in mathematical systems based on generalized composition. These systems will be called sequential categories. They have substitution as their fundamental operation.

1.1.26. DEFINITION. A sequential category is a system C = (ObC, Arc, dom, cod, subst) consisting of the following data: (1) A class ObC, whose elements are called objects. (2) A class Arc, whose elements are called arrows. (3) A function dom : Arc + M(0bC). (4) A function cod : Arc + M(0bC). (5) A function subst : Arc X,,, Arc + Arc, where M(0bC) stands for the free monoid generated by ObC, and where Arc xWxoArCand subst are defined as follows: The class Arc xmx,ArC is the disjoint union of the family of sets A r c x(,,

”)

Arc = {(f, g ) E A r c X ArC 1 codcf), = dom(g)”)

whose elements have the property that the p-th term of codcf) is the same as the v-th term of dom(g), with ( p , v) E w x w, and subst is the unique function determined by the universal property of disjoint unions from a family of functions

1.11

CATEGORICAL PRELIMINARIES

subst(p, v) : A r c x(,,.

.)

13

Arc +Arc.

The functions subst(p, v) are such that if domcf) = r, codcf) = Q yV, dom(g) = A y A , cod(g) = 0, and cod(& = dom(g), = y, then dom(subst(w, u ) ( g ,f ) ) = ArA and cod(subst(p, u ) ( g ,f)) = QOV. The relationship between f, g, and subst(p, v)(g,f ) is depicted in tree form by

f:r+@yV g:AyA-+O subst(w, u ) ( g ,f ) : A r A + QOV or simply by

r + a v ~A++@ A r A 3 QOV

if the labels f, g, and subst(p, v ) ( g , f ) are clear from the context. These data satisfy the following axioms: (SCl) For each y E ObC, there exists u l(y) : y + y E A r c such that (1) dom(l(y)) = cod(l(y)) = Y .

14

INTRODUCTION

[I.]

1.1.27. REMARK.The symbols appearing in axioms (SC3)-(SClO) have been arranged in a way that reveals the connection between these axioms: The deletion of I' and A from the domains of the arrows in (SC3)-(SC6) yields the domains of the arrows in (SC7)-(SClO), and the deletion of Y and R from the codomains of the arrows in (SC7)-(SClO) yields the codomains of the arrows in (SC3)-(SC6). Since sequential categories are a generalization of categories, we continue to use the terminology of category theory and define a functor between sequential categories as follows:

1.11

CATEGORICAL PRELIMINARIES

15

1.1.28. DEFINITION. A functor F : C + D is a pair (F, : ObC +ObD, Fa: A r c + ArD) of functions satisfying Axioms (Fl), (F2), and (F4) of a functor defined previously, together with the following generalization of Axiom (F3):

1.1.29. DEFINITION. A sequential category C is a subcategory of a sequential category D if ObC ObD, A r c ArD, if the functions dom, cod, and subst of C are the restrictions of the corresponding functions of D, and if all identity arrows of C are identity arrows of D. We call C a full subcategory of D if C is a subcategory of D, and if C ( A , B ) = D(A, B) for all A, B E ObC.

c

c

1.1.30. REMARK.It is clear that every category is sequential, and hence it makes sense to speak of a category as being a full subcategory of a sequential category. This is the only sense in which we shall have to use the concepts defined in 1.1.29.

In this monograph, we require sequential categories with arrows of the form subst(p, v)cf,g ) for p, v > 1 only as auxiliary structures. However, many familiar categories can be thought of non-trivially as sequential categories if we use some of their additional structure to define the functions subst(p, v). In Ens, for example, we can think of arrows f : A1 * * . A,,, +. B I * * . B,, as functions f whose domain consists of the Cartesian product of the sets A , , . . . ,A,,,, and whose codomain consists of the disjoint union of the sets B I ,. . . , B,, with subst really meaning substitution in the usual sense. Distributive lattices, considered as categories qua partially ordered sets, are other genuine sequential categories. Here the arrows f : A I . * * A,,, + B I . ’ * B, are pairs (inf(AI,. . . , A,,,), sup(B1,. . . ,B,,,)) with the property that inf(A1,. . . , A,,,) 5 sup(B1, . . . , B,,). It is clear from LORENZEN [ 195 11 that distributivity is necessary in order to give subst(p, v ) values for p > 1. The set of terms TL*of the language L*(X) defined in Chapter 13 yields another non-trivial sequential category C in which the functions

16

INTRODUCTION

11.2

subst(p, 1) have values for p > 1: Let A' = { * } be a fixed one-point set, A' = A = TL,,and A"" =..A" X A. Then we obtain a sequential category C by putting ObC ={A}, and letting Arc consist of 1(A) and the class {f' : A" 3 A I f E F" and n E w } whose elements are the functions defined by the equations f ( t , , . . . , t , ) = ft, . t,, with subst being substitution. e,.

1.2. Logical preliminaries

The basic notions of logic required in this monograph are language, deductive system, and deriuation. For the sake of simplicity and continuity of presentation, we define these concepts only in the generality needed for Chapters 2-12. Their extensions to quantilicational and infinitary logic required in Chapter 13 are introduced at that point. The definitions are formulated relative to a fixed, but arbitrary small category X. The language L(X) consists of the following data: (1) A class FL(X) whose elements are called formulas. (2) A class SeqL(X) whose elements are called (unlabelled) sequents. (3) A class LbSeqdX) whose elements are called labelled sequents. These data are defined inductively: 1.2.1. DEFINITION. Let I; = ObX U {N, A , v, 53, +,I,T, I},and let M ( Z ) be the free monoid generated by Z.Then FL(X) is the smallest inductive subset of M ( 8 ) satisfying the following three conditions: (1) ObX C FL(X). (2) I,T, 1 E FL(X). (3) If a,B E FL(X), then (aNP), (aA PI, (av P ) , (a.$P), and (a P ) E FL(X), where (an p ) , (aA p ) , (av p ) , (a=$ p ) , and (a p ) denote the strings u a p , A ap, v a@, =$ aP, and c$ ap, respectively. The elements of ObX and the formulas I,T and 1 are called the atomic formulas of L(X).

+

1.2.2. DEFINITION.Let M(FL(X)) be the free monoid generated by FL(X). Then SeqdX) = M(FL(X))X M(FL(X)).

Following Gentzen, and in view of their intended interpretation, we

1.21

LOGICAL PRELIMINARIES

17

write the elements (r,@) of SeqL(X) as r+@. In this notation, capital Greek letters stand for arbitrary elements of M(FL(X)),while lower case Greek letters denote elements of length 1, i.e., elements of FL(X). For the sake of simplicity we do not distinguish between a formula a and the sequence (a).We call r the antecedent and @ the succedent of the sequent r + @. 1.2.3. DEFINITION.Let Lb(h(X)) be the class of labels defined in Appendix A. Then LbSeqL(X) = Lb(&X)) x SeqL(X).

Extending the notation for sequents introduced in 1.2.2, we write the elements (f, (r,0)) of LbSeqL(X) as f : r+@. We call f the label of the sequent r + @. 1.2.4. DEFINITION. An unlabelled deductive system in the language L(X) is a substructure of the relational structure

A(X) = (SeqdX), (Al)-(A4), (Rl)-(Rl7)),

where A + B E ( A l ) iff A, B E ObX and X(A, B ) # 0, ( K , A, p ) E (RI) iff K = r + @ y Y , A = A y h + O , and p = AI'A+@WP, for some r, @, y, Y, A, A, 0 E M(FL(X)), etc. The relations (Ai) and (Rj) have the meaning assigned to them in Appendix B. 1.2.5. DEFINITION. A labelled deductive system in the language L(X) is a substructure of the relational structure

&X)

= (LbSeqdX), ( i l ) - ( i I 5 ) ,

(R1)-(RI I)),

where f : A + B E (A]) iff A, B E ObX and f E X(A, B), ( K , A, p ) E (Rl) iff K = f : A + B , A = g : B + C , a n d p = c o m p ( g , f ) : A + C , f o r s o r n e f, g, comp(g, f ) E Lb(&X)) and some A, B, C E FL(X), etc. The relations (h)and (Rj) have the meaning assigned to them in Appendix A. Finally, we introduce the concept of a derivation. For this purpose, we require the auxiliary notion of a tree, based on a non-empty set N ,

18

[1.2

INTRODUCTION

whose elements are called nodes, and a fixed one-point extension N * N U {m} of N. The following definition is adequate and convenient:

=

1.2.6. DEFINITION. A tree is a function T : N + N * with the property that T ( r ) = m for some r E N , and that for all x E N there exists a n E o such that T " ( x ) = r.

A tree is finite if N is finite, binary if T - ' ( x ) contains no more than two elements for all x E N*, and ordered if a well-ordering is given for each of the sets T - ' ( x ) . In Chapters 2-12, the concept of a tree will be synonymous with that of a finite, binary, ordered tree. The more general form of this notion required in Chapter 13 will be introduced at that point.

1.2.6. DEFINITION.A

{x, ~ ( x ) ., . . , T " ( X ) ) C N.

branch

of

a

tree

T

is

a

set

B:

=

A branch B: is maximal if T - ' ( x ) = 0 and T " ( x ) = r. The length of B: is n + 1. The height of T is the maximum of the lengths of the maximal branches of T . The width of T is the number of maximal branches of T.

1.2.6. DEFINITION. A derivation of an unlabelled deductive system xA(X) on a tree T : N + N* is a function f T : N + SeqdX) satisfying the following conditions: ( 1 ) If T - ' ( x ) = 0, then f 7 ( x )E 1 (Ai), for some i E {1,2,3,4}. (2) If T - ' ( x ) = { y } , then ( f T ( y )f,T ( x ) ) E 1 ( R i ) , for some i E (2, 3 , 4 , 5, 6, 7, 9, 1 1 , 13, 15, 17). (3) If T - ' ( x ) = { y , z } and if y Iz in the given well-ordering of { y , z}, then ( f T ( y ) , f T ( z ) , f T ( x ) 1) E ( R i ) , for some i E{1, 8, 10, 12, 14, 16}, where 1 ( A i ) and 1 ( R i ) denote the restrictions of ( A i ) and ( R i ) to xA(X). 1.2.7. DEFINITION. A derivation of a labelled deductive system x&X) on a tree T : N + N * is a function fT : N + LbSeqdX) satisfying the following conditions: (1) If T - ' ( x ) = 0, then f T ( x ) E 1 (Ai),for some i E (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1 1 , 12, 13, 14, 15, 16).

I .21

LOGICAL PRELlMINARlES

19

(2) If T - ' ( x ) = { y } , then (fT(y),fT(x))EF (Ri), for some i E {7,9, 1 I}. (3) If T - ' ( x ) = {y, z } and if y 5 z in the given well-ordering of {y, z } , then (fT(y),fT(z),f 7 ( x ) )E 1 (Ri), for some i E { I , 2. 3 . 4 , 5 . 6 . 8, lo}, where E ( A i ) and F ( R i ) denote the restrictions of (Ail and (Ri) to XA(X). 1.2.8. DEFINITION. A sequent r + @is derivable in a deductive system xA(X) if there exists a derivation of xA(X) such that r + @ =fr(r). with T(r)= 00. Similarly for f : r + @and x&X). fT

In much of our syntactic work below, we shall have to refer to the shapes of the trees underlying the elements of certain classes of derivations. For this purpose, we introduce the following concepts:

1.2.9. DEFINITION. The height of a derivation fT is the height of the underlying tree T , and the width of ft is the width of 7.

In practice, the tree T underlying a derivation fs is clear, up to isomorphism, from the configuration displaying the derivation. Since we are not interested in the nature of the nodes of 7 , this suffices for our purposes. Hence can defined the classes Der(&X)) and Der(A(X)) of derivations of &X) and of A(X) by an induction on height, without explicitly mentioning the trees involved. This is done in Appendices A and B below. Finally, we discuss several points of terminology, omission, and convention: (1) The relations (Ai) and (Aj) and their elements are, ambiguously, called axioms. The relations (Ri) and (Rj) are called rules of inference. Their elements are called applications or instances of ( R i ) and (Rj). Rule (Rl) is called the cut (rule). (2) The sequents above the line in the definition of a rule of inference in Appendices A and B are called the premisses and the sequent below the line is called the conclusion of the stated instance of the rule of inference. (3) An element of Der(A(X)) is said to be cut-free if it contains no instances of (Rl). A relation R on SeqL(X) is said to be an admissible rufe of inference of a deductive subsystem xA(X) of A(X) if its inclusion in xA(X) as an additional rule of inference leaves the class of derivable sequents of xA(X) unchanged.

20

INTRODUCTION

t1.2

(4) The formula y in (RI) is called a cut formula. The formulas a, p, a np, a A p, a v p, a! jp, and a p are said to be active, and the formulas in r, A, a, 9,and A are said to be passive in the stated instances of Rules (R2)-( R 17). ( 5 ) For mnemonic reasons, we use straight capital Greek letters, i.e., r, A, A, 8,and II, as antecedent symbols, and curved ones, i.e., @, q, 0 , a, and Y, as succedenf symbols. (6) For the purpose of applying structural rules to elements of M(FL(X)) in the antecedent or succedent of a sequent, we often treat such elements as if they were of length 1, provided that they are represented by a single capital Greek letter in the relevant sequent. (Cf., for example, Clause (C.3) in the statement of the cut elimination algorithm.) This practice is harmless and simplifies the exposition. We use the same device when interpretipg the elements of M(FL(X))as categorical objects. In both cases, the ambiguity is semantically justified by the coherence of the associativity isomorphisms in monoidal categories (cf. Theorem 2.6.1.1). (7) Identity arrows are often left unlabelled (cf. 4.6.9, for example), and on occasion several instances of structural rules of inference are notationally collapsed (cf. 5.4, for example). (8) We usually write L(X) for the class of formulas FL(X) of the language L(X) whenever the meaning is clear from the context, and assign the obvious meaning to the concepts of sublanguage, subformula, deductive subsystem, and subderivation. (9) The interpretations of elements of Der(A(X)) as arrows of suitable categories are stated only for the general case, i.e., the case where capital Greek letters represent non-empty sequences of formulas. The intended meaning of the remaining cases is clear from the context. In particular, as the interpretations of (A2), (A3), and (A4) suggest, a derivation of a sequent of the form r + i s intended to have the same meaning as any one of the obvious associated derivations of the sequent r +I,and a derivation of a sequent of the form +Q, is intended to have the same meaning as any one of the obvious associated derivations of the sequent T+ Q,, respectively 1- a.

+

CHAPTER 2

MONOIDAL CATEGORIES

The weakest structure on a category of logical interest is one whose object part is reminiscent of a monoid. This type of structure serves as a model for the proof-theoretical properties that are common to A and v, and to T and I,and that are independent of the symmetry of the operations A and v. 2.1. Definition

A monoidal category is a category C with the following structure: (1) A bifunctor ( - ) n ( - ) : C x C + C . (2) A distinguished object I E ObC. (3) Three natural isomorphisms a,A, and p, where a = { a ( A ,B, C) : A n ( B n C ) + ( A nB)n C E A r c I A, B, C E ObC}, A = {A(A): In A + A E A r c 1 A E ObC}, p = { p ( A ) : A nI+ A E A r c 1 A E ObC}.

These data satisfy three commutativity conditions, for ail A, B, C, D E ObC:

( A x ( B n C ) )n D

A 10: ( ( B XI C) I D )

(M2) A n (In B )

( A n I)x B

21

22

MONOIDAL CATEGORIES

[2.2

(M3) IwtILI

-1 2.1.1. REMARK.Axioms (Ml)-(M3) are known as the MacLane-Kelly coherence conditions for a,A, and p. They entail that all diagrams whose edges are constructed by means of x1, a,A, p, and 1 commute. 2.2. Examples

2.2.1. Any monoid (M, +, 0) becomes a monoidal category C if we put I = O , m # : n = m + n , O b C = M , a n d ArC={l(rn)(mEM}. 2.2.2. Any commutative monoid ( M , +, 0) becomes a monoidal category C if we put ObC = {M}, A r c = M, with f : M 4 M defined by f(x) = f + x , M x c M = M , a n d fxcg=f+g. 2.2.3. Any lower semilattice (S, A , T )with inf operation A and maximal element T becomes a monoidal category qua monoids, and so does every upper semilattice (S, v, I)with sup operation v and minimal element 1.

A category C has finite products if for any finite subset {A;1 i E n E o} of ObC there exists a P E ObC and a set {T; : P -+A; 1 i E n E o} of arrows of C with the property that for each B E ObC, and each f; : B --* Ai E A r c , there exists a unique arrow g : B -+ P E A r c such that comp(Tj, g ) = fj, for all j E n. The system ( P , m ) is called a product in C. If n = 0, then P is called a terminal object of C.

2.2.4. Any category C with finite products becomes a monoidal category if a unique terminal object is chosen for I,and if for each A, B E ObC, a unique product object P is chosen as A x1 B. A category C has finite coproducts if the category Cop has finite products. A coproduct object for n = O is called an initial object of C. 2.2.5. Any category C with finite coproducts becomes a monoidal category qua the monoidal structure on Copdefined in 2.3.4.

2.31

T H E C‘ATEGORY

Fm(X)

-33

2.2.6. The category RModR of R-R-bimodules and homomorphisms between them, for a fixed unitary ring R, has a natural monoidal structure. The usual tensor product with the action on the generators m @ n defined by the equation r ( m @ n ) r ’ = ( r m )@ (nr’) yields a bifunctor n with unit R. A detailed description of RModR may be found in LAMBEK [ 19661. 2.2.7. REMARK. In Ens, product objects are Cartesian products and coproduct objects are disjoint unions. Any one-element set is a terminal object, and the empty set is initial. In Cat, product and coproduct objects are analogous to those in Ens.

2.2.8. REMARK. The category Ens* of pointed sets and base-point preserving functions carries each one of the monoidal structures described in 2.2.4, 2.2.5, and 2.2.6. For any two pointed sets (P, * P ) and (Q, * a ) , the pair (P X Q, ( * P , * Q ) ) is a product object, rhe pair ( ( P + Q)/=, { * P , * Q}) obtained by forming the disjoint union of the sets P and Q and dividing out by the smallest equivalence relation containing ( * P , * Q ) is a coproduct object, and the pair ((P x Q)/=,6 ) is a tensor product object, where = is the smallest equivalence relation on P x Q satisfying the conditions (a. * Q) = (b, * Q ) and ( * P , c) = ( * P , d) for all a , b E P and all c , d E Q , and b = { ( p , q ) E P x Q ( p = * P or q = *Q}. For products and coproducts, the unit I is any one-element pointed set ( { p } , ~ ) and , for tensor products, it is any two-element pointed set ({P, q l , P > -

2.3. The category Fm(X) Small monoidal categories are the objects of a category mCat whose arrows are functors F with the property that F ( A n B) = F ( A )n F ( B ) , F(I)=I, F ( a ( A ,B, C ) )= a ( F ( A ) ,F ( B ) ,F ( C ) ) , F(A(A)) = A ( F ( A ) ) , and F ( p ( A ) )= p ( F ( A ) ) ,for all A, B, C E Obdom(F). There exists an obvious forgetful functor Um : mCat + Cat. We now describe the construction of a left adjoint Fm : Cat+mCat of Um. For this purpose, we require a language mL(X), a labelled deductive system md(X), and a congruence relation = on Der(mb(X)).

24

MONOIDAL CATEGORIES

[2.3

2.3.1. DEFINITION. The language of Fm(X) is the sublanguage mL(X) of L(X) generated by ObX, I, I, and ArX. 2.3.2. DEFINITION.The labelled deductive system of Fm(X) is the subsystem m&X) of &X) generated by Axioms (Al), (A2), (A3), (A4), (A6), (A7),(AS), (AS), and Rules (Rl) and (R2). 2.3.3. NOTATION.In keeping with the intended use of formulas as objects, we write dom(f) for A and codcf) for B, for any derivation f : A + B E Der(m&X)). 2.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(m&X)) satisfying the following conditions: (1) If f = g, then dom(f) = dom(g) and codcf) = cod(g). (2) If f = g and h = k, then comp(h,f) = comp(k, g). (3) If f = g and h = k , thenfwch=gnk. (4) If domcf! = A and cod(f) = B, then compcf, l(A))=f and comp(l(B), f ) = f. ( 5 ) comp(f,comp(g, h))=comp(compcf,g), h ) . (6) compCf I g, h I k) = compcf, h ) I comp(g, k). (7) comp(a-', a)= l(dom(a)) and comp(a, a-I) = l(dom(a-I)). (8) comp(a,fn(gIh))=comp((fwcg)n h, a). (9) comp(A-', A ) = l(dom(A)) and comp(A, A-') = l(dom(A-I)). (10) compcf, A ) = comp(A, l(1)wcf). (11) comp(p-l, p ) = l(dom(p)) and comp(p, p - ' ) = l(dom(p-l)). (12) compcf, p ) = comp(p, f~ l(1)). (13) comp(a, a)= comp(a I 1, comp(a, 1I a)). (14) comp(p I 1, a)= comp( 1, 1I A). (15) comp(1, A ) = p. 2.3.5. REMARK.In most of the above clauses, we have used the label of the concluding sequent of a derivation as a name for the derivation. In each case, the context resolves the resulting ambiguity. We continue this practice in the chapters below. We can now define the category Fm(X): (1) ObFm(X) = mL(X). (2) ArFm(X) = {af] I f E Der(m&X))}, where lJ.fl denotes the equivalence class determined by f.

2.31

T H E CATEGORY

Fm(X)

25

(3) For all derivable labelled sequents f : A + B, dom(llf1) = A and cod((Lf1)= B. (4) For all derivable labelled sequents f : A - B and g : B + C , comp(ugn, = Ucomp(g, (5) For all derivable labelled sequents f : A + B and g : C + D , a f l n ign = ( ~x1fgl. (6) For all A EObFm(X), ](A) =([l(A)IJ, where ] ( A ) : A + A is a derivation quoting Axiom ( A l ) or (A2). (7) The definitions of a, a - l , A, A - I , p , and p-l are analogous to the definition of the identities of Fm(X) in Condition ( 6 ) , with Axioms (A3), (A4), (A6), (A7), (h), and (A9) in place of and (A2). (8) The image of f : A + B E ArX in Fm(X) i\ [If]. This completes the description of Fm(X). By Conditions 2.3.4.1, 2.3.4.2, 2.3.4.4. and 2.3.4.5, Fm(X) is a category, and by Conditions 2.3.4.3, 2.3.4.6, and 2.3.4.7-15, Fm(X) is monoidal. Moreover, an easy induction on the construction of Fm(X) shows that every functor H : X + Um(M) extends to a unique arrow H : Frn(X) + M in rnCat. Hence we call Fm(X) the free nzonoidal category genertited by X. In order to complete the description of F m as a left adjoint of the forgetful functor Um, it remains to define it on the arrows of Cat.

u.m

f)n.

(A])

2.3.6. DEFINITION. Let H : C + D be an arrow of Cat, and Fm(C) and Fm(D) the free monoidal categories generated by C and D. Then Fm(H) : Fm(C)+ Fm(D) is the functor satisfying the following equations: (1) Fm(H)(A) = H ( A ) for all A E ObC. (2) Fm(H)(A n B ) = Fm(H)(A)xc Fm(H)(B) for all A, B E ObFm(C). (3) F m ( H ) ( I ) = I. (4) Fm(H)(aflJ) = [ H ( f ) ]for all f E A r c . ( 5 ) Fm(H)(l(A)) = l(Fm(H)(A)) for all A E ObFm(C). (6) Fm(H)(a(A, B, C)) = a(Fm(N)(A), Fm(H)(B), Fm(H)(C)) for all A, B, C E ObFm(C). (7) Fm(H)(A(A)) = A(Fm(H)(A)) for all A E ObFm(C). (8) Fm(H)(p(A)) = p(Frn(H)(A)) for all A E ObFrn(C). (9) Fm(H)(comp(g,f 1) = cornp(Fm(H)(I:),Fm(H)(f))for all c 0 r n p k - f ) E ArFm(C). (10) Fm(H)(f x1 g ) = Fm(H)(f) x1 Fm(H)(g) for all f. g E ArFm(C). The verification that Um and F m are adjoint functors is routine. We now show that there exists an alternative composition-free description of Fm(X) by means of an unlabelled deductive system mA(X).

26

MONOIDAL CATEGORIES

[2.5

2.4. The deductive system mA(X)

The unlabelled deductive system of Fm(X) is the subsystem mA(X) of A(X) generated by Axioms (Al), (A2), and the following restrictions of Rules (Rl), (R2), (R8), and (R9):

2.5. The semantics of Der(mA(X))

In this section, we interpret the derivations of mA(X) as arrows of Fm(X) and prove the completeness of Der(mA(X)) with respect to this semantics to the effect that every arrow of Fm(X) is representable by means of some derivation of mA(X). 2.5.1. DEFINITION.The interpretation of Der(mA(X)) in Fm(X) is the function S : Der(mA(X)) + ArFm(X) defined by the following conditions:

f

(1) S ( A + B ) = a f : A + B n .

(2) S(+ I)= l(1) : I+I.

2.51

THE SEMANTICS OF

21

Der(mA(X))

The interpretation S induces an equivalence relation = on Der(mA(X)) defined by f = g iff S(f)= S ( g ) . The following theorem shows that in some sense, Der(mA(X))/= and ArFm(X) are isomorphic: 2.5.2. THE COMPLETENESS THEOREM FOR Der(rnA(X)). For every f € Der(mi\(X)) there exists a g E Der(rnA(X)) such that S ( g ) = afn E ArFm(X).

PROOF.The proof is by an induction on the definition of the derivations of md(x). f ( 1 ) If f quotes Axiom (Al), let g be the derivation A 4B. (2) If f quotes Axiom (A2) and af] = l ( I ) , let g be the derivation -1, and if afn= I(AnB) and S ( h ) = 1(A) and S ( k ) = I(B), let g be the derivation h k A+A B+B AB+AnB AnB+AnB (3) If f quotes Axiom (A3), (A4), (A6), (A7), (h), or (A9), and S ( h ) = 1(A), S ( k ) = I(B), and S ( m ) = l(C), let g be the derivations

h k A+A B+B AB+An B C Z C ABC+ (An B) XI C A(BxIC)+ (An B) XI C An(BnC)+(AnB)nC ~~~

h A+A IA+A InA+A

h +I A+A A+InA

k

B+B C Z C h A+A BC+ Bn C ABC+An(BnC) (AXIB ) C + A N(Bn C ) (A~B)~C+AN(B~C) h A+A h AI+A and A + A + I AnI+A A+AnI

respectively. (4) If the last line of f consists of an application of (Rl), i.e., f is a derivation of the form

28

MONOIDAL CATEGORIES

P

[2.6

4

u:A+B v:B+C comp(v, u ) : A + C

and if Up] = S ( h ) and [ q ] = S(k), let g be the derivation h k A+B B+C A+C ( 5 ) If the last line of f consists of an application of (R2), i.e., f is a derivation of the form 4

P

u:A+B v:C+D U N V :A#C+BND

and if Up] = S ( h ) and ( q ] = S ( k ) , let g be the derivation A +hB

C +k D

2.5.3. COROLLARY. The category Fm(X) is isomorphic to a subcategory of the sequential category generated by the deductive system mA(X) and the interpretation S : Der(mA(X))+ ArFm(X). 0 2.6. The syntax of Fm(X)

The advantages of representing the arrows of Fm(X) by means of derivations of mA(X) in the place of derivations of m&X) are twofold: On the one hand, every derivation in mA(X) codes the name of the arrow it represents and we can therefore dispense with labels, except for those of some axioms, and on the other hand each arrow has a compositionfree representation: 2.6.1. THE CUT ELIMINATIONTHEOREMFOR mA(X). Every f € Der(mA(X)) is equivalent to a cut-free g E Der(mA(X)).

The proof of Theorem 2.6.1 is based on the following result first proved in MACLANE [19631:

2.61

THE SYNTAX OF

Fm(X)

29

2.6.1.1. THE COHERENCETHEOREMFOR mCat (MacLane). If X is discrete, then Fm(X) is simple. 0 PROOFOF THEOREM2.6.1. It is clear from the interpretation of the rules of inference of mA(X) that the coherence theorem for monoidal categories trivializes the semantic aspects of Theorem 2.6.1, and the result therefore follows at once from Clauses (C.l), (C.2.2), (C.13), (C.18.1-2), (C.24.1-2), (C.25.1-2), (C.34), and (C.40) of the cut elimination algorithm described in Appendix C. 0 Using Clauses (D. l ) , (DS), (D.6), (D.40), and (D.43) of the reducibility relation L defined in Appendix D, together with the Coherence Theorem for mCat, we can strengthen Theorem 2.6.1: 2.6.2. THE NORMALIZATIONTHEOREM FOR mA(X). Every f E Der(mA(X)) reduces to a unique equivalent normal g E Der(mA(X)). 0 Since it is clear from proof theory that, up to a change of axioms, any sequent in mL(X) has at most one normal derivation in mA(X), we have the following corollary: 2.6.3. THE CHURCH-ROSSERTHEOREMFOR mA(X). I f f = g , then there exists a normal h E Der(mA(X)) such that f 2 h and g 2 h. 0 2.6.4. COROLLARY.The word problem f o r the functor Fm is solvable. 0 It is clear that all normal derivations in mA(X) of a sequent A + B have the same underlying tree T,and are therefore unique and effectively determined by the syntax of Fm(X), relative to any fixed assignment of axioms of mA(X) to the top nodes of T. Hence Theorems 2.5.2, 2.6.1, and 2.6.3 characterize ArFm(X): 2.6.5. THE COMPUTABILITY THEOREMFOR Fm(X). Relative to X, the sets Fm(X)(A, B ) are computable f o r all A, B E ObFm(X). 0 2.6.6. COROLLARY.The embedding X + Fm(X) defined by f -+ afll is full and faithful.

30

MONOIDAL CATEGORIES

[2.6

PROOF,If f # g € X ( A , B ) , then f and g are normal derivations of A+ 18, an8 by Theorem 2.6.3, fZg. Hence afll Z O[gIE Fm(X)(A, B). On the other hand, it follows from Theorems 2.5.2, 2.6.1, and 2.6.2, that for every h E Fm(X)(A, B ) there exists a k E X(A, B ) such that h = Ik].

2.6,7,REMARK.Using the fact that any binary expression of n terms can be rebraclreted in 1 m=-( n+l

2n-1 n-1

)

ways, we observe that every arrow f :A, * - A, + B in the sequential category gewr@#ed by mA(X) determines 4 ( n ) distinct arrows in Fm(X). We also that even for discrete X, every object of Fm(X) has infinitely m n y isomorphic copies in Fm(X) since A = A M I = (A M I)MI= , , ,ctc., for all 4 E ObFm(X).

.

CHAPTER 3

SYMMETRIC MONOIDAL CATEGORIES

We now study the effect of the symmetry of the operations A and v on the considerations of Chapter 2. The appropriate class of categorical models for this purpose is the class of small symmetric monoidal categories. 3.1. Definition

A symmetric monoidal category is a monoidal category C with the following additional structure: (4) A natural isomorphism u,where CT

= {cT(A,B ) : Ax

B + B x A E Arc I A, B E ObC}.

The category C satisfies three additional commutativity conditions, for all A, B, C E ObC: LI

(M4) A x ( B x C)----*

( A xB ) x C -

AxB 31

I7

C x (A x B )

32

SYMMETRIC MONOIDAL CATEGORIES

13.3

3.1.1. REMARK.Axioms (Mlt(M6) are known as the MacLane-Kelly coherence conditions for a,A, p, and u.Their independence was proved [1963], they entail that in KELLY[1964]. As was first shown in MACLANE all diagrams whose edges are constructed by means of N, a,A, p, a,and 1, and none of whose vertices contains two occurrences of the same object, with the possible exception of I, commute.

3.2. Examples 3.2.1. The monoidal categories in Examples 2.2.2, 2.2.3, 2.2.4, 2.2.5 and 2.2.8 all have a natural symmetric structure. 3.2.2. For any commutative ring K, the category mod of left K modules with the usual tensor product as bifunctor and the ring K as distinguished object, carries a symmetric monoidal structure, and so does the category ModK of right K-modules. 3.2.3. COUNTER-EX~~MPLES. In addition to the trivial counter-examples provided by the non-commutative monoids in Example 2.2.1, the following construction, suggested by Michael Barr, shows that not every monoidal category of R-R-bimodules admits a symmetric structure: Let RModR be the monoidal category of R-R-bimodules of Example 2.2.6, where R = k [ x ] is a ring of polynomials over a field k, and let RMRand RNR be two R-R-bimodules whose actions are determined by the conditions RM1 RR, N R = RR, & ( m , x) = m for all m E M , and +(x, n) = 0 for all n E N. Then the tensor product M @ R N = 0, since m @ n = & ( m , x ) @ n = m @+(x, n) = m 8 0 = 0, and N @ R M= R. Hence RModR is not symmetric monoidal with respect to the tensor product.

3.3. The category Fsm(X) Small symmetric monoidal categories are the objects of a category smCat whose arrows are the arrows F of mCat satisfying the property that F(u(A,B ) ) = a ( F ( A ) ,F ( B ) ) ,for all A, B E Obdom(F). There exists an obvious forgetful functor Usm : smCat+ Cat, and we now extend the definition of Fm to construct a left adjoint functor Fsm : Cat+ smCat of Usm.

3.41

T H E DEDUCTIVE SYSTEM

smA(X)

33

3.3.1. DEFINITION.The language of Fsm(X) is the sublanguage smL(X) of L(X) generated by ObX, I, I, and ArX. 3.3.2. DEFINITION.The labelled deductive system of Fsm(X) is the subsystem smb(X) of b(X) generated by Axioms (Al), (A2), (A3), (A4), ( A 9 , (A6), (A7), (h), (A9), and Rules (81) and (R2). 3.3.3. REMARK.A comparison with 2.3.1 and 2.3.2 shows that smL(X) = mL(X), and that smb(X) results from m&X) by the inclusion of Axiom (A5). 3.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(smb(X)) satisfying the conditions of the equivalence relation defined in 2.3.4 and the following additional requirements: (16) compCf I g, a )= comp(a, g I f). (17) comp(comp(a, a),a)= comp(comp(a I 1, a),1 I a). (18) comp(h, a)= p. (19) comp(c+,a)= l(dom(a)). The category Fsm(X) is defined analogously to the category Fm(X) with Clause (7) in the definition of Fm(X) now also mentioning Axiom (AS).We call the category Fsm(X) the free symmetric monoidal category generated by X. The values of Fsm on the arrows of Cat are defined as in 2.3.6, with the following additional clause: (1 1) Fsm(H)(a(A, B)) = a(Fsm(H)(A), Fsm(H)(B)) for all A, B E ObFsm(X). As in Chapter 2, the verification that Usm and Fsm are adjoint functors is routine. We now extend the composition-free description of Fm(X) to a composition-free description of Fsm(X). 3.4. The deductive system smA(X) The unlabelled deductive system of Fsm(X) is the subsystem smA(X) of A(X) generated by Axioms (Al), (A2), and the following restrictions of Rules (Rl), (R2), (R4), (R8), and (R9):

34

SYMMETRIC MONOIDAL CATEGORIES

[3.5

3.4.1. REMARK.The deductive system smA(X) results from mA(X) by the inclusion of Rule (R4) as additional rule of inference.

3.5. The semantics of Der(smA(X))

We extend the interpretation of Der(mA(X)) in ArFm(X) to an interpretation S : Der(smA(X))+ ArFsm(X) by means of Clauses (1)-(6) of 2.5.1 and the following additional definition:

Analogously to the identification in 2.5, we put f = g iff Scf) = S ( g ) , and obtain a bijection Der(smA(X))/= = ArFsm(X): 3.5.1. THE COMPLETENESS THEOREMFOR Der(smA(X)). For every f E Der(smb(X)) there exists a g E Der(smA(X)) such that S ( g ) = afJl E ArFsm(X).

3.61

THE SYNTAX OF

Fsrn(X)

35

PROOF. The theorem follows from the proof of Theorem 2.5.2, with Case (3) augmented as follows: If f quotes (h), and S ( h ) = 1(A) and S ( k ) = l ( B ) , let g be the derivation h k B+B A + A BA-BxlA AB+BnA AMB+BIA

0

3.5.2. REMARK.In the proof of Theorem 3.5.1, it was tacitly assumed that the notation reveals the active formulas of instances of (R4). In cases of ambiguity, the active formulas will be highlighted by means of dots enclosing them. Thus

h .AA.A+ B .AA.A + B

and

h A.AA. --* B A.AA.+ B

denote different derivations. Since A ( 1 ) = p ( 1 ) in Fsm(X), such distinctions are unnecessary in the case of (R2). 3.5.3. COROLLARY.The category Fsm(X) is isomorphic to a subcategory of the sequential category generated by the deductive system smA(X) and the interpretation S : Der(smA(X))+ ArFsm(X). 0

3.6. The syntax of Fsm(X)

The cut elimination theorem for mA(X) extends to smA(X) and affords the same syntactic advantages. 3.6.1. THE CUT ELIMINATION THEOREM FOR smA(X). Every f € Der(smA(X)) is equivalent to a cut-free g E Der(smA(X)).

The proof of Theorem 3.6.1 uses the following result:

3.6.1.1. THE COHERENCETHEOREMFOR smCat (MacLane). If X is

36

SYMMETRIC MONOIDAL CATEGORIES

[3.6

discrete, then for all A, B E ObFsm(X) with the property that they contain no object of X more than once, Fsm(X)(A, B ) contains at most one element. 0 PROOF OF THEOREM3.6.1. The theorem follows from the proof of Theorem 2.6.1, together with Clauses (C.20.1-4) and (C.36) of the cut elimination algorithm described in Appendix C, provided that the mentioned clauses preserve equivalence. But an inspection shows that nowhere do they depend on the identity of formulas other than the two instances of the cut formula. Hence the required equivalences are immediate consequences of the coherence theorem for symmetric monoidal categories. This proves the cut elimination theorem for smA(X). 0 The following counter-example shows that the coherence theorem for smCat does not hold unconditionally: 3.6.2. COUNTER-EXAMPLE. Let A = B N B, with B E ObX. Then the set Fsm(X)(A, A) contains two distinct elements: 1(B N B ) and a ( B , B).

Using Clauses (D.l), (D.3), (DS), (D.6), (D.24), (D.26), (D.27), (D.40), and (D.43) of the reducibility relation L defined in Appendix D, together with the coherence theorem for smCat, we can strengthen the cut elimination theorem: 3.6.3. THE NORMALIZATION THEOREM FOR smA(X). Every f E Der(smA(X)) reduces to a unique equivalent normal g E Der(smA(X)). 0

The usefulness of normal derivations derives from the fact that they are effectively calculable by the cut elimination and normalization algorithms, and that this process makes the equality relation in Fsm(X) decidable: 3.6.4. THE CHURCH-ROSSER THEOREMFOR smA(X). If f = g , then there exists a normal h E Der(smA(X)) such thaf f 2 h and g 2 h.

PROOF.Since Fsm(X) is free on X and since

L

preserves equivalence, it

3.61

THE S Y N T A X OF

Fsm(X)

37

is sufficient to show that distinct normal derivations f, g : A + a represent distinct arrows in Ens. In the light of Theorem 3.6.1, we may assume that X is discrete. By Clauses (D.6), (D.27), (D.40), and (D.43) of 2 , we may assume that f and g contain no instances of (R9), and by Clauses (D.I), (D.3). (D.5), and (D.6) that f and g contain no instances of (R2). Thus if I is the only atomic subformula of a,then A is empty because of the absence of instances of (R2), and by Theorem 3.6.1, f quotes an axiom iff g quotes an axiom. Under these conditions, neither f nor g contains an instance of (R4). By Clause (D.26), the same is true if both f and g end with an instance of (R8). In these cases, the result therefore follows from Theorem 2.6.3. Two possibilities remain: 1. Derivation f ends with (R8), and therefore contains no instances of (R4),and g ends with (R4). 2. Both f and g end with (R4). It is clear from the nature of (D.24) and (D.26) that the following examples are typical: (1) f and g are the derivatives A+A B+B AB+AnB

g A + A B+B AB+AnB BA+AnB

with A = B E ObX. (2) f and g are among the derivations A-+A B+B AB+AnB C+C ABC -+ ( A I B ) n C BAC + ( A n B ) n C

A+A B+B C+C AB+AxB ABC + ( A n B ) x C ACB + ( A n B ) n C

A+A B+B AB-+AnB C+C ABC+ ( An B ) n C BAC + ( A n B ) x C BCA -+ ( A x B ) n C

A+A B+B AB+AnB C+C ABC -+ ( A n B ) 11: C ACB+ ( A nB)n C CAB -+ ( A n B ) n C

38

SYMMETRIC M O NO I DAL CATEGORIES

A+A B+B AB+AMB C+C A B C + ( A x( B)x( C

[3.6

A+A B+B AB+AnB C+C ABC + (Ax(B ) MC ACB + ( A x( B)x( C CAB + ( A 10: B ) n C CBA + ( A x( B ) XI C

with A = B = C E ObX. Let M be an infinite set, consider Ens as a symmetric monoidal category with respect to Cartesian products, and let FM : Fsm(X) +Ens be the unique functor which preserves the symmetric monoidal structure of Fsm(X) exactly and agrees with the constant functor ConstM : X-Ens on X. Then it follows from the definition of the interpretation S that F M ( S ( f )#) F M ( S ( g ) )Since . M is infinite, the functor FM will separate all similar derivations containing any finite number of instances of (R8). 0 3.6.5. COROLLARY. The word problem f o r the functor Fsm is solvable. 0

It is clear that all normal derivations in smA(X) of a sequent A + B have the same width and are effectively determined by the syntax of Fsm(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees. Hence Theorems 3.5.1, 3.6.1, and 3.6.3 characterize ArFsm(X): 3.6.6. THE COMPUTABILITY THEOREMFOR Fsm(X). Relative to X, the sets Fsm(X)(A, B ) are computable f o r all A , B E ObFsm(X). Cl 3.6.7. COROLLARY. The embedding X + Fsm(X) defined by f + afl is full and faithful.

PROOF.Similar to the proof of Corollary 2.6.6. 0 3.6.8. REMARK.If X is discrete, and Al, . . . , A , are n not necessarily distinct objects of X, then a calculation in Ens, regarded as a symmetric monoidal category with respect to Cartesian products, shows that any arrow f : A1 . A. --* B in the sequential category generated by X deter-

- -

3.61

THE SYNTAX OF

Fsm(X)

39

mines @(n) n ! distinct arrows of Fsm(X). Moreover, Fsm(X)(A, B ) is empty for all A, B E ObFsm(X), if A and B do not contain the same number of occurrences of an object of X. As in the case of Fm(X), the number of objects isomorphic to any given object of Fsm(X) is infinite by virtue of the presence of I.

CHAPTER 4

CARTESIAN CATEGORIES

In this chapter, we study the proof-theoretical properties that are particular to A and T. The natural class of categorical models for these considerations is the class of small Cartesian categories. 4.1. Definition

A Cartesian category is a category C with the following structure: (1) A bifunctor ( - ) A ( - ) : C X C + C . (2) A distinguished object T E ObC. (3) Two adjunctions a,,and ar, where a,,= {a,,(A, B, C) : C(A, B

A

C) + C(A, B) X C(A C) E ArEns I A, B, C E ObC},

and

a,={a,(A):C(A,T)-*{*}EArEnsA E ObC}. 4.2. Examples

4.2.1. A category C is Cartesian iff it has finite products.

4.2.2. COUNTER-EXAMPLE. Example 2.2.2 above shows that not every symmetric monoidal category carries a Cartesian structure. 4.3. The category Fc(X)

Small Cartesian categories are the objects of a category cCat whose arrows are functors F with the property that F(A A B) = F(A) A F ( B ) , 40

4.31

THE CATEGORY

FC(X)

41

F(T) = T, and F(&'(A)( *)) = a;'(F(A))( * ) for all A, B E Obdom(F), and that a,(F(A), F ( B ) , F ( C ) ) ( F ( f )= ) (F(g), F ( h ) ) for all A, B, C E Obdom(F) and all f, g, h E Ardom(F) for which a,(A, B, C)(f) = (8, h ) . There exists an obvious forgetful functor Uc : cCat+Cat. We now modify the definition of Fm and construct a left adjoint Fc of Uc.

4.3.1. DEFINITION.The language of Fc(X) is the sublanguage cL(X) of L(X) generated by ObX, T, A , and ArX.

4.3.2. DEFINITION.The labelled deductive system of Fc(X) is the subsystem c&X) of &X) generated by Axioms (A]), (A2), (AlO), (A12), (A13), and Rules (Rl) and (R3). 4.3.3. DEFINITION. The relation = is the smallest equivalence relation on Der(cd(X)) satisfying the following conditions: ( 1 ) If f = g, then dom(f) = dom(g) and codcf) = cod(g). (2) If f = g and h k, then comp(h,f) = comp(k, g). (3) If f = g and h = k, then (f, h ) = (8, k ) . (4) If domcf) = A and cod(f) = B, then comp(f, 1(A)) -f and comp(l(B), f ) = f. ( 5 ) comp(f,comp(g, h))=comp(compcf,g), h ) . (6) comp(m, (f, g)) = f. (7) comp(.rr,, cf, 8)) = g . (8) (comp(m, h ) , comp(.rr,, h ) ) = h. (9) If cod(f) = T, then f = T. We can now define the category Fc(X): (1) ObFc(X) = cL(X). (2) ArFc(X) = Der(c&X))/=. (3) For all derivable labelled sequents f : A + B, dom(ef1) = A and cod(ef1) = B. (4) For all derivable labelled sequents f : A + B and g : B+ C, comp(Ug1, ef1) = Ucomp(g, f11. ( 5 ) For all AEObFc(X), l ( A ) = [ l ( A ) j , where 1(A): A + A is a derivation quoting Axiom (A1) or (A2). (6) The definitions of T, nA,and .~r,are analogous to that of the identities of Fc(X) in Condition 5, with Axioms (AlO), (A12), and (A13) in place of Axioms ( A l ) and (A2).

-

42

CARTESIAN CATEGORIES

[4.3

(7) For all derivable labelled sequents f : A + B and g : A + C, (Vn,

ugn) = ucf, g)n.

(8) For all derivable labelled sequents f : A + B and g : C+D, af] A c)),comp(g, T p v , c)m (9) For all A, B E ObFc(X), a,(A A B,A, B)(l(A A B))= (TA(A, B), TJA, B))and a;'(A)( * ) = T(A). (10) The image of f : A + B E ArX in Fc(X) is &fj. This completes the description of Fc(X). Analogously to the previous cases, we call Fc(X) the free Cartesian category generated by X. In order to prove that Fc(X) is indeed Cartesian it suffices to observe that since ugn = u(com~cf,TA(A,

TAUA g ) = T A ( ( f T A ,

gTp)) = f T A

and

Tpcf A

g ) = T p ( c f T ~ gTpTTp)) , = gTp,

it follows that ~ ~ ( A cg )f h ) = f ( ~ ~ and h )

rP(cf A g ) h ) = g(Tp,h).

Hence a,,is natural. It also follows that

4.3.4. DEFINITION.Let H : C + D be an arrow of Cat, and Fc(C) and Fc(D) be the free Cartesian categories generated by C and D. Then Fc(H) :Fc(C)+ Fc(D) is the functor satisfying the following equations: (1) Fc(H)(A) = H(A) for all A E ObC. (2) Fc(H)(A A B)= Fc(H)(A) A Fc(H)(B) for all A, B E ObFc(C). (3) Fc(H)(T) = T. (4) Fc(H)(&fl)= [Hcf)l for all f E Arc.

4.41

THE DEDUCTIVE SYSTEM

d(x)

43

( 5 ) Fc(H)( 1(A)) = l(Fc(H)(A)) for all A E ObFc(C). ( 6 ) Fc(H)(comp(g, f ) ) = comp(Fc(H)(g), Fc(H)(f)) for all comp(g, f )

E ArFc(C).

(7) Fc(H)((f, g ) ) = (Fc(H)Cf), Fc(H)(g)) for all f, g E ArFc(C). (8) Fc(H)(T(A)) = T(Fc(H)(A)) for all A E ObFc(C). (9) Fc(H)(rA(A, B)) = rA(Fc(H)(A),Fc(H)(B)) for all A, B E ObFc(C). (10) Fc(H)(r,,(A, B ) ) = r,,(Fc(H)(A), Fc(H)(B)) for all A, B E ObFc(C).

The verification that Uc and Fc are adjoint functors is routine. We now give a composition-free description of Fc(X) by means of an unlabelled deductive system cA(X).

4.4. The deductive system cA(X)

The unlabelled deductive system of Fc(X) is the subsystem cA(X) of A(X) generated by Axioms (Al), (A3), and the following restrictions of Rules (Rl), (R2), (R3), (RlO), and (R11):

4.4.1. REMARK.Apart from the more general form of Rule (R2) in

cA(X), the significant difference between the systems mA(X) and cA(X) lies in the formulation of Rules (R8) and (R10). It turns out that although the category Fc(X) is far from simple, even for discrete X, the chosen form of (R10) is precisely the form which guarantees the completeness of the subsystem of cA(X) generated by (R2) and (R10) with respect to ArFc(X), so that, like (Rl), Rule (R3) is an admissible rule of inference of cA(X), required only for the elimination of cuts in the normalization of derivations.

44

[4.5

CARTESIAN CATEGORIES

4.5. The semantics of Der(cA(X))

We now interpret the derivations of cA(X) in ArFc(X) and prove the completeness of Der(cA(X)) with respect to this semantics. This interpretation requires the following canonical arrows of Fc(X), determined by a,: (1) TA(A,B ) : A A B + A for all A , B E ObFc(X), where

( 2 ) S ( A ) : A +A

A

( 3 ) a ( A , B,C ) : A where

A for all A E ObFc(X), where

A

( B A C ) + ( A A B ) A C for all A, B,C E ObFc(X),

4.5.1. DEFINITION. The interpretation of Der(cA(X)) in Fc(X) is the function S : Der(cA(X))+ ArFc(X) defined by the following conditions:

f

( 1 ) S ( A + B ) = D : A + B]I. ( 2 ) S(+ T)= 1(T): T+T.

4

4.51

T H E SEMANTICS OF

=

(r A ff ) A A

45

Der(cA(X))

(InS)nl

(r A (aA a))A A

d

Analogously to the previous cases, we define f = g iff S(f)= S ( g ) , and obtain the desired bijection Der(cA(X))I= = ArFc(X): 4.5.2. THE COMPLETENESS THEOREMFOR Der(cA(X)). For every f € Der(cd(X)) there exists a g € Der(cA(X)) such that S ( g ) = a f n E ArFc(X).

PROOF. The proof is similar to that of Theorem 2.5.2.

f

( 1 ) If f quotes Axiom (Al), let g be the derivation A + B. (2) If f quotes Axioms (A2)and Dj = 1(T), let g be the derivation -+T, and if 1(A A B ) and S ( h ) = ](A) and S ( k ) = 1(B), let g be the derivation h k A-+A B+B AB+A AB+B AB+AAB AAB-AAB

vn=

(3) If f quotes Axioms (AlO), (A12), or (A13), and S ( h ) = 1(A) and S ( k ) = 1(B), let g be the derivations +T A+T

respectively.

h A+A AB+A AAB+A

k B+B AB+ B AAB+B

4

[4.5

CARTESIAN CATEGORIES

(4) If the last line of f consists of an application of (kl), i.e., f is a derivation of the form .

u : A +P B v : B +4 C comp(u, u ) : A + C and if [ p ] = S ( h ) and [ q ] = S ( k ) , let g be the derivation

h k A+B B+C A+C ( 5 ) If the last line of f consists of an application of (R3), i.e., f is a derivation of the form P

4

u:A+B u:A+C ’ (u, v ) : A +B A C and if [ p ] = S ( h ) and [ q ] = S ( k ) , let g be the derivation h A+B

k A+C

4.5.3. REMARK.In the proof of Theorem 4.5.2 it was assumed, as in the proof of Theorem 3.5.1, that the notation reveals the active formulas of an instance of a rule of inference. However, notational ambiguities can arise in Der(cA(X)) involving instances of (R2) and (R3). Once again, dots enclosing the active formulas will be used to resolve these ambiguities. Thus,

f

A+B .A.A + B ’

f

A+B A.A. + B’

g

.AA.A + B .A.A + B ’

g

A.AA. + B A.A.+B ’

denote, respectively, different derivations.

4.5.4. COROLLARY.The category Fc(X) is isomorphic to a subcategory of the sequential category generated by the deductive system cA(X) and the interpretation S : Der(cA(X))+ ArFc(X). 0

4.61

T H E S Y N T A X OF

FC(X)

47

4.6. The syntax of Fc(X) As in the previous cases, our basic tool for the study of the syntactic properties of Cartesian categories is the effective proof that the cut-free derivations of cA(X) characterize the arrows of Fc(X).

4.6.1. THE CUT ELIMINATION THEOREM FOR cA(X). Every f € Der(cA(X)) is equivalent to a cut-free g E Der(cA(X)).

PROOF.By Clauses (C.l), (C.2.1), (C.2.3), (C.3), (C.14), (C.18), (C.19), (C.26), (C.27), (C.34), (C.33, and (C.42) of the cut elimination algorithm described in Appendix C, every derivation of cA(X) containing an instance of (Rl) reduces to a cut-free one. It remains to show that the required reduction steps preserve equivalence. (1) Case (C.l) is trivial. (2) Case (C.2.1) is a consequence of the naturality of a,, since the commutativity of

for example, entails the commutativity of

and we therefore have the equation comp(g, wp)= wp(f A 8). (3) Case (C.2.3) is similar to the previous case, and so is Case (C.3), since for derivations in cA(X), @ = 9 = 0 in the succedents of the premisses of the instance of (R10) appearing in this reduction. (4) The equivalence required for Case (C.14) is a consequence of the functoriality of A , which makes the following diagram commute:

48

[4.6

CARTESIAN CATEGORIES

( 5 ) The equivalences required for Cases (C. 18.1-2) are trivial consequences of the associativity of composition, and so are the equivalences required for Cases ((2.19.1-2). (6) Case (C.19.3) is a consequence of the naturality of a,', since the commutativity of

[A, A] x [A, A] = [A, A [A. f l x [A. fl

1

[A, Bl x [A, B ] = [A, B

A

A]

A

Bl

for example, entails the commutativity of f

A

(7) Case (C.26) is also a consequence of the naturality of a;',since the commutativity of

1

[A, B l X [ A , Cl SE [A, B If.Bl x U,cl

A

C]

1u.BAcI

[D, B ] x [D,C] = [D,B A Cl for example, entails the commutativity of

D

(8) The equivalences required for Cases (C.27), (C.34), (C.33, and ((2.42) are immediate from the associativity of composition.

4.61

THE S Y N T A X OF

FC(X)

49

This proves the cut elimination theorem for cA(X). 0 The deductive system cA(X) has the following additional syntactic properties by which we can strengthen the cut elimination theorem: 4.6.2. LEMMA.I f a = T in Fc(X), then + a is derivable in cA(X).

PROOF. Since a is terminal, the set Fc(X)(T, a )= { *}. By Theorem 4.5.2, the sequent T + a is therefore derivable in cA(X). It therefore follows from the cut elimination theorem that the sequent + a is also derivable in cA(X). 0

4.6.3. COROLLARY. If a = T , then T is the only atomic subformula of a.

The next lemma follows by an induction similar to that described in Appendix C, together with the observation that for all a +a, p + p , a

A

f

g

p + a A p, r A + 4, A a Z + $ E Der(cA(X)),

(l)a+a

P+P

@+ff

Ap

f f A\P+ff

A

P

E f f

A

@ + a Ap

4.6.4. LEMMA.For every cut-free f E Der(cA(X)) there exists an equivalent cut-free g E Der(cA(X)) containing n o instances of (R3) and containing only instances of (R2), if any, in which the active formulas are atomic.

On the basis of the cut elimination theorem and the above lemmas, we

50

CARTESIAN CATEGORIES

[4.6

define a derivation f E Der(cA(X)) as normal if it is normal in the sense of Appendix D, and satisfies three additional conditions: (1) f contains no instances of (R3). (2) Unless f is a derivation mentioned in Case (3) below, f contains only instances of (R2) whose active formulas are atomic. (3) If f derives r + a and T is the only atomic subformula of a,then f is of the form g

+a

r j f f(a), where g is the unique derivation of + a by means of (A3) and (RlO), and where (a)denotes the instances of (R2) required to derive T + a from + a.

By combining the results of Theorem 4.6.1, Lemma 4.6.2, and Lemma 4.6.4, and extending the reducibility relation L of Appendix D in the obvious way, we obtain our key theorem: 4.6.5. THE NORMALIZATION THEOREM FOR cA(X). Every f E Der(cA(X)) reduces to a unique equivalent normal g E Der(cA(X)).

PROOF.We must show that the reductions in Conditions (D.l), (D.7), (D.8), (D.49), (D.53), and (D.81.1) preserve equivalence. But for (D.11, (D.8), and (D.53) the argument is the same as in 4.6.1.2, and for (D.7) and (D.49) it is the same as in 4.6.1.7. The equivalence required for (D.81.1), finally, is clear from the fact that for all A, B,C E ObFc(X), TA(A,B A C ) = TA(A,C ) TA(A,B)A 1(C) a(A,B,C ) . 0

-

-

The next theorem shows that for normal derivations f, g E Der(cA(X)), f = g iff f = g, and that the normalization process therefore yields an effective characterization of the class of commutative diagrams of Fc(X): 4.6.6. THE CHURCH-ROSSER THEOREMFOR cA(X). If f = g, then there exists a normal h E Der(cA(X)) such that f L h and g L h.

PROOF.Since Fc(X) is free on X, and since L preserves equivalence, it suffices to show that distinct normal derivations f,g : A + a represent distinct arrows in Ens. In the light of Theorem 4.6.1, we may assume that X is discrete.

4.61

T H E SYNTAX OF

FC(X)

51

Since f and g are normal, th ey contain no instances of (R3), all active formulas of instances of (R2) are atomic, and by Clauses (D.8) and (D.49), we may assume that f and g contain no instances of (R1I ) . If T is the only atomic subformula of a, then Condition (3) of the definition of normality entails that f = g . Furthermore, Theorem 4.6.1 guarantees that f quotes an axiom iff g quotes an axiom, and that in this case again f = g iff f = g. Three possibilities remain: ( I ) Both f and g end with an instance of (RIO). (2) Both f and g end with an instance of (R2). (3) Derivation f ends with an instance of (R10) and derivation g ends with an instance of (R2). It is clear from the nature of ( D . l )and (D.7) that the following examples are typical: ( I ) f and g are the derivations 111

r

I1

A+B A-C A-BAC

and

A+B A 5 C A+BAC

and

A+A A.A.+ A

and not both 111 = r and iz = s. (2) f and g are the derivations

-

A-A .A.A A

with A Z T . ( 3 ) f and g are among the derivations A+A A+A .A.A-+A A.A.+A AA-AAA

with A T T .

A-A A+A A+AAA A.A.+ A A A

A+A A-A A.A.+A .A.A-A AA+AAA A+A A+A A+AAA .A.A+ A A A

Treating Ens as a Cartesian category with respect to Cartesian products and using the functor FM : Fc(X)+Ens of Cartesian categories defined analogously to the functor FM in the proof of Theorem 3.6.4, we see from the definition of the interpretation S that in all cases, F M ( S ( f )#) F M ( S ( g ) ) .

0 4.6.7. COROLLARY. The word problem for the functor Fc is solvable. 0

52

CARTESIAN CATEGORIES

[4.6

A reflection on the properties of normal derivations shows that all normal derivations of a sequent A -+ B have the same width and are effectively determined by the syntax of Fc(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees. Hence Theorems 4.5.2, 4.6.1, and 4.6.5 characterize ArFc(X): 4.6.8. THECOMPUTABILITY THEOREM FOR Fc(X). Relative to X, the sets Fc(X)(A, B ) are computable f o r all A, B E ObFc(X). 0 4.6.9 COROLLARY. The embedding X + Fc(X) defined by f + afl is full and faithful.

PROOF.Similar to the proof of Corollary 2.6.6. 0 4.6.9. REMARK.The presence of the diagonal arrows S(A) : A + A A A in Fc(X) increases the size of the horn sets of Fc(X) in comparison with that of the horn sets of Fsm(X). For example, if X is discrete and A E ObX, then Fsm(X)(A",A") has n ! elements, whereas Fc(X)(A", A") has n" elements. Here A' = A, A* = (A N A) or (A A A), A"" = ((A") N A) or ((A") A A), etc. The two elements of Fsm(X)(A N A, A NA) are 1(A N A) and u(A, A), and the four elements of Fc(X)(A A A, A A A) are 1(A A A), u(A, A), comp(S(A), m ( A , A)), and comp(S(A), TJA, A)). These arrows are represented by the derivations

A+A A+A AA+ANA AnA-AKtA

A-A A+A AA+AnA AA+AnA ANA+ANA

A+A A+A A.A. + A .A.A + A AA+AAA AAA-AAA

A+A A+A .A.A + A A.A. + A AA+AAA AAA+AAA

A+A A+A A.A. + A A.A. + A AA+AAA AAA-AAA

A+A A-A .A.A + A .A.A + A AA+AAA AAA+AAA

4.61

T H E S Y N T A X OF

FC(X)

53

respectively. A further consequence of the joint presence of the diagonal arrows 6 ( A ) : A+ A A A and the projections m ( A , B ) : A A B + A and r P ( A ,B ) : A A B + B is that for a discrete X, the sets Fc(X)(C, D ) are empty iff D contains an object of X not contained in C.

CHAPTER 5

BICARTESIAN CATEGORIES

In this chapter, we study the proof-theoretical properties of A , v, T, and 1 that are independent of distributivity. The appropriate class of categorical models for this purpose is the class of small bicartesian categories. 5.1. Definition

A bicartesian category is a Cartesian category C with the following additional structure: (4) A bifunctor (-) v (-) : C X C + C. (5) A distinguished object 1 E ObC. (6) Two adjunctions a. and a&,where a. = {au(A, B, C ) : C(A v B, C)+ C(A, C) X C ( B , C) E ArEns I A, B, C

E ObC},

and

I

a&= {&,(A): C(1, A ) + { * } E ArEns A E ObC}.

5.2. Examples 5.2.1. A category C is bicartesian iff it has finite products and finite

coproducts. Hence, in particular, lattices with smallest and largest elements are bicartesian categories.

5.2.2. COUNTER-EXAMPLE. The set of all functions f : [0, 1]+ R satisfying the condition f ( t x + (1 - t ) y ) > t f ( x )+ (1 - t ) f ( y ) for 0 < t < 1, bounded above by the upper half of the unit circle centred at (f,O), 54

5.31

T H E CATEGORY

Fbc(X)

5.5

partially ordered by f s g iff f ( x ) s g ( x ) for all x E [0, 11, with f A g = {min(f(x), g(x)) 1 x E [0, 11) is a lower semilattice with largest element that is not a lattice. Hence we have a natural example of a Cartesian category which is not bicartesian. 5.3. The category Fbc(X)

Small bicartesian categories are the objects of a category bcCat whose arrows are functors F satisfying the conditions of arrows in cCat and have the additional property that F ( A v B )= F ( A ) v F ( B ) , F(1)= I, and F ( a ; ' ( A ) (*)) = a ; l ( F ( A ) (*)) for all A, B E Obdom(F), and that a,(F(A), F ( B ) , F ( C ) ) ( F ( f )=) ( F ( g ) ,F ( h ) ) for all A, B,C E Obdom(F) and all f, g, h E Ardom(F) for which a,(A, B,C)(f)= (g, h ) . There exists an obvious forgetful functor Ubc : bcCat+ Cat. We now extend the definition of Fc and construct a left adjoint Fbc of Ubc. 5.3.1. DEFINITION. The language of Fbc(X) is the sublanguage bcL(X) of L(X) generated by ObX, T, A , I, v, and ArX. 5.3.2. DEFINITION.The labelled deductive system of Fbc(X) is the

subsystem bc.&(X)of &X) generated by Axioms (AI), (A2), (AlO), (A1I ) , (A12), (A13), (A14), (A15),and Rules ( R I ) , (R3), and (R4).

5.3.3. REMARK.A comparison with 4.3.1 and 4.3.2 shows that bcL(X) and bc&X) result from cL(X) and c&X) by the inclusion of I,v, and (A1 l), (A14), (A15), and (R4), respectively. 5.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(bc&X)) satisfying the conditions of Definition 4.3.3, and the following additional requirements: (10) If f = g and h = k, then [f, h ] = [g. k l . (1 1 ) comp([f, 81, vn*)= f. (12) comp([f, gl, 6) = g. (13) [comp(k, d), comp(k, v31= k. (14) If d o m u ) = I,then f = 7*.

We now define the category Fbc(X) by modifying and extending the

56

BICARTESIAN CATEGORIES

15.3

definition of Fc(X): (1) ObFbc(X) = bcL(X). (2) ArFbc(X) = Der(bcb(X))/=. (3) As in 4.3.3. (4) As in 4.3.3. (5) As in 4.3.3. (6) As in 4.3.3, with the inclusion of T*, d , and T:, determined by Axioms ( A l l ) , (A14), and (AlS). (7) As in 4.3.3. (8) As in 4.3.3. (9) As in 4.3.3. (10) The image of f : A + B E ArX in Fbc(X) is afn. ( 1 1 ) For all derivable labelled sequents f : A + C and g : B + C, ugni = gin. (12) For all derivable labelled sequents f : A + B and g : C + D, af.1v !dl = U[comp(.rm*(B,D ) ,f), com~(.rr,*(B,D ) ,g)Ib (13) For all A, B E ObFbc(X), &,(A, B, A v B)( l(A v B))= (TA*(A,B ) , &A, U ? ) and &'(A)( * ) = T*(A). This completes the description of Fbc(X). We call the category Fbc(X) the free bicartesian category generated by X. The calculations required to show that it is indeed bicartesian are similar to those in 4.3.3. On the arrows of Cat, the functor Fbc is defined by adapting and extending Definition 4.3.4 thus:

run,

urf,

5.3.5. DEFINITION. Let H : C + D be an arrow of Cat, and Fbc(C) and Fbc(D) be the free bicartesian categories generated by C and D. Then

F%c(H) : Fbc(C)+ Fbc(D) is the functor satisfying the following equations: (1)-(lo) As in 4.3.4, with Fbc in place of Fc. (11) Fbc(H)(A v B ) = Fbc(H)(A) v Fbc(H)(B) for all A, B E ObFbc(C). (12) Fbc(H)(I) = 1. (13) Fbc(H)(lf,gl) = [Fbc(H)Cf), Fbc(H)(g)l for all f , g E ArFbW). (14) Fbc(H)(.r*(A)) = T*(F~c(H)(A)) for all A E ObFbc(C). (15) Fbc(H)(nf(A, B ) ) = d(Fbc(H)(A), Fbc(H)(B)) for all A, B E ObFbc(C). (16) F%c(H)(r,*(A, B)) = r?(Fbc(H)(A), Fbc(H)(B)) for all A, B E ObFbc(C).

5.41

T H E DEDUCTIVE SYSTEM

bcA(X)

57

Once again, the verification that Ubc and Fbc are adjoint functors is routine.

5.4. The deductive system bcA(X) We now expand the deductive system cA(X) to an unlabelled deductive system bcA(X) generating a sequential category which contains an isomorphic copy of Fbc(X) and has the desired syntactic properties. The proof-theoretical interest of bcA(X) lies in the fact that it illustrates the connection between cut eliminability and the distributivity of A and v . We explain this remark by examining the nature of the cut-free representations in A(X) of the distributivity arrows

f : A v ( B A C)+(A v B ) A ( A v C),

h :( A A B ) v ( AA C)+ A

A

( B v C),

g : ( A v B ) A( A v C ) + A v (B

k :A

A

A

c),

( B v C)+ ( A A B ) v ( AA

C),

existing in any sequential category qua distributive lattices as described in 1.1.30. A+A B+B A+A c+ c A-AB BC+AB A-AC BC+AC A+AvB BAC+AVB A+AvC BAC+AVC A v ( B A C )+ A v B A v ( B A C)+Av C A v ( B A C)+(A v B ) A ( Av C)

B+B

(1)

C+C

A+A BC+B BC+C A+A A+A(B A C) BC+BAC A+ A(B A C ) BA+A(BAC) BC+A(BAC) B ( A v C)+ A(B A C) A ( A v C)+ A(B A C) ( A v B ) ( A A C)+ A(B A C ) ( A v B ) A ( A v C)+ A A ( B A C)

(2)

c+ c

A+A A+A B+B AB+A AC+A AB+BC AC+BC AAB+A AAC+A AAB+BvC AAC+BVC (AA B)v (AA B v (AAB)v(AAC)+A ( A A B ) v ( A A C ) + A A ( B v C)

c)+

c

(3)

58

BICARTESIAN CATEGORIES

[5.4

B+B C-PC A+A B+BC C+BC A-A A(B v C ) + A BvC+BC A(B v C ) + A A(B v C ) + BA A(B v C)+ BC A(B v C ) + A ( A A C ) A(B v C ) + B(A A C ) A(B v C)+ ( A A B ) ( A A C ) A A ( B v c)+ ( A A B ) v ( A A

(4)

c)

Derivations (l), (Z), (3), and (4) represent the arrows f, g, h, and k, respectively, with obvious abbreviations. The generality of Rules (R10) and (R12) required in the above derivations may be classified as follows:

The greater generality of (R10) in Derivation (2) can be avoided by replacing the single instance of (R13) towards the end of the derivation be three instances of (R13) higher up in the derivation. We notice that onIy Derivations (2) and (4) require antecedents, respectively succedents, of length greater than 1. Since either one of the inequalities represented by arrows g and k forces a lattice in which it holds to be distributive, it is clear that the length of the succedents of instances of (R10) and the length of the antecedents of instances of (R12) must be restricted to being no greater than 1. On the other hand, the arrows g and k are still representable by these restricted forms of (RIO) and (R12) if the cut rule (Rl) is admitted in full generality: Let a = A A (Bv C ) , P = A ( A A B ) v ( A A C), and let

P

a+

A

( ( AA

4

B)v C), y = A A (C v ( A A B ) ) , 6 = r

P, P+ Y, Y-*

be the following derivations:

6

5.41

T H E DEDUCTIVE SYSTEM

bcA(X)

59

B+B C+C A+A B+B B+BC C+BC AB+A AB+B BvC+BC AB+AAB _- ~~A(B v C)+ B ( A A B ) C A(B v C ) B + ( A A B)C A(B v C)A(B v C)+ ( A A B ) C ( A A B)C ~

~

A+A A(B v C)+ A A A ( B v C)+ A P = A

A

( A A ( B v C))(A A ( B v C))+ ( ( A A B ) v CN(A A B ) v C ) A A (B v C)+(A A B )v C ( B v C)+ A A ((AA B ) v

c)

A+A B+B AB+B AB+A AB+AAB AB+ C ( A A B )

c+c

C + C(AA B ) B+ C v ( A A B ) C + C v ( A A B ) A+A ( AA B ) v C+ C v ( AA B) A ( ( A A B ) v C)+ A A ( ( A A B ) v c ) + C v ( A A B) A A ( ( A A B ) v C)+ A A A ( ( A A B ) v C)+ C v ( A A B ) q= A A ( ( A A B ) v C)+ A A ( C v ( A A B ) ) A

A

A+A B+B AB+A AB+B AB+AAB AAB+AhB C+C C+C A+A C + C ( A A B ) A A B+ C ( A A B ) AC+A AC+C C v ( A A B)+ C ( A A B ) AC+Ar\C A(C v ( A A B))+ C ( A A B ) ( A A C ) A ( C v ( A A B ) ) C + ( A A B ) ( A A C ) A(C v ( A A B ) ) A ( Cv ( A A B ) ) + ( A A B ) ( A A C ) ( A A B ) ( A A C ) ( A A ( C v ( A A B ) ) ) ( AA ( C v ( A A B)))+ ((A A B ) v (A A C))((A A B ) v ( A A C ) ) A A (C v ( A A B ) ) + ( A A B ) v ( A A C)

Then the derivation

represents the arrow k. The arrow g is represented similarly. The cut

60

BICARTESIAN CATEGORIES

p.5

rule (R1) is therefore not admissible in full generality as a rule of inference of bcA(X). Hence we have a non-trivial example of a subsystem of A(X) without a cut elimination theorem: 5.4.1. COUNTER-EXAMPLE. The deductive system consisting of Axioms

(Al), (A3), and (A4), and Rules (RI), (R2), (R3), (RS), (R6), (RIO), and (R12), with the succedents in (RIO) and the antecedents in (R12) restricted to sequences of length 1, does not admit a cut elimination theorem. Fortunately, we can formulate the deductive system bcA(X) by means of two special cases of (Rl) for which a cut elimination theorem holds. The unlabelled deductive system of Fbc(X) is the subsystem bcA(X) of A(X) generated by axioms (Al), (A3), (A4), and the following restrictions of (Rl), (R2), (R3), (RS), (R6), (RlO), (Rll), (R12), and (R13): (R1)

r-+y A ~ A - + Q , ArA+Q,

rjQyq y r+mq

+

~

5.5. The semantics of Der(bcA(X))

We extend the interpretation of Der(cA(X)) in ArFc(X) to an interpretation of Der(bcA(X)) in ArFbc(X) by means of the following canonical arrows of Fbc(X), determined by au: (4) d ( A , B) : A + A v B for all A, B E ObFbc(X), where

a d A , B, A v B)(l(A v B ) ) = (T?(A,B ) , d ( A , B ) ) . ( 5 ) S*(A): A + A v A for all A E ObFbc(X), where

oG'((l(A), I(A))) = S*(A).

5.51

THE SEMANTICS OF

Der(bcA(X))

( 6 ) (a*)-I(A,B, C ) : ( A v B ) v C + A v ( B v C) for ObFbc(X), where

61

all

A , B, C E

a,I(,r?(A v B , C)&(A, B ) , (u,'(T?(Av B, C).rr,*(A,B ) , r , * ( AV B, C ) ) ) = ( a * ) - ' ( AB, , C).

5.5.I . DEFINITION. The interpretation of Der(bcA(X)) in Fbc(X) is the function S : Der(bcA(X))+ ArFbc(X) satisfying Conditions (1)-(7) of 4.5.1 and the following additional equations: (8) S(l+) = l(1): I+ 1.

(Ca v 0)v

*

f

As in Fc(X), the equivalence classes of Der(bcA(X)) obtained by defining f = g iff S c f ) = S ( g ) are plentiful enough to classify the arrows of Fbc(X): 5.5.2. THE COMPLETENESSTHEOREM FOR Der(bcA(X)). For every f E Der(bc&X)) there exists a g E Der(bcA(X)) such that S ( g ) = vl E

ArFbc(X).

62

BICARTESIAN CATEGORIES

[5.5

PROOF.We modify and extend the proof of Theorem 4.5.2. (1) As in 4.5.2. (2) As in 4.5.2, with -the following addition: If efn = 1(1), let g be the derivation l+, and if 1Lfl=1(A v B), let g be the derivation

h k B+B A+A A+AB B+AB AvB+AB AvB+AvB (3) As in 4.5.2, with the following addition: I f f quotes Axioms (All), (A14), or (AlS), let g be the derivations

I+ l+A

h A+A A+AB A+AvB

k B+ B B+AB B+AvB

respectively. (4) As in 4.5.2. (5) As in 4.5.2. (6) If the last line of f consists of an application of (R4), i.e., f is a derivation of the form P 4 u:A+C u:B+C [u, u ] :A v B + C and if 1Ipn= S ( h ) and [qn = S ( k ) , let g be the derivation

h k A+C B+C AvB+C

0

5.5.3.REMARK.We repeat Remark 4.5.3 concerning the use of dots in order to display the instances of (R5) and (R6) unambiguously. 5.5.4. COROLLARY. The category Fbc(X) is isomorphic to a subcategory of the sequential category generated by the deductive system bcA(X) and the interpretation S : Der(bcA(X))+ ArFbc(X). 0

5.61

THE S Y N T A X OF

Fbc(X)

63

5.6. The syntax of Fbc(X)

We extend Theorem 4.6.1 to bcA(X) and show that the arrows of Fbc(X) have a composition-free description:

5.6.1. THE CUT ELIMINATION THEOREMFOR bcA(X). Every f € Der(bcA(X)) is equivalent to a cut-free g E Der(bcA(X)). PROOF. By Theorem 4.6.1, with Clause (C.26) generalized to allow non-empty Q and 9,together with Clauses (C.4), (C.7), (C.8), (C.9), (C.12), (C.15), (C.21), (C.22), (C.29), (C.371, (C.38), (C.43), and (C.44) of the cut elimination algorithm, f reduces to a cut-free derivation g . It remains to show that the reduction steps preserve the meaning of f. Most cases can be disposed of by duality: (C.7) is dual to (C.2), (C. 15) is dual to (C.14), (C.21) is dual to (C.34), ((2.22) is dual to (C.35), (C.29) is dual to (C.42), ((2.37) is dual to (C.18), (C.38) is dual to (C.19), (C.43) is dual to (C.26), and (C.44) is dual to ((2.27). Hence we are left with Cases (C.4), (C.8), (C.9), and (C.12), and the task of re-examining Case ((2.26). A brief reflection shows that (C.8) is dual to (C.4), (C.9) is dual to (C.3), and the two subcases arising in (C.12) are analogous, respectively dual, to (C.2.1). In the case of (C.4), we must show that the following reduction preserves equivalence: g

+@

r -+f

Q

~

~

g

f r-+QaB9 p+o

r+Qa@z

P +@

g

+@ a+@

Since g is cut-free by hypothesis, we may take 0 to be T. Hence the above derivations are equivalent provided that the diagram A v ( B v C)(AvB)vC ( I V T ) V T

IVT

A vT

IVS*

64

[5.6

BICARTESIAN CATEGORIES

commutes for all A, B, C E ObFbc(X). By Axiom (Ml) of monoidal categories, Diagram (1) commutes iff Diagram (2) commutes, where (2) results from (1) by the replacement of a*(A, B, C) by (a*)-'(A, B, C). But by the uniqueness of terminal arrows, Diagram (2) commutes iff

- 1 I 1

A v ( B v C)

IV(TVT)

A v (T v T)

IVS*

A

(a*)-'

(3)

(AvB)vC (IVT)VT

iva*

5.61

THE S Y N T A X OF

Fbc(X)

65

in the proof of Theorem 4.6.1. By the naturality of a,', the right-hand side of Case (3) is equivalent to the derivation

r+ayvsq

YVs+'(YAp

r+@aApq

and by the joint naturality of aa and arrthis derivation is equivalent to the left-hand side of Case (3). This proves the cut elimination theorem for bcA(X). 0 Using the completeness theorem for Der(bcA(X)) and arguing as in the proof of Lemma 4.6.2, we obtain an analogous result for bcA(X):

5.6.2. LEMMA.If a = T and p derivable in bcA(X). 0

= 1 in

Fbc(X), then + a and l3+

are

In order to enable us to establish an analogue of Corollary 4.6.3 for bicartesian categories, we require additional preliminaries: 5.6.3. LEMMA.A = A v I= A

A

T for all A E ObFbc(X).

PROOF.The existence of the required isomorphisms follows at once from the Yoneda lemma since

66

BICARTESIAN CATEGORIES

and

[ X ,A v TI

[ X ,A ] x [ X ,TI

[5.6

[ X ,A ]

for all X , Y E ObFbc(X). In contrast to the situation in cA(X), it is no longer true in bcA(X) that we can take all active formulas in instances of (R2) to be atomic. Since Fbc(X) is non-distributive, the derivation A+A A(B v C ) + A with A, B , C E ObX, for example, cannot be replaced by the derivation A+A A+A AB+A AC+A A(B v C )+ A A dual argument shows that the active formulas of instances of (R5) cannot be taken to be atomic. However, a slightly weaker version of Lemma 4.6.4 and its dual still holds for bcA(X):

5.6.4. LEMMA.For every cut-free f EDer(bcA(X)) there exists an equivalent cut-free g E Der(bcA(X)) containing no instances of (R3) and (R6), and no instances o f (R2) and (R5) whose active formulas are o f the form (Y A p and a v p, respectively. . 0

For the purpose of the proof of our final lemma, we regard the category Ens of 2.2.7 as a bicartesian category, with the empty set 0 as an initial and a fixed one-element set { * } as a terminal object. Similarly, we regard the category comRng of commutative rings with 1 and ring homomorphisms as bicartesian, with the ring of integers Z as an initial and the trivial ring 0 as a terminal object. (The product objects of comRng are Cartesian products with coordinate-wise operations (cf. 6.2.2 for the definition of addition, for example), and the coproduct objects are tensor products of rings, taken over Z (cf. LANG[1965]).) 5.6.5. LEMMA.The objects l ~ and l T v T are neither initial nor terminal objects of Fbc(X).

5.61

T H E S Y N T A X OF

Fbc(X)

67

PROOF. Let Const 0 : X + comRng and Const { * } : X + Ens be the constant functors with object values Const O(A) = 0 and Const { * } ( A )= { * } for all A E ObFbc(X),and let FO: Fbc(X) + comRng and F{*): Fbc(X) + Ens be the induced functors of bicartesian categories. Then F o ( l A I)= Z x Z in comRng, and F+,(T v T) = {*} + {*} in Ens. Obviously, Z x Z + 0. Suppose that Z x Z = Z,and let

be a product diagram determined by this isomorphism. Then I @ ) = comp(h, T A )= comp(h, n,,), and therefore T A = np Since ~ A ( ub), = a and n J a , b ) = b, we have a = b for all a, b E Z . A contradiction. A cardinality argument establishes that { * } + { * } is isomorphic to neither 0 nor {*}. 0 5.6.6. COROLLARY. The initial and terminal objects of Fbc(X) are characterized by the following properties: (1) I f a 1T and p = I , then T and 1 are the only atomic subformulas of a and p. (2) a ~ p = T ifa=p=T. (3) a v p = T if a ( @= ) T and P ( a ) = I . (4) ( ~ ~ p = l i f ~ ( ~ ) ~ T u n d p ( co rui f) a= =I p , pI. (5) a r v p = l i f a r = p = I . 0

5.6.7. COROLLARY. If a = T, there exists a unique derivation g of + a consisting at most of instances of (A3), (RS), (RIO), and (R13), and n o instance of (R5) in g has a n actiue formula of the f o r m a v p. 0

5.6.8. COROLLARY. If p =I,there exists a unique derivation h of p + consisting a t most of instances of (A4), (R2), (R1 I), and (R12), and n o instance of (R2) in h has a n active formula of the f o r m a A p. 0 We now extend the definition of the normality of derivations of cA(X) to the derivations of bcA(X) by defining f E Der(bcA(X)) to be normal if it is normal in the sense of Appendix D, and satisfies four additional requirements:

68

BICARTESIAN CATEGORIES

[5.6

(1) f contains no instance of (R3) and (R6). (2) Unless f is a derivation mentioned in Cases (3) and (4) below, f contains no instances of (R2) and (R5) whose active formulas are of the form (Y v p and (Y A p, respectively. (3) If f derives r+@, and if one of the disjunctions of the formulas of @ is isomorphic to T, then f is of the form g

+@

r+ (p(')r where g is the unique derivation of + @ compatible with Corollary 5.6.7, and where (a)consists of the instances of (R2) required to derive r+@ from +@. (4) If f derives r+@, and if one of the conjunctions of the formulas of r is isomorphic to I and no disjunction of the formulas of @ is isomorphic to T, then f is of the form

where h is the unique derivation of + compatible with Corollary 5.6.8, and where ( 7 ) consists of the instances of (R5) required to derive r+@ from r + . By combining Theorem 5.6.1 with the preceding lemmas and corollaries, and extending the reducibility relation 2 of Appendix D in the obvious way beyond the extension required in Chapter 4, we obtain the desired analogue of Theorem 4.6.5 for Der(bcA(X)): 5.6.9. THE NORMALIZATIONTHEOREM

FOR bcA(X). Every f E Der(bcA(X)) reduces to a unique equivalent normal g E Der(bcA(X)).

PROOF.The theorem follows from Theorem 4.6.5 and the normalization algorithm defined in Appendix D, provided the reductions in Conditions (D.4), (D.10), (D.17), (D.21), (D.34), (D.36), (D.37), (D.38), (D.59, (D.591, and (D.62) preserve equivalence. But this is clear: (1) The equivalences required for (D.4), (D.10), (D.17), (D.21), (D.361, and (D.55) are immediate consequences of the associativity of composition.

5.61

THE SYNTAX OF

Fbc(X)

69

(2) The equivalences required for (D.34) are a consequence of the functoriality of v . (3) The equivalences required for (D.37) and (D.59) follow from the naturality of am. (4) The equivalences required for (D.38) follow from the naturality of a*.

(5) The equivalences required for (D.62) are consequences of the coherence of a*,i.e., Theorem 2.6.1.1. This proves the normalization theorem for bcA(X). 0 In view of the duality of A and v in bcA(X), Theorem 4.6.6 therefore generalizes immediately to bcA(X). Hence we have an effective characterization of commutativity in Fbc(X): 5.6.10. THECHURCH-ROSSER THEOREM FOR bcA(X). I f f = g, then there exists a normal h E Der(bcA(X)) such that f 2 h and g L h. 0 5.6.11. COROLLARY. The word problem f o r the functor Fbc is sol-

vable. cl

As in the previous cases, all normal derivations of a sequent A + B in bcA(X) have the same width because of the restriction on the antedents of instances of (R12) and succedents of instances of (R10) to single formulas, and are effectively determined by the syntax of Fbc(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees. Hence Theorems 5.5.2, 5.6.1, and 5.6.9 characterize ArFbc(X): 5.6.12. THE COMPUTABILITY THEOREMFOR Fbc(X). Relative to X, the sets Fbc(X)(A, B ) are computable f o r all A, B E ObFbc(X). 0 5.6.13. COROLLARY. The embedding X + Fbc(X) defined by f full and faithful.

PROOF. Similar to the proof of Corollary 2.6.6. 0

+uf]

is

CHAPTER 6

DISTRIBUTIVE BICARTESIAN CATEGORIES

We now investigate the effect of the distributivity of A over v on the results of Chapter 5 . The class of categorical models required for this purpose is a subclass of the class of small bicartesian categories.

6.1. Definition

A distributive bicartesian category is a Cartesian category C with the following additional structure: (4) A bifunctor (-) v (-) : C x C + C. ( 5 ) A distinguished object IE ObC. (6) Two adjunctions a8 and a,,where as = {as(A, B, C, D) : C(A A (B v C), D)+C(A

A

B, D) X C(A A C, D ) I A, B, C,D E ObC},

E ArEns

and a,= {a,(A) : C(I, A)+ { *} E ArEns 1 A E ObC}.

The adjunctions a,,,aI, and ag of a distributive bicartesian category C determine canonically an adjunction am with respect to which C is bicartesian: For each Y E ObC, the adjunctions a, and aTyield a natural isomorphism v : C(-, Y )+C(-, T A Y) whose components are given by the string C(X,Y )= { * } x C(X,Y )3 C(X, T) x C(X,Y )= C(X, T A Y).By Corollary 1.1.21 of the Yoneda lemma, the arrows v ( Y ) ( l ( Y ):)Y -+ T A Y are isomorphisms, for all Y E ObC.The adjunctions awand a89 together with the arrows v( Y)(1( Y)), define the desire adjunction ao,with components C(A v B, C)= C(T A (Av B),C)= C(T A A,C)X C(T A B, C)= C(A, C)X C(B,C). 70

6.31

THE CATEGORY

Fdbc(X)

71

6.2. Examples

6.2.1. A category with finite products and tributive bicartesian iff products distribute products. Thus any distributive lattice with element is distributive bicartesian, and so are

finite coproducts is disisomorphically over coa largest and a smallest Ens and Cat.

6.2.2. COUNTER-EXAMPLES. Any non-distributive lattice is bicartesian, but not distributive bicartesian. A less trivial example is provided by the category of abelian groups. Let A and B be two abelian groups, written additively, let A x B be the abelian group obtained by defining ( a , b ) + ( c , d ) = ( a + c , b t d ) , and let r A : A x B + A , r , , : A x B + B , rT : A + A x B, and r $ : B + A x B be the homomorphisms of groups satisfying the equations m ( a , b ) = a, r,,(a, b ) = b, r X ( a )= ( a ,0 ) , and .rr$(b)= (0, b). Then the object A x B, together with the arrows rAard r,,is a product, and the same object A x B , together with the arrows rT and rz is a coproduct in the category of abelian groups. Moreover, the trivial group is both initial and terminal. Hence the category of abelian groups is bicartesian. But distributivity would require that for all A, B, C, A x ( B x C ) = A x ( B + 12)s( A x B ) + ( A x C ) = ( A x B ) x ( A x C ) . This is obviously false. The bicartesian category Ens, fails to be distributive for similar reasons. 6.3. The category Fdbc(X)

Small distributive bicartesian categories are the objects of a category dbcCat whose arrows are functors F satisfying the conditions of arrows in bcCat and have the additional property that a s ( F ( A ) , F ( B ) , F( C ) , F ( D ) ) ( F ( f ) ) = ( F ( g )F, ( h ) ) for all A , B, C, DEObdom(F) and all f, g, h E Ardom(F) for which w ( A ,B,C, 0)Cf) = (8, h ) . There exists an obvious forgetful functor Udbc : dbcCat+ Cat. We now modify the definition of Fbc and construct a left adjoint Fdbc of Udbc.

6.3.1. DEFINITION. The language of Fdbc(X) is the sublanguage dbcL(X) of L(X)generated by ObX, T, A , I,v, and ArX.

6.3.2.DEFINITION. The labelled deductive system of Fdbc(X) is the

72

DISTRIBUTIVE BlCARTESlAN CATEGORIES

[6.3

subsystem dbc&X) of b(X) generated by Axioms (A]), (A2), (AlO), (All), (A12), (A13), (A14), (A15), and Rules (Rl), (R3), and (85). 6.3.3. REMARK.A comparison with 5.3.1 and 5.3.2 shows that dbcL(X) = bcL(X), and that dbc&X) results from bcd(X) by the replacement of (R4) in bcd(X) by (R5). 6.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(dbcd(X)) satisfying the conditions of Definition 4.3.3, and the following additional requirements: (10) If f = g and h = k, then (f, h ) = (8, h). (1 1) comp((f, g), 1 A nP) = f. (12) comp((f, g ) , 1 A .rr3 = g. (13) (comp(k, 1 A a?),comp(k, 1 A 7~:)) = k. (14) If domu) = I,then f = T * .

We define the category Fdbc(X) by modifying and extending the definition of Fc(X): (1) ObFdbc(X) = dbcL(X). (2) ArFdbc(X) = Der(dbc&X)) /=. (3) As in 4.3.3. (4) As in 4.3.3. (5) As in 4.3.3. (6) As in 4.3.3, with the inclusion of T * , T?,and IT$, defined by means of axioms ( A l l ) , (A14), and (Al5). (7) As in 4.3.3. (8) As in 4.3.3. (9) As in 4.3.3. (10) The image of f : A + B E ArX in Fdbc(X) is (1 1) For all derivable labelled sequents f : A A B + D and g : A A C + 0, ngn) = gin. (12) For all derivable labelled sequents f : A + B and g : C 40, af] v Us1 = Ucomp((p, q ) , (7,l))lL where

cvn,

vfll.

nu,

p = comp(.rr?, comp(f, m ) ) and q = comp(.rrz, comp(g, m)).

(13) For all A, B,C E ObFdbc(X),

w(A, B,C, A and

A

(Bv C ) )= (1(A) A d ( B , C ) , 1(A) A T % ( Bc)), , cu;'(A)( *) = T*(A).

6.41

T H E DEDUCTIVE SYSTEM

dbcA(X)

73

This completes the description of Fdbc(X). The definition differs non-trivially from that of Fbc(X) only in Clauses (2), (1 I), (12), and (13). We call the category Fdbc(X) the free distributive bicartesian category generated by X. Routine calculations show that Fdbc(X) is indeed distributive bicartesian. On the arrows of Cat, we define the functor Fdbc a s follows: 6.3.5. DEFINITION. Let H : C + D be an arrow of Cat, and Fdbc(C) and Fdbc(D) the free distributive bicartesian categories generated by C and D. Then Fdbc(H) : Fdbc(C)+ Fdbc(D) is the functor satisfying the following equations: (1)-(12) As in 5.3.5, with Fdbc in place of Fbc. all f. g E (13) Fdbc(H)((f,g))= (Fdbc(H)(f),Fdbc(H)(g)) for ArFdbc(C). (14)-(16) As in 5.3.5, with Fdbc in place of Fbc. The verification that Udbc and Fdbc are adjoint functors is routine.

6.4. The deductive system dbcA(X)

It is clear from the discussion in Section 5.4 that a sequential construction of the category Fdbc(X) requires at least one of Rules (RI), (RIO), and (R12) in full generality. Since we intend this construction to be stable under cut elimination, our choice is narrowed down to (R10) and (R12). In order to be able to use arguments of duality freely in the proof of the cut elimination theorem, we therefore admit both these rules in full. The nature of the cut elimination algorithm then forces us also to postulate all of (RI) as a rule of inference of dbcA(X). The unlabelled deductive system of Fdbc(X) is the subsystem dbcA(X) of A(X) generated by Axioms (Al), (A3), (A4), and Rules (Rl), (R2), (R3), (RS), (R6), (RIO), (R1 I), (R12), and (R13):

74

[6.5

DISTRIBUTIVE BICARTESIAN CATEGORIES

6.5. The semantics of Der(dbcA(X))

We now extend the interpretation of Der(bcA(X)) in ArFbc(X) to an interpretation of Der(dbcA(X)) in ArFdbc(X). For this purpose, we require the canonical arrows defined in 4.5 and 5.5, with the following additions: (7) &(A, B, C ) : A A ( B v C)-*(A A B ) v (A A C) for all A, B, C E ObFdbc(X), where &(A, B, C ) = v ( ( AA B ) v (A A C))(1((A A B ) v (A A C))), and where v is the natural isomorphism determined by the composition C((A

A

B ) v (A A C ) ,D)-% C(A A B, D ) X C(A A C, D )

C(A A ( B v C ) ,D )

with C = Fdbc(X). (8) S , ( A , B , C ) : ( A V B ) A C - , ( A A C ) V ( B A C for ) ObFdbc(X), with 6, defined by the compositions (A v B ) A

c"- c A (A v B)-

%

all

A,B,CE

(cA A) v (cA B )

(9) Sf(A, B, C ) :(A v B ) A (A v C ) + A v ( B A C ) for all A , B , C E

6.51

THE SEMANTICS OF

Der(dbcA(X))

75

ObFdbc(X), with i3f defined by the compositions (Av B )A (AA C )

\

80

( A A ( A v C ) )v ( B A ( A v C ) )

ITfVl

A v ( B A ( Av C))

1

( A v ( B A C ) )v ( B A ( A v C ) )

(Av (BA

IVsA

c))v ( ( B A A ) v ( B A C ) )

A v (B A C)

(10) S;(A, B, C ) : ( A v C ) A ( B v C ) + ( A A B ) v C for all A, B,C E ObFdbc(X), with i3; defined by the compositions

(AA B )v C ( 1 1 ) K ( A , B, C, D, E ) : ( A A ( ( B v C ) V D ) ) A E + ( B V ( ( A A C ) A E ) )

v D for all A, B, C , D, E E ObFdbc(X), with

positions

K

defined by the com-

76

-

[6.5

DISTRIBUTIVE BICARTESIAN CATEGORIES

( A A ( ( B v C ) v D ) )A E

\

8~A 1

( ( AA ( B v C))v ( A A D ) )A E

I

(8,

V I)A

1

( ( ( AA B ) v ( A A C ) )v ( A A D ) )A E

I

1

( ( ( AA B ) v ( A A C ) )A E ) v ( ( AA D )A E ) 8PVl

( ( ( AA B ) A E ) v ( ( AA C ) A E ) ) v ( ( AA D ) A E )

I

(IA

v 1) v =A

6.5.1. REMARK.As the notation suggests, the arrows SA and SX, respectively S, and S z , are related by duality. The diagrams defining S, and 8; can be obtained from one another by the interchange of A and v, and the reversal of all arrows, with their labels dualized. The duality of SA and SX follows from the commutativity of the diagram obtained by dualizing the defining diagram of SX above.

6.5.2. DEFINITION.The interpretation of Der(dbcA(X)) in Fdbc(X) is the function S : Der(dbcA(X))+ ArFdbc(X) satisfying Conditions ( l ) , (2), (4), (6),(7), (8), (lo), (1 l), and (13) of 5.5.1, and the following generalizations of Conditions (3), (5), (9), and (12): (31, (9)

(@ v 0 )v 9

6.51

T H E SEMANTICS OF

Der(dbcA(X))

(a v (a/i8))v

I1

*

We note that the present interpretation of (R1) links the two previous special cases, i.e., comp(S(g), (1 A S ( f ) )A 1) and comp(( 1 v S ( g ) )v 1 , S(f)), by means of K , and the present interpretations of (R10) and (R12) result from the earlier ones by composition with comp(SX v 1,6$) and comp(b,, SA A 1). Not unexpectedly, Der(dbcA(X)) classifies the arrows of ArFdbc(X): Let f = g in Der(dbcA(X)) iff S(f)= S ( g ) . Then we obtain a bijection Der(dbcA(X))/= = ArFdbc(X): 6.5.3. THE COMPLETENESS THEOREMFOR Der(dbcA(X)). For every

f E Der(dbcd(X)) there exists a g E Der(dbcA(X)) such that S ( g ) = l[f 1E ArFdbc(X).

PROOF.The proof is identical to that of Theorem 5.5.2, except for Case (6), which we modify as follows: (6) If the last line of f consists of an application of (R5). i.e., f is a derivation of the form P 9 U:AAB+D u:AAC+D ' (u, v ) : A A (B v C ) + D

78

DISTRIBUTIVE BICARTESIAN CATEGORIES

[6.6

and if !PI= S ( h ) and UqJJ=S ( k ) , let g be the derivation A+A B+B AB+A AB+B h AB+AAB AAB+D AB+D

A+A C+C AC+A AC+C k AC+AAC AAC+D AC+D A(B v C)+ D 0 A A ( B v C )+ D

6.5.4. REMARK.We repeat Remark 4.5.3 concerning the use of dots in order to display the instances of (R5)and (R6) unambiguously. 6.5.5. COROLLARY.The category Fdbc(X) is isomorphic to a sub-

category of the sequential category generated by the deductive system dbcA(X) and the interpretation S : Der(dbcA(X))+ ArFdbc(X). 0

6.6. The syntax of Fdbc(X)

The syntactic advantages of the system dbcA(X) over the system dbch(X) once again lie in the fact that the derivations of dbc(X) code their own labels and provide a cut-free description of ArFdbc(X): 6.6.1. THE CUT ELIMINATION THEOREMFOR dbcA(X). Every f E Der(dbcA(X)) is equivalent to a cut-free g E Der(dbcA(X)).

PROOF.In view of the greater generality of Rules (Rl), (RlO), and (R12) in dbcA(X), we must reexamine most cases dealt with in the proof of Theorem 5.6.1. In particular, we must review Cases (C.3), (C.4), (C.8-9), (C. 12), (C.14-15), (C.18-19), (C.21-22), (C.26-29), (C.34-35), (C.37-38), and (C.41-44). Several cases can be disposed of by means of duality: (C.8) is dual to (C.41, ((2.9) is dual to (C.3), ((2.15) is dual to (C.14), (C.21) is dual to (C.341, ((2.22) is dual to (C.39, (C.37) is dual to (C.18), (C.38) is dual to (C.19), ((2.41) is dual to (C.28), (C.42) is dual to (C.29), (C.43) is dual to (C.26), and (C.44) is dual to (C.27).

6.61

FdbdX)

THE SYNTAX OF

79

Hence it suffices to consider Cases (C.3), (C.4), (C.12), (C.14), (C.18), (C.19), (C.26), (C.271, (C.281, (C.29), (C.34), and (C.35). But these cases are easy consequences of the commutativity of the following diagrams, for all A, B, C E ObFdbc(X), and all f, g E ArFdbc(X): IAr

AAB-CAD

CAB A

A

A

A

( B v C)-

61

( AA B)v ( AA C )

%

( B v C)-(A

4

BvC
0, then no sequential analogue of E + F is derivable in smclA(X), and for q > 0, the above counting formula is easily proved by an induction on P I * . + p n + q. In these considerations, it is of course understood that n > 2. The absence of the formula N ( p , q ) from the above estimate may at first seem surprising. However, it follows from the interpretation of Der(smclA(X)) in ArFsmcl(X) and the cut elimination algorithm for smclA(X) that comp(h, f ) = f for all f : E + F constructed above and all N ( 2 q , 2q) arrows h : F + F constructed from l(B), for example. The following calculation is typical of the general case, and uses the fact that COmp(KA 1(I), K A * ) = 1(A*): +

a

122

SYMMETRIC MONOIDAL CLOSED CATEGORIES

[8.6

The derivations involved in Counter-examples 8.6.4 and 8.6.7 are not constructible in mclA(X) since both depend on the greater generality of (R15) and since those in 8.6.7 also depend on the presence of (R4). We now examine the consequences of this greater deductive power in detail. By virtue of Theorem 8.6.1, we may again assume that X is discrete. We begin by drawing attention to two important restrictions on (R4) and (R15) in normal derivations: 1. By (D.24), all active (R4) formulas contain an atomic subformula distinct from I. 2. If FA+ (Y jp is derived from raA+ fi by (R15) and r is non-empty, then, by (D.85), (Y contains an atomic subformula distinct from I and so does some formula to the left of a in r. (1) The greater generality of (R15) allows identical antecedent formulas with subformulas distinct from I to be moved to the succedent from any position. This gives rise to the following types of distinct normal derivations: A+A A+A .A.A + A K( A A + A j (A n A) k .A.A+ A A+AjA’

A+A A+A A.A.+AnA A + A .$(A n A) k A.A. + A A+AjA’

with k denoting internal composition, in the sense of Chapter 7. (2) In combination with (R14), Rule (R4) yields a variety of somewhat more subtly distinct new normal derivations: A+A

f3

f2=

k AA+A A + A AA(A~A)+A*

f4=

k AA+A A + A AA(A .$ A)+ A AA(AjA)+A

k AA+A

f,= AA(A+A)+A’ k A + A AA+A A(A 3 A)A + A = AA(A 3A)+ A

8.61

T H E S Y N T A X OF

A+A

k

AA+A

FsmcKX)

A+A

123

k

AA+A

with k again denoting internal composition. For suitable A, the intuitive meaning of the last six derivations in Ens is as follows:

fdx, Y, f ) = comp(f(y), x), fdx, Y,f ) = f(comp(y, XI), f d y r x, f ) = f(comp(x, Y)), fdx, Y, f ) = comp(y, f(x)), fdx, Y. f ) = comp(x, f ( 11,~ fh(xIY, f ) = comp(f(x), Y ). The unique strict functor FM : Fsmcl(X) + Ens of symmetric monoidal

closed categories defined analogously to the functor Fbf in the proof of Theorem 3.6.4 separates the normal derivations in ( I ) and ( 2 ) above for some A. It does so, for example, if A = (B 3 B )and B E ObX. On the other hand, it fails to distinguish the first two derivations in ( 1 ) if A = (B+I), with B E ObX, for example, since FM(Bj I)is terminal in Ens. Once again, the functor FA : Fsmcl(X) + V defined analogously to the functor FA in the previous chapter comes to our rescue and distinguishes all derivations in (1) and (2). (3) In addition, we have the type of derivations in which (R4), (R9), and (R14) combine to produce new distinct normal derivations. The following example is typical: k AA+A AA+A A+A AAA+A A ( A 3 ( A A A ) )+ A ~~

~

k

AAAA A+A AAA+A A(A 3 ( A A A)) A

Here again, for sake of definiteness, k denotes internal composition. (4) Similarly, (R4), (R9), and (R15) can combine to produce new non-equivalent normal derivations as follows:

k'

f=

A.AA. -+ A A.AA. + A A ( A n A)+ A A+(AnA)jA

k' .AA.A + A .AA.A + A ( An A)A +A g = A + ( A nA ) 3 A

124

[8.6

SYMMETRIC MONOIDAL CLOSED CATEGORIES

with k' representing a double composition arrow of Fsmcl(X). The remaining ten non-equivalent derivations are constructed in the same way by means of the other permutations of AAA. Again we have nonequivalence by virtue of the non-equality FA(Scf))# F,(S(g)),for all f and g built in this way. (5) The joint presence of (R4),(R14),and (R15)also produces new non-equivalent derivations: The schemes k k AA+A AA+A AA(A3 A ) A+ A

and

k

k

AA+A AA+A AAA(A 3 A )+ A

together determine twelve non-equivalent derivations of the sequent AA(A=$ A )+ A A. Because o f (D.65)some , of these derivations do not involve an instance of (R4).The following examples are typical:

+

k A.A.+A k AA+A A+A+A AA(A +A)+ A 3 A

k A.A.+A k AA+A A+A*A AA(A A )+ A 3 A AA(A3 A )+ A 3 A

+

k k AA+A AA+A A.A.A(A =$ A )+ A AA(A 3 A )+ A 3 A

k k AA+A AA+A A.A.A(A 3 A)+ A AA(A A . 3 . A)+ , A AA(A3 A)+ A 3 A

k k AA+A AA+A A.A.(A 3 A ) A+ A A ( A3 A ) A+ A =$ A AA(A 3 A )+ A 3 A

k k AA+A AA+A A.A.(A 3 A ) A+ A

+

AA(A +A)+ A 3 A AA(A3 A)+ A 3 A

with k denoting internal composition, as usual. Here too, the functor FA distinguishes all twelve derivations. In general, an inspection of the rules of inference shows that for normal derivations, the symmetric aspects of Fsmcl(X) manifest themselves in six distinct ways in smclA(X):

8.61

THE SYNTAX OF

Fsmcl(X)

125

(1) f and g are the derivations

wn a(T), r +m a (a) and -

A+ a

A+ a

in the notation of (D.24), with a# T. (2) f and g are the derivations

(3) f and g are the derivations

(4) f and g are the derivations

( 5 ) f and g are the derivations

(6) f and g are the derivations

An induction on the construction of S c f ) and S ( g ) in smcld(X) shows that in view of the restrictions on (R4) and (R15)in normal derivations, the V-valued functor FA distinguishes f and g, for all normal m, n, r, and s.

I26

SYMMETRIC MONOIDAL CLOSED CATEGORIES

[8.6

A similar induction also shows that the functor FAis faithful with respect to Rules (R8)and (R14), in the sense discussed in the previous chapter. It is clear from the preceding discussion and the relevant clauses of the reduction algorithm of Appendix D, that if f , g : I'+ a are two normal derivations, then f ends with (R2) iff g ends with (R2); f ends with (R9) iff g ends with (R9); f ends with (R8) iff g ends with (R8) or (R4); f ends with (R14) iff g ends with (R14), (R15), or (R4); f ends with (R4) iff g ends with (R4), (R8),(R14), or (R15). The given examples are typical of the syntactic possibilities involved. Thus the previous induction on the construction of Scf) and S ( g ) in smcli\(X) establishes that f = g iff FA(Scf)) = FA(S(g)) iff f = g , and we have proved the decidability of =: 8.6.9. THE CHURCH-ROSSERTHEOREMFOR smclA(X). I f f = g, then there exists a normal h E Der(smclA(X)) such that f 1 h and g 2 h. 0 8.6.10. COROLLARY.The word problem f o r the functor Fsmcl is sol-

vable. 0

Since instances of Rule (R4) do not affect the width of the derivations of a sequent A + B in smclA(X), it is clear from Chapter 7 that all normal derivations of A + B have the same width, and are effectively determined by the syntax of Fsmcl(X) relative to any fixed assignment of axioms to the top nodes of the underlying trees of these derivations. Hence Theorem 8.5.2, 8.6.1, and 8.6.2 characterize ArFsmcl(X): 8.6.11. THE COMPUTABILITY THEOREMFOR Fsmcl(X). Relative to X, the sets Fsmcl(X)(A, B ) are computable for all A, B E ObFsmcl(X). 0 8.6.12. COROLLARY.The embedding X + Fsmcl(X) defined by f full and faithful.

+ i f 1 is

PROOF. Similar to the proof of Corollary 2.6.6. 8.6.13. COROLLARY.If the objects A , B 1 , .. . ,B, of Fsmcl(X) have atomic subformulas distinct from I,and if Il(C,D)l(denotes the cardinality of Fsmcl(X)( C, D ) , then (1) l((A'2"',A(2m+1))l) = ll(A(2n+'), A""')ll = 0.

8.61

T H E SYNTAX OF

FsmcKX)

127

where A(*"),etc., have the same meaning as in 8.6.5. 0 8.6.14. COROLLARY.If X is discrete, then the sets FsmcI(X)(A, B ) are

finite for all A, B E ObFsmcl(X). 0

CHAPTER 9

CARTESIAN CLOSED CATEGORIES

In the light of the foundational contributions of Lawvere and others within the framework of elementary topoi, Cartesian closed categories constitute one of the mathematically most important types of structure studied in this monograph, especially since it is now known that an elementary topos is simply a Cartesian closed category C with a distinguished object R and a natural isomorphism Sub(-) = c(-,R) which classifies the subobjects of an object A in terms of the characteristic functions on A. The consequences of this additional axiom are not examined until Chapter 13 since the involved ideas of the completeness and the cocompleteness of a category are only marginally deductive. In the present context, therefore, Cartesian closed categories serve merely as Since such the natural models for the joint proof theory of T, A , and categories are automatically symmetric monoidal closed it is interesting to note that, proof-theoretically, the passage from symmetric monoidal closed to Cartesian closed categories involves both a loss and a gain. The terminality of T leads to the identification of all those derivations which were kept distinct by Counter-examples 8.6.4 and 8.6.5. On the other hand, the joint presence of the unit S of the adjunction a,,and the counit E of the adjunction CXA requires a class of derivations for the construction of the free Cartesian closed categories which is so rich in structural properties that it possesses infinitely many non-equivalent cut-free derivations of certain sequents.

+.

9.1. Definition

A Cartesian closed category is a Cartesian category C with the following additional structure: (4) A bifunctor (-) (-) : Copx C + C. ( 5 ) An adjunction a ~where ,

+

(YA

= {(YA(A, B, C) : C(A A

B, C ) + C(B, A j C) E ArEns I A, B, C E ObC}. 128

9.31

THE CATEGORY

Fccl(X)

129

9.1.1. REMARK.A comparison with Definition 7.1 shows that Definition 9.1 results from Definition 7.1 by the replacement of the word monoidal by Cartesian and the replacement of #: by A in Condition (5). 9.2. Examples

9.2.1. All monoidal closed categories mentioned in Examples 7.2.1, 7.2.2, 7.2.5, and 7.2.6 are Cartesian closed.

9.2.2. For any small category C, the category Funct(C"',Ens) becomes Cartesian closed if we define ( F AG ) ( A ) = F ( A ) x G ( A ) for all A E ObC"', T=Const{*} for some fixed one-point set {*}, and define the functor F j G by the condition (Fj G ) ( A )= Nat(G A C(-, A ) , F) for all A E ObCoP.The definition of F j G is based on the fact that this functor is set-valued and that by the Yoneda lemma we must therefore have a natural bijection (F G ) ( A )= Nat(C(-, A ) , F j G ) for all A E ObCO', and the fact that the adjunction CXA demands that Nat(G A C(-, A ) , F ) = Nat(C(-, A ) , F jG ) .

+

9.2.3. COUNTER-EXAMPLE. The monoidal closed category Ens* is not Cartesian closed since any terminal object of Ens* is also initial, and since any Cartesian closed category with this property is equivalent to a discrete one-object category. PROOF.Let C be a Cartesian closed category. Then A = A A T for all A E ObC by the Yoneda lemma since C(X, A ) = C(X, A ) x C(X, T) = C(X, A A T) for all X E ObC. Suppose that T = 1.Then it follows from the closed structure of C that C(A, B ) = C ( A A T, B ) = C ( A A I,B ) = C(1, A B ) = { * } for all A, B E ObC. 0

+

9.3. The category Fccl(X) Small Cartesian closed categories are the objects of a category cclCat whose arrows are functors F satisfying the conditions of arrows in cCat and have the additional property that F ( A B) = F ( A )j F ( B ) , and F ( B ) ,F ( C ) ) ( F ( f )=) F(cxA(A, B, C)(f)) for all A, B , C E that CXA(F(A), Obdom F and all f : A A B + C E Ardom F. There exists an obvious

+

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CARTESIAN CLOSED CATEGORIES

[9.3

forgetful functor Uccl : cclCat+ Cat, and we now construct a left adjoint Fccl : Cat + cclCat of Uccl. 9.3.1. DEFINITION.The language of Fccl(X) is the sublanguage cclL(X) of L(X) generated by ObX, T, A , j,and ArX. 9.3.2. DEFINITION.The labelled deductive system of Fccl(X) is the subsystem ccld(X) of d(X) generated by Axioms (Al), (A2), (AlO), (A12), (A13), and Rules (Rl), (R3), (RlO), and (Rll). 9.3.3. REMARK.A comparison with 4.3.1, 4.3.2, 7.3.1, and 7.3.2 shows that cclL(X) results from cL(X) by the inclusion of j,and from mclL(X) by the replacement of I by T, and M by A , and that ccld(X) results from cd(X) by the inclusion of Rules (RlO) and (R1 1). Moreover, the rules of inference of ccl&X) result from those of mcld(X) by the replacement of Rules (R2), (R6), and (R7) by (R3), (RlO), and (Rll), respectively. 9.3.4. DEFINITION. The relation = is the smallest equivalence relation on Der(ccld(X)) satisfying Conditions (1)-(9) of 4.3.3, and Conditions (20)(27) of 7.3.4, with A in place of M, and with f~ g =(cornp(.f,m), comp(g, d). 9.3.5. REMARK.The definition of f A g conforms with Clause (8) in the definition of Fc(X). Moreover, by the identity laws for categories, Conditions (23) and (24) reduce to (23) comp(cr(1, I), (VA, comp(mCf), wP)))-f. (24) %(COmP(EA(1, (VA, c o m p k r p ) ) ) ) g. We now define the category Fccl(X) as follows: (1) ObFccl(X) = cclL(X). (2) ArFccl(X) = {If! I f E Der(cclb(X))}, where Ufll denotes the equivalence class determined by f. (3)-(10) As in 4.3.3 with Fccl(X) in place of Fc(X). (14) For all derivable labelled sequents f : A + B and g : C + 0,

ufn+

ugn = I I ~ A ( ~ A g))n. C ~ ~ (IS) For all A, B, C E ObFccl(X), and all derivable labelled sequents f : A A B + C, = I ~(.f)ll. A This completes the description of Fccl(X). We call this category the free Cartesian closed category generated by X. The numbering of the

(Ufn

9.41

THE DEDUCTIVE SYSTEM C C b ( X )

131

defining conditions of Fccl(X) has been arranged to allow us to define the bicartesian closed category Fbccl(X) of Chapter 10 below by simply combining the defining conditions of Fbc(X) and Fccl(X). The values of the functor Fccl on the arrows of Cat are defined as in 4.3.4, together with Clauses (12)-(14) of the definition of Fmcl in 7.3.5. As always, we omit the mechanical calculations which verify that Uccl and Fccl are adjoint functors. 9.4. The deductive system cclA(X)

The unlabelled deductive system of Fccl(X) is the extension of the deductive system cA(X) obtained by including Rules (R4), (R14), and (R15) as additional rules of inference. Thus cclA(X) is the subsystem of A(X) generated by Axioms (Al), (A3), and the following restrictions of Rules (Rl), (R2), (R3), (R4), (RlO), ( R l l ) , (R14), and (R15):

9.4.1. REMARK.Rule (R4) is required because of the asymmetry of Rule (R14) and that fact that Fccl(X) is symmetric monoidal: By the Yoneda lemma, the adjunction [ X , A A BI = [ X ,A ] x [ X ,BI = [ X , BI X [ X ,A1 = [ X , B A A] determines a natural isomorphism a ( A , B) : A A B + B A A for all objects A, B E ObFccl(X) which is easily seen to satisfy Axioms (M4)-(M6) of 3.1.

132

[9.5

CARTESIAN CLOSED CATEGORIES

9.5. The semantics of Der(cclA(X))

We now extend the interpretation of Der(cA(X)) in ArFc(X) to an interpretation of Der(cclA(X)) in ArFccl(X) by combining Definition 4.5.1 with Clause (7) of 3.5, Clause (8) of 7.5.2, and Clause (9) of 8.5.1. We summarize the process for ease of reference: For all A, B, C E ObFccl(X), we define ( 1 ) aAA A B, A, B)(l(A A B ) ) = ( ~ A ( A B ), , mP(A.B ) ) . (2) G1((l(A),l(A))) = S(A). (3) CY;~(CZ;'(T~(A, B A C),T A ( B C)mp(A, , B A C ) ) ,mJB, C)mp(A,B A C ) ) = a ( A ,B, C ) . (4) (mp(A, B ) , ~ A ( AB ,) ) = d A , B ) . (5) €(A, B ) = &'(A, A 3 B, B)(I(A 3 B)). (6) q ( A , B ) = AM, B, A A B)(l(A A B)). With the help of these canonical arrows of Fccl(X), we define the function S : Der(cclA(X))+ ArFccl(X) as follows: (1)-(7) As in 4.5.1.

f

\

rupA-t4)=((rAp)Aa)hA~(rA(p @) s(Tpa A + 4

(9)

1

A a ) ) A A (I

A

cr) A I

9.51

THE SEMANTICS OF

Der(cclA(X))

133

t-

This completes the interpretation of Der(cclA(X)) in ArFccl(X). The equivalence classes of Der(cclA(X)) obtained by defining f = g iff S(f)= S ( g ) classify ArFccl(X): 9.5.1. THE COMPLETENESS THEOREM FOR Der(cclA(X)). For every f E Der(ccl&X)) there exists a g E Der(cclA(X)) such that S ( g ) = [f] E

ArFccl(X).

PROOF.We combine the proofs of Theorems 4.5.2 and 7.5.3 by replacing #: by A in 7.5.3, and replacing the derivation

k A+A BISB h AB+ArxB A#:B+C AB+C B+A+C in 7.5.3.7 by the derivation

k A+A B ~ B AB+A AB+B h AB+AAB AAB+C AB+C 9.5.2. COROLLARY. The category Fccl(X) is isomorphic to a subcategory of the sequential category generated by the deductive system cclA(X) and the interpretation S : Der(cclA(X))-+ArFccl(X). 0

134

CARTESIAN CLOSED CATEGORIES

[9.6

9.6. The syntax of Fccl(X)

The category Fccl(X) shares with all other categories discussed in this monograph the property that its arrows have a composition-free description: 9.6.1. THE CUT ELIMINATIONTHEOREM FOR cclA(X). Every f € Der(cclA(X) is equivalent to a cut-free g E Der(cclA(X)).

PROOF.By the proof of Theorem 4.6.1 and Conditions ((2.3, (C.16), (C.20), (C.30), (C.31), (C.36), and (C.45) every derivable sequent has a cut-free derivation. It therefore remains to show that the reductions involved in the above clauses preserve equivalence. In the case of (C.16) and (C.31) the required argument is identical to that used in the proof of Theorem 8.6.1. In the case of (C.20) and (C.36) it is similar to that used in the proof of Theorem 3.6.1, and in the case of (C.30) and ((2.45) it is similar to that used in the proof of Theorem 7.6.1. The equivalences required for (CS), finally, follow at once from the commutativity of the diagram

for all A, B, C, D E ObFccl(X) and all h E ArFccl(X). 0 Using the additional clauses (D.2), (D.3), (D.1 l), (D.12), (D.15), (D.16), (D.18), (D.19), (D.22), (D.23), (D.24), (D.28), (D.29), (D.32), (D.33), (D.52), (D.56), (D.57), (D.64), (D.65), (D71), (D.73), (D.79, (D.77), and (D.79), we can extend Theorem 4.6.5 to cclA(X). But since the reductions given by these clauses do not guarantee the uniqueness of terminal arrows, we first require a lemma: 9.6.2. LEMMA.If a =T, then the sequent + a is derivable in cclA(X). PROOF.By the completeness theorem, the sequent T + a is derivable.

9.61

T H E SYNTAX OF F C C l ( X )

I35

By the cut elimination theorem a65 ObX. Hence a is either T, or of the form p A y or p j y. There therefore exist cut-free derivations +T

T-T T+B

T+v

The lemma now follows from an induction on the height of derivations. 0 Before being able to proceed to the description of the normal derivations of cclA(X), we must give a syntactic characterization of the terminal objects of Fccl(X). In contrast to the situation in Fc(X), Fbc(X), and Fdbc(X), the objects of the underlying category X may now also occur as atomic subformulas of the terminal objects of Fccl(X). Fortunately, the class of terminal objects of Fccl(X) has nevertheless an effective description: Let L(T) be the sublanguage of cclL(X) generated by the conditions (1) T E L(T). (2) If a, p E L(T), then (aA p ) E L(T). (3) If a f L(T) and p f cclL(X), then ( p j a)E L(T). 9.6.3. LEMMA.a

=T

in Fccl(X) i$

Q

E L(T).

PROOF.This result follows by an easy induction on formulas from the Yoneda lemma and the fact that the adjunctions

The definition of the normality of derivations in cclA(X) takes advantage of two additional properties of Der(cclA(X)):

136

CARTESIAN CLOSED CATEGORIES

[9.6

9.6.4. LEMMA.For every cut-free f EDer(cclA(X)) there exists an equivalent cut-free g E Der(cclA(X)) containing no instances of (R2), (R3), and (R4) whose active formulas are o f the form a A p.

PROOF.The lemma follows from an induction similar to that described in Appendix C, using Theorem 3.6.1.1 and the equivalences determined in Der(cclA(X)) by Conditions (D.19.3), (D.29.4), and (D.Sl.l), i.e.,

etc., for all f E Der(cclA(X)) deriving sequents with suitable antecedents. 0

9.6.5. LEMMA.Every cut-free f EDer(cclA(X)) is equivalent to a cutfree g € Der(cclA(X)) containing no instances of (R14) with the property that the active formulas in their right premisses are isomorphic to T. PROOF.Similar to that of the previous lemma, using equivalences of the form k k AA+ 4 AA+ 4 h T+a ApA++ Aa+pA+c$ A r a j P A + + -Ara+PA+c$ 0 We now define a derivation f EDer(cclA(X)) to be normal if it is normal in the sense of Appendix D, and satisfies four additional conditions: (1) All subderivations o f f not containing instances of (R14) and (R15) are normal in the sense of 4.6, with the condition that the active formulas of instances of (R2) are atomic replaced by the condition that the active formulas of instances of (R2) are either atomic or of the form ff

*P.

(2) Unless f is a derivation mentioned in (4) below, f contains no

9.61

T H E S Y N T A X OF

Fccl(X)

I37

instances of (R2) and (R3) whose active formulas are of the form a A p. (3)f contains no instance of (R14) whose active formula in the right-hand premiss is isomorphic to T. (4) If f derives r+ a and a = T, then f is of the form

where g is the necessarily unique derivation of + a by means of (A3), (R2), (RlO), (Rll), (R15) which is normal in the sense of Appendix D, and where (a)consists of the instances of (R2) required to derive r+a from + a. By combining the results of Theorems 4.6.5 and 9.6.1 with the preceding lemmas and extending the reducibility relation L of Appendix D in the obvious way to allow for the reductions required in the normality Conditions (1)-(4) above, we now obtain our desired result: 9.6.6. THE NORMALIZATION THEOREMFOR cclA(X). Euery f € Der(cclA(X)) reduces to a unique equivalent normal g E Der(cclA(X)).

PROOF.In view of our previous results, it remains to verify that the additional clauses of the normalization algorithm and Condition (3) in the definition of normality are compatible with our definition of equivalence. (1) The equivalences required for Cases (D.2), (D.3), (D. IS), (D. 16). (D.18), (D.19), (D.24), (D.28), (D.29), (D.71), (D.73), (D.79, (D.77). and (D.79) follow from Theorem 4.6.5. (2) The calculations required for Cases (D.32), (D.33), (D.56), (D.57), (D.64), and (D.65) are similar to those required for their analogues in the proofs of Theorem 7.6.2 and 8.6.3. (3) Clause (D.11) requires the following equivalence:

f f I: A+B D+E A+B f? CA-B D-E - A(B+D)+E C A ( B 3 D)+ E CA(B 3 D)+ E By the interpretation of derivations in ArFccl(X) this equivalence holds

138

[9.6

CARTESIAN CLOSED CATEGORIES

provided that the diagram

(C A A ) A (BJD)-B

comp(f. wA)

A

(BJD)

commutes. But this is immediate from the functoriality of (4) Clause (D. 12) requires the following equivalences:

A .

f

AB+C A B +f C ADB+C - B+A+C DB -+ A 3 C = DB + A J c

TAZB

TAS B

.T.TA + B - A+TJB TA + T J B - TA+ T J B Equivalence (1) is clear from the naturality of following diagram commute, for C = Fccl(X):

I

C(AhB,C) c(IA mp c)

E

I

(YA,

which makes the

C(B,AJC) C(wP A 3 C )

C(A A (D A B ) , C ) = C(D A B, A + C). Equivalence (2) follows from the naturality of ( Y ~ ,i.e., the commutativity of the above diagram with T A A in the place of A A B, B in the place of C, and A in the place of T A , and the fact that by the coherence of A, 1(T) A h(T, A) = A(T, T A A). (5) The equivalences required for (D.22) and (D.23) are established analogously to those required for (D.ll) and (D.12). (6) The equivalence required for Condition (3) in the definition of normality is an easy consequence of the fact that for all f,g E ArFccl(X), comp(m, 1 A f) = T ~and , comp(.rr,, g A 1) = T~

9.61

THE SYNTAX OF

139

FCCl(X)

(7) The equivalences required for (D.52) follow at once from the naturality of a*. (8) The additional equivalences required for (D.81) and (D.83) follow easily from the interpretation of (R15) and the fact that for all terminal objects A E ObFccl(X), a ( A , A) = 1(A A A). This proves the normalization theorem for cclA(X). Before being able to prove the Church-Rosser theorem for cclA(X), we consider the entirely new phenomenon of the infinity of certain hom sets of Fccl(X), even for a discrete finite category X. 9.6.7. COUNTER-EXAMPLE. If A is not isomorphic to T, then the following process generates infinitely many non-equivalent derivations of the sequent A(A 3 A ) + A: A+A (1) A)+ A A(A j A+A A+A A)+ A A(A j A+A

A+A A+A A(A+AA)+A

(3)

A(AjA)+A

A+ A

A+A A(A

+

A+A A+A A(A+A)+A A)(A I$ A ) + A

A(A

etc.

+ A)+

(4)

A

PROOF.Consider Ens as a Cartesian closed category in the sense of 9.2.1, and let F, : Fccl(X)+ Ens be the unique functor which preserves the Cartesian closed structure of Fccl(X) and agrees with the constant functor Const w : X + Ens on X, and let f. be the arrow of Fccl(X) represented by Derivation (n) above, for all n z 1. Then (1)

Foul) = T A ,

140

CARTESIAN CLOSED CATEGORIES

[9.6

(2) FIucf2)= (3) F,,,cf3) = E * € A 1 * * 1 A 8, (4) Fwcf4)=E - E A 1 .a * A 1 - a 1 A (1 A 8) 1 A 8, etc. Let a E o,and let S : w --* o be the successor function. Then Fwcfn) (a, S ) = S"+'(a)= a + ( n + 1). Hence Focfn)= Focfm) iff n = m. 0 €9

9.6.8. REMARK.In contrast to all previously considered categories, the category Fccl(X) contains arrows that are no longer components of generalized natural transformations. By the definition of generalized natural transformation (cf. EILENBERG and KELLY[ 1966a]), the arrows f i defined above are generalized natural iff the following diagrams commute in Fccl(X), for all A EObFccl(X), and all f : A + A E ArFccl(X):

A

A

(A

+ A)-

fAl

A

A

(A

+ A)

As easy calculation in Ens shows that Diagrams (1) and (2) commute only if A = T, or if i = 1 or 2. Next we examine in general the syntactic possibilities for non-equivalent normal derivations of the same sequent in cclA(X). In the light of Theorem 9.6.1, we may assume for this purpose that the underlying category X is discrete. Suppose, therefore that f,g : r + a are two normal derivations. If a! = T, then it is easily seen from the definition of 2 that f = g. Hence we need to consider only derivations representing non-terminal arrows of Fccl(X). By Condition (3) in the definition of normality and Lemma 9.6.3, the active formulas in the conclusions of instances of (R14) inf and g do not represent terminal objects of Fccl(X). Syntactically, it is clear from the nature of the rules of inference and the

9.61

T H E SYNTAX OF

FcCl(X)

141

clauses of the normalization algorithm that we must consider the following possibilities: The derivation f ends with (R2) and the derivation g ends with (R2), (R3), (R4), (R14), or (R15); f ends with (R3) and g ends with (R3), (R4), (RIO), or (R14); f ends with (R4) and g ends with (R4), (R14), or (R15); f ends with (R10) and g ends with (Rl0);f ends with (R11) and g ends with (RI 1);f ends with (R14) and g ends with (R14) or (R15): f ends with (R15) and g ends with (R15). The following examples, in which A S T, are typical of the general cases: m A+ A .A.A + A

n A+ A A.A. + A

(R2)

(R2)

e A ( A j A )+ A ( A3 A ) A+ A m A+A A ( A j A)+ A

r

(R2)

(R4)

S

A+A A+A A ( A 3 A)+ A

(R14)

AZA ASA k AA+A AA+A AA+AAA A+AAA

033)

A A G A AA+A AA+AAA A+AAA

(R3)

[9.6

CARTESIAN C L O S E D CATEGORIES

AZA

k

r

(R3)

A A + A AA+P AA+AAA A+AAA

AZA AZA

2

A ( A 3 A ) ( A3 A )5 A A ( A 3 A)+ A

S

A+A A+A A+AAA

(R3)

A ( A 3 A)+ A

(R14)

k AA+ A A ( A 3 A): A .AA.(A A , 3 . AMA , , 3 . A)+ . W4) .AA.(A 3 A ) ( A3 A ) + A k AA+ A A ( A 3 A): A AA.(A 3 A ) ( A3 A).+ A (R4) AA.(A A ) ( A3 A). + A

*

k' .A.AA -+ A AA+AjA

AZA

(R4)

k' A.A.A + A AA+A+A

(R 10)

r S A+A A+A A+AAA

n

A+A A+AAA

(R15)

(R 10)

and not both m = r and n = s.

k m AA+A A+A A A ( A 3 A )+ A

(R14)

k AA+A k AA+A A+AJA A A ( A3 A)+ A 3 A

n k A+A AA+A A A ( A 3 A)+ A

(R14

(R14)

k k AA+A AA+A A.A.A(A 3 A ) + A A A ( A 3 A)+ A 3 A

(R

9.61

I43

T H E S Y N T A X OF F C C l ( X )

where k and k’ denote internal composition and double composition, and where e, e2,and e3 are the following derivations: A+A A+A e= A ( A I$ A ) + A ’

03 I

A A(A

== A +

+

A + A A-+A A(A+A)+A A ( A 3 A ) ( A A)-+ A

e2 = A + A

+

A+A A+A A + A A(A+A)+A A ( A 3 A ) ( A3 A ) + A A ) ( A3 A ) ( A3 A)-+ A

An induction on the construction of S(f)and S ( g ) in ccl&X) shows that F , ( S ( f ) )# F , ( S ( g ) ) for all f and g, with F, denoting the functor described in 9.6.7, even if m = n in (l), (6), (9), and (15); m = r = s in (4) and (8); and also if e, e2,e3,k , and k’ are replaced by other derivations, as long as these do not represent terminal arrows of Fccl(X). Thus the functor F, is faithful with respect to the rules of inference and for non-terminal derivations we have f = g iff F,(S(f))= F,(S(g)) iff f = g. The reducibility relation = therefore decides 2 : 9.6.9. THE CHURCH-ROSSERTHEOREMFOR cclA(X).

exists a normal h E Der(cclA(X)) such that f

2

Iff = g, then there 2 h. 0

h and g

9.6.10. COROLLARY.The word problem f o r the functor Fccl is sol-

vable. 0

Although Counter-example 9.6.7 shows that a sequent A + B may have infinitely many normal derivations, even for discrete X, it is clear from the definition of normality that for any given width n E o, the sequent A -+ B has only finitely many distinct normal derivations relative to any fixed assignment of axioms to the top nodes of the underlying trees of these derivations, and that the syntax of Fccl(X) effectively determines these derivations. Hence Theorems 9.5.1, 9.6.1, and 9.6.6 characterize ArFccl(X): 9.6.11. THE COMPUTABILITY THEOREMFOR Fccl(X). Relative to X, the sets Fccl(X)(A,B ) are computable f o r all A , B E ObFccl(X).

144

CARTESIAN CLOSED CATEGORIES

[9.6

PROOF.Assume that the hom sets of X are finite. Then Fccl(X)(A, B) is the disjoint union of the finite computable sets Fccl(X),(A, B ) of arrows from A to B whose normal representations in Der(cclA(X)) have width n. 0 9.6.12. COROLLARY. The embedding X full and faithful.

+ Fccl(X)

PROOF.Similar to the proof of Corollary 2.6.6.

defined by f + UfJj is

CHAPTER 10

BIC ARTESIAN CLOSED CATEGORIES

In this chapter, we correlate our results on Cartesian, bicartesian, distributive bicartesian, and Cartesian closed categories, and present a categorical semantics for the proof theory of the full intuitionist propositional calculus. As we proceed, the reason for restricting ourselves to intuitionist theories in this monograph will finally become clear, and we shall be able to substantiate the remark made in Chapter 1 that, in contrast to Heyting algebras, Boolean algebras have no non-trivial categorical generalizations (cf. SZABO[1974b]). The natural class of models for our purposes is the class of small bicartesian closed categories. Analogously to the situation in lattices where the existence of relative pseudo-complements entails distributivity (cf. RASIOWAand SIKORSKI [1970]), the closed structure of bicartesian closed categories forces such categories to be distributive bicartesian, and has the further consequence of trivializing all objects of the form A A 1 since in Fbccl(X), A A 1 = 1 for all A E ObFbccKX). The latter property requires the identification of a variety of derivations of bcclA(X) whose counter-parts in bcA(X) were previously kept distinct.

10.1. Definition

A bicartesian closed category is a bicartesian category C with the following additional structure: (7) A bifunctor (-) j (-) : Copx C + C. (8) An adjunction CYA, where CYA

= {cYA(A, B,

C) : C(A A B, C ) + C(B, A 3 C) E ArEns 1 A, B, C E ObC}.

The following elementary properties of bicartesian closed categories will be needed below: 145

146

BICARTESIAN CLOSED CATEGORIES

t10.1

10.1.1. LEMMA. Any bicartesian closed category is distributive biCartesian.

PROOF. The composition [A A (Bv C ) , Y] = [Bv C, A jY] = [B, A j Y] x [C, A. Y] = [A A B,Y] X [A A C, Y] of the appropriate components of the adjunctions aAand a, defines the required adjunction

+

ff8.

0

10.1.2. LEMMA.The following isomorphisms exist in any bicartesian closed category, for all objects A, and all initial objects 1 and terminal objects T: (1) A v I = A . (2) A A T = A. (3) A ~ 1 = 1 . (4) T + 1 = 1 . (5) l j l = T .

PROOF.By the Yoneda lemma, the components (1) [A, Yl = [A, Yl X [I, Y l = [A v I,Yl, (2) [ X ,A] = [ X ,A] X [ X ,TI = [ X ,A A TI, (3) [A A 1,Yl = 11,A YI, (4) [ X ,1 3 = [T A X,1 1 EZ [ X ,T 3 I], (5) 1+1]= [ A I x,11 [I 11, of the adjunctions a,,,a,, aA,a,,and a, determine the desired isomorphisms. 0

+

[x,

10.1.3. LEMMA.For any initial object 1 of a bicartesian closed category C and all A E ObC, any f : A +I€Arc is an isomorphism.

PROOF.Consider the product diagram

in which f

=

7rAhand 1(A) = rPh.By Lemma 10.1.2 and the initiality of I,

10.11

147

DEFINITION

T,,is an isomorphism with inverse ~ *A ( A). 1 Hence ~ * =f T*mh = h, and therefore 1(A) = 'rr,T*f. Moreover, the initiality of 1 forces l(1)= f'rr,T*. Hence f is an isomorphism. 0

10.1.4. COROLLARY. For all A , B EObC, the sets C ( A , B 31)are either empty o r singletons. 0

C ( A , l ) and

In any bicartesian closed category C we can define a negation operator ObC relative to a fixed initial object 1 by the equation 1 A = A j1.For all A, B E ObC, this operator determines five unique arrows (1) 8p1(A, B) : l ( A v B)+ ( 1 A ) A ( l B ) , (2) Sp2(A, B) : A v B + l ( ( 1 A ) A ( T B ) ) , (3) Sp3(A, B) : ( 1 A ) A ( 1 B ) -+ l ( A v B), (4) 8p4(A, B) : ( 1 A ) v ( l B ) + l ( A A B), (5) Sps(A, B) : A A B + - I ( ( - I A )v ( T B ) ) , which we call the De Morgan arrows of C. They are defined as follows: (1) S ~ I ( AB) , is the value of 1 ( l ( A v B)) under the bijection 1 : ObC+

[ l ( A v B), l ( A v B)]

[A v B, l ( l ( A v B))]

= [A, l ( l ( A v B))] x [ B ,l ( l ( A v B))1 = [ 1 ( A v B), 1 A ] x [ l ( A v B ) ,l B ]

= [ 1 ( A v B ) , ( 1 A ) A (1B)I.

(2) Sp2(A, B) is the value of l ( ( 1 A ) A ( 1 B ) ) under the bijection

(3) Gp3(A, B) is the value of 6 p ~ ( AB) , under the bijection [A v B, l ( ( 1 A ) A ( l B ) ) ]

[ ( l A ) A ( l B ) , l ( A V B)].

148

BICARTESIAN CLOSED CATEGORIES

[10.1

the bijection

[ ( l A ) A (A A B),1 1 X [ ( l B ) A (A A B),11 = [A A B,l ( l A ) ] X [A A B, 1 ( 1 B ) ] = [ l A , l ( A A B)] X [lB, l ( A A B)] = [ ( l A ) v ( l B ) , l ( A A B)].

(5) 6p5(A,B) is the value of Sp4(A, B) under the bijection [ ( l A ) v ( l B ) , l ( A A B)]

[A A B,l ( ( 1 A ) v (lB))].

10.1.5. REMARK.The uniqueness of the De Morgan arrows follows at once from Corollary 10.1.4, and the adjunction [A, B 3 C] = [B, A C] is obtained from the symmetric monoidal product structure of C by composing [A, B =$ Cl = [BA A, CIS [A A B,Cl = [B,A 3 Cl. In all bicartesian closed categories, the arrows Spl and Sp3 are isomorphisms by virtue of their uniqueness. We now explore the consequences of the invertibility of Spz, Sp4, and 6115. For this purpose, we define a Heyting algebra to be a bicartesian closed category C in which the sets C(A, B) U C(B, A) are either empty or singletons for all A, B E ObC, and a Boolean algebra to be a Heyting algebra in which A = l ( 7 A ) for all A E ObC.

+

10.1.6. REMARK.The pre-order defined by A 5 B iff C(A, B) # 0 makes such categories into Heyting algebras, respectively Boolean algebras, in the usual sense. Heyting algebras are also known as Brouwerian lattices, implicative lattices, relatively pseudo-complemented lattices with smallest element, and pseudo-Boolean algebras. The present use of the term Heyting algebra is the usual one adopted by workers in the theory of elementary topoi (cf. FREYD [ 19721). Since relatively pseudo-complemented lattices always possess a largest element (cf. 7.2. l), but may fail to have a smallest element (the dense open subsets of the real line R with the usual topology, for example, form a relatively pseudo-complemented lattice, with A 3 B being the interior of the union of B and the complement of A), the names Heyting algebra, Heyting lattice, Brouwerian lattice, and implicative lattice have also been used for relatively pseudo-complemented lattices without a smallest element (cf. GRATZER[19711, CURRY [1963], BIRKHOFF[1%71, and CURRY [1963],

10. I ]

DEFINITION

149

respectively). Moreover, the term Brouwer lattice in CURRY[1963] is used dually as a name for a subtractive lattice. The term Brouwerian lattice in the present sense of Heyting algebra is used in ABBOTT[ 19691. The connection between intuitionist logic and relatively pseudo-complemented lattices which has eventually led to this nomenclature was first discovered by Ogasawara in 1939 (cf. CURRY[1963]). The name pseudo-Boolean algebra, finally, is used in RASIOWAand SIKORSKI [ 19701.

10.1.7. THEOREM.Any bicartesian closed category C in which 8414 is invertible is equivalent to a Heyting algebra.

PROOF.It suffices to show that C(A, B ) is either empty or a singleton for all A, B E ObC, since any endofunctor on C which is constant precisely on isomorphic objects determines a Heyting algebra H as a subcategory of C which is equivalent to C in the sense of Definition 1.1.19. By Lemma 10.1.2 and the invertibility of 8p4, we have the isomorphism T = I j I = (IA I)jI = (Ij I)v (Ij I)= T v T. Hence [X, A] = [ X A (T v T), A ] = [T v T, X j A ] = [T, X j A ] X [T, X j A] = [ X A T , A ] X [ X A T , A ] = [ X A T , A A A ] = [ X , A A A ] for all X , A E ObC. By the Yoneda lemma, we therefore have A = A A A. Let A, B,ObC, and f, g E C(B, A). Via the isomorphism A = A A A we have a product diagram 1

A-A-A

1

in which f = comp( 1, h ) = g . Hence C(B, A) = { *}. 0 The following example shows that the invertibility of 8414 does not imply that the Heyting algebras constructed in the proof of the previous theorem are Boolean: 10.1.8. COUNTER-EXAMPLE. Let H be the linearly ordered set 15 a IT, considered as a Heyting algebra qua lattices, with its relative pseudocomplementation given by the following table:

150

[10.1

BICARTESIAN CLOSED CATEGORIES

It is easy to check that ( 1 A )v ( 1 B )= l ( A A B ) in H for all elements A and B. 0 There exist of course Heyting algebras in which Sp4 is not invertible since the sequent 1 ( A A B)+ ( 1 A )v ( 1 B ) is not intuitionistically valid. In the 5-element Heyting algebra H with maximal branches T 5 a 5 y 5 T and 1 Ip 5 y 5 T , for example, l ( a A p ) = 1 1 = T, and ( l a )v

(1P) = p

v a = y.

10.1.9. THEOREM. Any bicartesian closed category C in which is invertible is equivalent to a Boolean algebra.

Sp2

or 6j.a

PROOF.In the light of the remarks in the proof of Theorem 10.1.7 and the definition of Boolean algebras, it suffices to show that C is a simple category in which A = l ( 1 A )for all A E ObC. By the invertibility of Sp2 and Lemma 10.1.2, we obtain an isomor= l ( ( 1 A )A T ) = l ( 1 A )for all phism A = A v 1 = l ( ( 1 A )A (11)) A E ObC. By the invertibility of Sps and Lemma 10.1.2, we obtain a similar isomorphism A = A A T = l ( ( 1 A )v ( 1 T ) )= l ( ( 1 A )v I)= l ( 1 A )for all A E O b C . Hence in either case, we have [B, A ]= [ B ,l ( l A ) ]= [ ( l A )A B, I] for all A, B E ObC. B y Corollary 10.1.4, the category C is therefore simple. 0 Finally, we give an alternative characterization of Boolean algebras in terms of an adjunction relating A , v, and j: 10.1.10. THEOREM. Any bicartesian closed category C in which there exists an adjunction C ( A A B, C v D ) = C(B, C v ( A D ) ) f o r A, B, C, D E ObC, is equivalent to a Boolean algebra.

+

10.11

DEFINITION

151

PROOF.It is easily verified that a sequent cur+ @p is valid in a Boolean algebra B iff the sequent r+@cu+p is valid in B. Hence the stated adjunction is compatible with the structure of a Boolean algebra. By Theorem 10.1.7 and the definition of a Boolean algebra it is therefore sufficient to show that the given adjunction makes 8p4 invertible and furthermore produces an isomorphism A = l ( 1 A ) for all A E ObC. Using the symmetry of the monoidal product and coproduct structures of C, we obtain three adjunctions [ X , ( 1 A ) v ( l B ) ] = [(A A B ) A X , I v I] = [(A A B ) A X , 1 1 = [ X ,l ( A A B)], and

[A, A] = [A

A

T, A v I ] = [T, A v ( l A ) ] ,

[ T A , T A ] = [A A ( T A ) , I ] = [ ( T A ) A A, I ] = [A, l ( l A ) ] ,

for all X , A, B E ObC. By the Yoneda lemma, 8p4 is therefore invertible, and it remains to construct an arrow f : l ( l A ) + A. The simplicity of C and the nonemptiness of [ l A , l A ] guarantee that f is an isomorphism. Let f be the composite arrow defined by the following commutative diagram: l(1A)

(1.7)

(l(1A))A T

I"g

( l ( 1 A ) ) A (A v ( 1 A ) )

where g is the unique arrow produced by the second adjunction above, has and where (-,-), T , rrp, and E are used in the sense of Chapter 9. the meaning assigned to it in Chapter 6, and h is the isomorphism which exists by Lemma 10.1.2. Then f E A r c . 0

152

BICARTESIAN CLOSED CATEGORIES

110.3

10.2. Examples

10.2.1. Every relatively pseudo-complemented lattice L= ( L , A , T, v, I,+) with inf operation A , largest element T, sup operation v, smallest element I, and relative pseudo-complementation 3, i.e., Heyting algebra, is a bicartesian closed category, and so is, a fortiori, every finite distributive lattice and every Boolean algebra. 10.2.2. All Cartesian closed categories mentioned in Examples 9.2.1 and 9.2.2, with the exception of 7.2.1, are bicartesian closed. 10.2.3. All elementary topoi E mentioned at the beginning of Chapter 9 are bicartesian closed. The coproduct structure of E is definable in terms of the properties of the subobject classifier R (cf. PARE [1974] and LAMBEKand RATTRAY[ 19751). 10.2.4. COUNTER-EXAMPLE. The relatively pseudo-complemented lattice of dense open subsets of the real line R, with the usual topology, mentioned in 10.1.6, is Cartesian closed, but not bicartesian closed since it fails to have an initial object, i.e., minimal element.

10.3. The category Fbccl(X)

Small bicartesian closed categories are the objects of a category bcclCat whose arrows are functors F satisfying the conditions of arrows in bcCat and cclCat. As in all previous cases, there exists an obvious forgetful functor Ubccl : bcclCat- Cat, and in this section, we construct a left adjoint Fbccl : Cat- bcclCat of Ubccl. 10.3.1. DEFINITION.The language of Fbccl(X) is the sublanguage and ArX. bcclL(X) of L(X) generated by ObX, T, A , I, v,

+,

10.3.2. DEFINITION.The labelled deductive system of Fbccl(X) is the

subsystem bccld(X) of &X) generated by Axioms (Al), (A2), (AlO), (A1 l), (A12), (A13), (A14), (Al5), and Rules (Rl), (R3), (R4), (RlO), and (Rll). 10.3.3. REMARK.A comparison With 5.3.1, 5.3.2, 9.3.1, and 9.3.2 shows

10.41

T H E DEDUCTIVE S Y S T E M

bcclA(X)

153

that bcclL(X) results from bcL(X) by the inclusion of 3,and from cclL(X) by the inclusion of 1 and v, and that bccl&X) results from bc&X) by the inclusion of Rules (RlO) and (Rll), and from ccld(X) by the inclusion of Axioms (A1I), (A14), (A15), together with Rule (R4). 10.3.4. DEFINITION.The relation = is the smallest equivalence relation on Der(bccl&X)) satisfying Conditions (1)-(9) of 4.3.3, Conditions (10)(14) of 5.3.4, and Conditions (20)-(27) of 7.3.4, with A in place of n, and with f A g = (comp(f, nA),compk, no)). 10.3.5. REMARK.In effect, 10.3.4 combines the defining conditions of ArFbc(X) and ArFccl(X), i.e., the conditions of 5.3.4 and 9.3.4. We now define the category Fbccl(X) by combining the definitions of Fbc(X) and Fccl(X): (1) ObFbccl(X) = bcclL(X). (2) ArFbccl(X) = (If1 I f E Der(bccl&X))}, where If1 denotes the equivalence class determined by f. (3)-(13) As in 5.3.4, with Fbccl(X) in place of Fbc(X). (14)-(15) As in the definition of Fccl(X), with Fbccl(X) in place of Fccl( X) . We call the category Fbccl(X) the free bicartesian closed category generated by X. The values of the functor Fbccl on the arrows of Cat are defined by combining the definitions of the functors Fbc and Fccl. Again we omit the routine verification of the adjointness condition connecting the functors Ubccl and Fbccl.

10.4. The deductive system bcclA(X)

The unlabelled deductive system of Fbccl(X) is the extension of the deductive system dbcA(X) obtained by including Rules (R4), (R14), and (R15)as additional rules of inference. The necessity of using dbcA(X) as the base system follows from Lemma 10.1.1 which shows that every bicartesian closed category is in fact distributive bicartesian. Thus bcclA(X) is the subsystem of A(X) generated by Axioms (Al), (A3), (A4), and the following rules of inference:

154

BICARTESIAN CLOSED CATEGORIES

[10.4

10.4.1. REMARK.The observation made in Remark 9.4.1 concerning the requirement of Rule (R4)continues to apply. The dual effect of this remark is the fact that Rule (R7), which is an admissible rule of inference of bcclA(X) by virtue of the derivation

I'+@apV

r+@avpq

a+a p+p ~ + p a @+pa

avP+Pa

r+ @pa*

and is prima facie necessary as a rule of bcclA(X) because of the symmetry of the coproduct structure of Fbccl(X), is dispensable as a primitive rule of inference. The cut elimination algorithm for Der(bcclA(X)) is also independent of (R7). The situation is different for (R6). Like Rule (R3), it is required for the cut elimination algorithm, but unlike (R3), it is not needed for the normal representation of arrows of Fbccl(X). The restriction to single formulas in the succedent of the left premisses of instances of (R14) is syntactically convenient since it facilitates our work and does not affect the cut elimination and normalization algorithms. The restriction is however semantically unnecessary. The following commutative diagram shows that the generalization of (R14)to

r+@aQ

ApA+O

Ara 3 P A + @@W

represents a permissible construction in Fbccl(X):

10.41

THE DEDUCTIVE SYSTEM

2 2

c
= f T ( r )with , ~ ( r=)w .

13.4. The semantics of Der(A*(X))

Small quantifier-complete categories are the objects of a category qcCat whose arrows are functors @ satisfying the conditions of arrows ) V(@(F)), in bcclCat, and have the additional property that @ ( V ( F ) = @ ( 3 ( F )= ) 3 ( @ ( F ) ) , and that a v ( @ ( A ) ,Q>(F))(@(f)) = (@(f(n))) and crg(@(F),@ ( A ) ) ( @ ( g ) )= [ @ ( g ( n ) ) ] ,for all F E Funct(o, dam(@)), and all f, g , f(n), g ( n ) E Ardom(@) connected by the equations w ( A , F ) c f )= ( f ( n ) ) and cr3(F, A ) ( g ) = [ g ( n ) ] , and where @ ( F )E Funct(o, cod(@)) is the functor whose object values are of the form @ ( F ( n ) ) .There exists an obvious forgetful functor Uqc : qcCat+ Cat, and the construction of Fbccl(X) adapts easily to the present situation and produces a left adjoint Fqc : Cat+ qcCat of Uqc: (1) We form the sublanguage qcL*(X) of L*(X) generated by ObX, T, I,A , v, 3,A , and V . (2) For all F = { F ( n )I n E o} qcL*(X), we .put V ( F ) = A { F ( n )1 n E o}, 3 ( F ) = V { F ( n )1 n E 0 ) . (3) We augment the class of labels of bccl(X) by allowing all new formulas of qcL*(X) in the label schemes for bcclL(X), and include as additional labels all elements of the form r,,(V(F),F ( n ) ) , r X F ( n ) , 3(F)), c f ( O ) , . . . ,f(n),.. .) and If(O),. . . , f ( n ) , . . .I, for each V ( F ) , 3 ( F )E qcL*(X), F ( n ) E F, and arbitrary labels f(n). (4) We augment Axioms (A2), (810), (All), (A12), (A13), (A14), and (815) by allowing A , B, C E qcL*(X), and adjoin two new axioms:

13.41

(A17)

THE SEMANTICS OF

Der(A*(X))

207

If V ( F )E qcL*(X) and F ( n ) E F, then a,(V(F),F ( n ) ): V ( F )+ F ( n ) is an axiom.

(Al8)

If 3(F)EqcL*(X) and F ( n ) E F , then a ? ( F ( n ) ,3 ( F ) ): F ( n ) + 3 ( F ) is an axiom.

(5) We augment Rules (Rl), (R3), (R4), (RlO), and (R11) by allowing the premisses and conclusions to be sequents in the extended language qcL*(X), and adjoin two new infinitary rules of inference: (R 12)

f(0) : A + F(O),. . . , F ( n ) : A + F ( n ) ,. . .

(f(n)): A + V ( F ) g ( 0 ) : G(O)+ B , . . . ,g ( n ) : G ( n ) + B,. . . k ( n ) l : %GI-, B

(R13)

where (f(n)) abbreviates (f(O), . . . ,f ( n ) ,. . .) and [ g ( n ) ] abbreviates [g(O),. . . ,g ( n ) , . . .I. The described extension of bccl&X) yields a deductive system h*(X) and thus a class of derivations Der(h*(X)) representing the arrows of Fqc(X). More precisely, ObFqc(X) = qcL*(X), and ArFqc(X) = Der(d*(X))/=, with = being the obvious extension of the equivalence relation defined in 10.3.4. The definition of Fqc on the arrows of Cat is obtained by routinely modifying that of Fbccl. We call the category Fqc(X) the free quantifier-complete category generated by X. The objects of qcCat constitute the desired models for Der(A*(X)). We now describe the details of this semantics. It consists of an interpretation S of the derivations of A*(X) as arrows of the free quantifier-complete category Fqc(AtL*(X)) generated by the category AtL*(X), i.e., by the category whose objects are the atomic formulas of L*(X) and whose arrows are those of X, together with the additional identities required to make AtL*(X) into a category. By the freeness of Fqc(AtL*(X)), any functor F : AtL*(X)--* Uqc(C) extends to a unique functor F' : Fqc(AtL*(X)) + C of quantifier-complete categories, and thus induces an interpretation of Der(A*(X)) in any small quantifiercomplete category C. As a preliminary, we interpret the formulas of L*(X) as objects of Fqc(AtL*(X)). The definition proceeds by a transfinite induction on

208

QUANTIFIER-COMPLETE CATEGORIES

[13.4

13.4.1. REMARK.A new phenomenon at the level of predicate and infinitary formulas is the possibility of several distinct formulas representing the same object of a category. Thus for a ( x ) E AtL*(X), for example, S((VSn)a(Sn))= S ( ( V S m ) a ( S m ) ) = A { a x [ T l 1 7 E TL*), S(EISn)a(Sn)) = s ( ( 3 6 m ) a ( ~ m ) = ) V{ax[TI 1 E TL*),

etc. The function S induces an equivalence relation on L*(X) obtained by defining a = P iff S(a)= S(P), and it is easily seen that the resulting equivalence classes correspond to the objects of Fqc(AtL*(X)): L*(X)/= = ObFqc(AtL*(X)). We now define the interpretation S : Der(A*(X))+ ArFqc(AtL*(X)) by transfinite induction and put f = g iff Scf) = S ( g ) . As in Chapters 2-12 above, we do not distinguish notationally between an object of Fqc(AtL*(X)) and a formula or finite sequence of formulas in a sequent representing such an object. This simplifies the notation and the context easily eliminates any ambiguities arising from this practice. The function S satisfies the conditions of its namesake in Section 10.5, together with the following additional clauses:

where F, = {S(ax[~]) I T E TL*}.fX[7] is the derivation obtained from f by the replacement of every occurrence of x in f by T, and (Scfx[.r]))is the arrow obtained from the corresponding sequence by applying aG'. The definition of fX[7] makes sense by virtue of the restriction on variables in Rule (R18).

13.41

THE SEMANTICS O F

Der(A*(X))

209

with Fa having the same meaning as in (1) and nr(V(Fa))being the t-th projection, i.e., the t-th component of the natural transformation A defined in 13.1.17.

with F, having the same meaning as in (1) and .rrT(3(Fa))being the t-th coprojection, i.e., the t -th component of the natural transformation K defined in 13.1.18.

where F, and fX[7] have the same meaning as in (1). The definition of makes sense by virtue of the restriction on variables in Rule (R21). The arrow 6* is the codiagonal of @, i.e., the value under a;’ of the sequence of identities of a, and the arrow 6, is an infinitary analogue of the left distributivity defined in 6.5, i.e., 6, is the value of l ( r A 3(Fa)) under the following string of isomorphisms:

fX[7]

210

[13.4

QUANTIFIER-COMPLETE CATEGORIES

where (S(f(n))) has the same meaning as in (1).

where rnhas the same meaning as in (2). 4-

where r?:has the same meaning as in (3).

f (0) f(1) f( n ) rA(0)A + @, r A ( 1)A + @, . . . , TA(n)A + @, r(V A(n))A + @

...

13.51

T H E SYNTAX OF

Fqc(AtL*(X))

21 1

where a,, [ S C f ( n ) ) ]and , 6* have the same meaning as in (4). The obvious transfinite extension of the proof of Theorem 10.5.1 yields the desired completeness of Der(A*(X)): 13.4.2. THE COMPLETENESS THEOREM FOR Der(A*(X)). For every f E ArFqc(AtL*(X)) there exists a g E Der(A*(X)) such that S ( g ) = f. 0 13.4.3. COROLLARY. The category Fqc(AtL*(X)) is isomorphic t o a subcategory of the sequential category generated b y the deductive system A*(X) and the interpretation S : Der(A*(X))+ ArFqc(AtL*(X)). 0 13.5. The syntax of Fqc(AtL*(X))

We end this chapter by giving a composition-free characterization of the arrows of Fqc(AtL*(X)), outlining a normalization theorem for A*(X), and describing a weak Church-Rosser theorem for finitary derivations. As corollaries, we obtain the decidability of the restriction of the equality relation on ArFqc(AtL*(X)) to finitarily representable arrows, and the fullness and faithfulness of the embedding X-+ Fqc(AtL*(X)). We prove the cut elimination theorem for A*(X) by a transfinite induction along the lines of the proof of Theorem 10.6.1. The proof therefore provides an effective procedure for reducing finitary derivations to cut-free ones. Because of the undecidability of quantificational logic, our methods fail to extend Theorem 10.6.2 to a computability theorem for all finitarily representable arrows of Fqc(AtL*(X)). In order to be able to prove the cut elimination theorem for A*(X), we must extend the definitions of the degree and rank functions of Appendix C to ordinal-valued partial functions on Der(A*(X)). For this purpose, we let Ord stand for the class of all ordinals, and first define the complexity of the formulas of L*(X). 13.5.1. DEFINITION. The degree of a formula a E L*(X) is an ordinal deg(a) determined by the following conditions: (1) If a E AtL*(X), then deg(a) = 0. (2) If a = P A y, P v y , or P y. then deg(a) = sup{deg(P)+ 1, deg(y) + 1).

+

212

QUANTIFIER-COMPLETE CATEGORIES

[13.5

(3) If a = (VS)Pt[S1 or (35)Pt[51, then deg(a) = deg(P(t))+ 1. (4) If a = ( A A(n)) or ( V A(n)), then deg(a) = sup{deg(A(n)) + 1 I n E a). With the help of the degrees of formulas, we now define the degree and rank functions deg, rnk : Der(A*(X))4 Ord relative to a cut-free derivation f, and to an arbitrary derivation g ending with an instance of (Rl), i.e., a derivation of the form

13.5.2. DEFINITION.Let rnk,(g) be the supremum of the lengths of all branches of h ending with r+ @ y q whose elements contain the cut formula y in the succedent, and rnk,(g) be the supremum of the lengths of all branches of k ending with AyA-8 whose elements contain the cut formula y in the antecedent, and let m be the total number of consecutive applications of (R6) at the end of Derivation h, and n be the total number of consecutive applications of (R3) at the end of Derivation k. Then the ordinals degCf), rnkCf), deg(g), and rnk(g) are defined by the following equations: (1) degu) = 0. (2) rnk(f) = 1. (3) deg(g) = deg(y) + 1. (4) rnk(g) = A * p + p, where A = rnkA(g) ( m+ l), and p = rnk,(g) (n + 1).

-

-

23.5.4. REMARK.The factor p in the term A * p in (4) above guarantees that rnk(g) is greater than both rnkA(g)and rnk,(g), even if rnkA(g)is finite and rnk,(g) is infinite. Since this possibility did not arise in the finitary case, it was sufficient, in Appendix C, to define rnk(g) as A + p. Next we define the required extension of the relation > of Appendix C for the proof of the cut elimination algorithm for A*(X). For this purpose, we must first restrict Der(A*(X)) beyond the restrictions imposed in 10.4.2. Again the restrictions are semantically unnecessary, but are required in the cut elimination procedure. A brief reflection on the consequences of these restrictions shows that they do not affect the truth of Theorem 13.4.2.

13.51

THE SYNTAX OF

Fqc(AtL*(X))

213

13.5.5. Terminology. Following Gentzen (cf. SZABO[1969]), we call the free variable x in the active formula a(x) of an instance of (R18) and (R21) the eigenvariable of that instance of (R18) and (R21). 13.5.6. Restriction on eigenvariables. A derivation f belongs to Der(A*(X)) iff all its eigenvariables are distinct. In KLEENE [ 1962, 19721, derivations satisfying the condition of 13.5.6 are called pure variable proofs. Their existence is clear from the restriction on variables in Rules (R18) and (R21), and the fact that every derivation f can be converted to an equivalent pure variable proof by the renaming of free variables follows from the interpretation of Der(A*(X)) in ArFqc(AtL*(X)) and the infinity of the set FV of free variables of L*(X). 13.5.7. Restrictions on (Rl). Two derivations of the form

belong to Der(A*(X)) iff @ = Q = 0. 13.5.8. REMARK.Without Restriction 13.5.6, a reduction of the form

might be illegitimate because of the restriction variables in (R18), and without Restriction 13.5.7, a reduction of the form

f I--+

g

A yA -+ a(x) @ Y V AYA-+ W5)ax[51 > ATA @(V5)ax[61Q -+

r+f aYq

g

A ~ A - + ~ ( X )

A r A + @a(x)Q A r A + @(V6)ax[61Q

214

QUANTIFIER-COMPLETE CATEGORIES

[I33

might be illegitimate because the expression on the right-hand side violates the form of (R18). Its interpretability in Fqc(AtL*(X)) would require the existence in ArFqc(AtL*(X)) of infinitary analogues g , : A (A v A(n))+ A v ( A A(n))

of the distributivity arrows g :(A v B ) A ( A v C ) + A v (B A C) of Fbc(X) described in 5.4. The following counter-example, taken from RASIOWA and SIKORSKI [1970], shows that not even at the level of non-Boolean Heyting algebras can we expect the existence of such arrows: 13.5.9. COUNTER-EXAMPLE. Let X be the space of real numbers with the usual topology, let Open(X) be its associated category of open subsets, and put A = ( - l , O ) U ( O , l ) and A ( ~ I ) = 1( - ~ + ~. , ~ + ~

)‘

Then A U A ( n ) = ( - 1 , l ) for all n E m , and A.,,A(n)=O (since the singleton (0) is not open). Hence A v A A(n) = 0 and A(A v A(n)) = (-1.1). Since 0c (-1, l), we have an arrow A v A A(n)+ A (A v A ( n ) ) , but there is obviously none in the opposite direction. We now augment the relation > of Appendix C. We require twentysix additional reductions, classified again, as in Appendix C, according to the rank of the derivation to be reduced. We simplify the notation by since writing (V5)a[5] and (35)a[5] in place of (V4)aI[5J and (35)a1[51 the term f is always clear from the context, and we write fx[t] for the derivation which results from the derivation f by the replacement of every occurrence of the eigenvariable x in f by the term t. (R18, R19)

> (R18, R2)

(C.47)

13.51

THE SYNTAX OF

Fqc(AtL*(X))

215

(R5, R19)

(C.50) (R20,R2) Dual to (C.49).

(C.5 1)

(R5, R21) Similar to ((2.49).

(C.52)

(R22, R2)

>

A A -0 ~ ArA+ 0 (C.54)

(R5, R23) Similar to (C.49).

(C.55)

(R24, R25) Dual to (C.53).

(C.56)

(R24, R2) Similar to (C.51).

(C.57)

216

QUANTIFIER-COMPLETE CATEGORIES

[13.5

(R5, R25)

r+f

do) s(n) .. AA(O)A+ 0 , .. . ,AA(n)A+ 0,. @( V A(n))V A(vA(n))A+ 0 > AFA+ (PO* (C.58)

(-,R19) Similar to (C.59)

(C.60)

(-,R20) Similar to (C.59).

(C.61)

Similar to (C.60).

((2.62)

(-, R21) (-.R22)

ArA + ( A A(n))'

f

.

I,

g(0) f g(n) AyA+A(O) r+y AyA+A(n) ArA+ A(0) ,. . . , ArA+ A(n) ,. . . ArA+(AA(n))

r+y

>

(C.63)

(-,R23) Similar to (C.60).

(C.64)

(-,R24) Similar to (C.61).

(C.65)

(-,R25) Dual to ((2.63).

(C.66)

(R19,-) Dual to (C.61).

(C.67)

(R20,-) Similar to (C.61).

(C.68)

13.51

THE SYNTAX OF

Fqc(AtL*(X))

217

(R21,-) Similar to (C.60).

(C.69)

(R23, -) Similar to (C.64)

(C.70)

(R24, -) Similar to (C.65).

(C.7 1)

(R25, -) Similar to (C.63).

(C.72)

This concludes the definition of the extension of > required for the proof of the cut elimination theorem for A*(X). The equivalences of the above pairs of derivations under the interpretation of Der(A*(X)) in ArFqc(AtL*(X)) follow easily from considerations analogous to those required for proofs of equivalence of their finitary counter-parts in Der(bcclA(X)). Taken in conjunction with Theorem 13.4.2, a transfinite induction on the ranks and degrees of the derivations of A*(X) ending with instances of (Rl) establishes the desired composition-free describability of ArFqc(AtL*(X)): 13.5.10. THE CUT ELIMINATION THEOREMFOR A*(X). Every f € Der(A*(X)) is equivalent to a cut-free derivation g E Der(A*(X)). 0 13.5.11. REMARK.In Theorem 13.5.10 and in the remainder of this chapter, Der(A*(X)) denotes the class of derivations of A*(X) satisfying the restrictions on derivations imposed in 10.4.2, 13.5.6, and 13.5.7. Finally, we extend the reducibility relation z of Appendix D to a reducibility relation on the set of finitary derivations of A*(X) by specifying the following hierarchy of application of the rules of inference, whenever there exists a proof-theoretical choice:

5 5 13 5 101201 1 2 5 2 1 I1 9 1 1 4 1 3 5 11 5 2 1 4 , 151 1 2 5 2 1 I1 9 1 1 4 1 3 5 11 12114,

18~12121~19~141131115214,

where i 5 j says that Rule (Ri) has priority over Rule (Rj), in the sense of Appendix D. Let 2 be the obvious extension to Der(A*(X)) of its namesake in Appendix D, generated by the prescribed hierarchy of application of the finitary rules of inference of A*(X) and the analogues of Conditions

218

QUANTIFIER-COMPLETE CATEGORIES

[13.5

(D.l-85) for Rules (R18)-(R25), and extend the concept of the normality of derivations of Chapter 10 to Der(A*(X)) in the obvious way. Then an argument similar to that required for the proof of Theorem 10.6.9 shows that the relation L determines an algorithm for normalizing the finitary derivations of A*(X): 13.5.12. THE NORMALIZATION THEOREMFOR A*(X). Every finitary derivation f E Der(A*(X)) reduces to a unique equivalent normal derivation g E Der(A*(X)). 0

The syntactic benefits of Theorems 13.4.2, 13.5.10, and 13.5.12 for Fqc(AtL*(X)) derive from the decision procedure, relative to X, which they entail for the commutativity of all diagrams of Fqc(AtL*(X)) whose vertices are representable by elements of FinL*(X): An argument analogous to that required for the proof of Theorem 10.6.10 established the desired procedure: 13.5.13. THE CHURCH-ROSSER THEOREMFOR A*(X). If f and g are two equivalent finitary derivations of A*(X), then there exist equivalent normal derivations hf, h, € Der(A*(X)) such that f 2 hf, g 2 h,, and hf and h, differ at most in their bound variables. 0 13.5.14. COROLLARY.The embedding X + Fqc(AtL*(X)) defined by

f + If1 is full and faithful.

PROOF.Similar to the proof of Corollary 2.6.6. 0

APPENDIX A

THE LABELLED DEDUCTIVE SYSTEM

d(X)

A . l . The class Lb(&X))

Relative to a fixed, but arbitrary small category X, the class Lb(&X)), whose elements are the labels of the sequents of the language L(X) underlying &X), is defined inductively: ( L b l ) If f € ArX, then f € Lb(&(X)). (Lb2) If A E FL(X) and A E ObX, then 1(A) E Lb(A(X)). (Lb3) If A E FL(X),then A(A), A-'(A),p(A), p - ' ( A ) ,T ( A ) ,and T*(A)E Lb(d(X)). (Lb4) If A, B E FL(X),then a ( A , B ) , m ( A , B),TJA, B ) , %-?(A,B ) , and a*,(A,B )E Lb(&X)). (Lb5) If A, B, C E FL(X), then a ( A , B, C ) and & ' ( A , B , C) E Lb(&X)). (Lb6) If f E Lb(&X)), then so do a ~ ( fand ) ap(f). (Lb7) If f , g E Lb(&XN, then so do comp(f, g ) , f n g , EAU,g ) , and eP(f,g).

(f, g ) , [f, gl, (f, 81,

(Lb8) There are no other labels. A.2. The axioms of &X) The axioms of

d(X) are defined inductively, relative to the category X:

( A l ) If A, B E ObX, and f E X(A, B), then f : A + B is an axiom. (A21 If A E FL(X) and A e ObX, then 1(A): A + A is an axiom.

(A3 ~f

A, B, c E F ~ ( x ) , then a ( ~B,, C ) : A U(B axiom. 219

c)+( A ~ B )cXis an

220

THE LABELLED DEDUCTIVE SYSTEM

d(x)

fA.3

(A4) If A, B, C E Ft(X), then cu-'(A, B, C ) : (Am B) m C + Am (B m C ) is an axiom. (AS) If A, B E FL(X), then a ( A , B ) : A m B + B m A is an axiom. (A6) If A E FL(X), then h(A) :I N A + A is an axiom.

(A7) If A E FL(X), then h-'(A) : A + I m A is an axiom. (A8) If A E FL(X), then p(A) : A m I + A is an axiom.

(As) If A E Ft(X), then p-'(A) : A +

A m 1 is an axiom.

(AlO) If A E Ft(X), then T(A) : A + T is an axiom. ( A l l ) If A E FL(X), then T*(A) : I+ A is an axiom. (A12) If A, B E FL(X), then TA(A,B ) : A

A

B + A is an axiom.

(A13) If A, B E FL(X), then .rr,(A, B) : A

A

B + B is an axiom.

(A14) If A, B E FL(X), then d ( A , B) : A + A v B is an axiom. (Al5) If A, B E FL(X), then TZ(A, B) : B + A v B is an axiom.

(A16) There are no other axioms. A.2.1. REMARK.The restriction in (A2) to formulas A E ObX ensures the required uniqueness of the identity arrows of the categories constructed by means of &X). A.3. The rules of inference of d(X)

(R1)

f:A+B g:B+C comp(g, f ) : A + C

f:A+B g:C+D (R2) f n g : A m C + B m D (R4)

f:A+C

g:B+C [f, g ] : A v B --* C

f:A+B g:C+D (R6) a(f,g ) : A m (B J C ) + D

f:A+B (R3) (R')

g:A+C (f,g):A + B A C

f:AAB+D g:AAC+D ( f , g ) : A A ( B v C)+D f:AmB+C (R7) a ~ ( f B) :+ A J C

A.41

THE CLASS

f:A+B g:C+D (R') E p ( f , g ) : ( A D ) n C + B

+

(Rlo)

22 1

f:AnB+C (R9) a , ( f ) : A + C + B

f:A+B g:C+D g ) :A A ( B C ) + D

+

EA(f,

Der(b(X))

(R")

f:AAB+C a,i(f):B+A+C

A.3.1. REMARK.The use of identical labels in the conclusions of Rules (R6) and (RlO), respectively (R7) and (Rl l), is convenient and harmless since neither pair of rules is required simultaneously in practice. A.4. The class Der(d(X))

Relative to the category X determining the axioms of &X), the class Der(d(X)), whose elements are the derivations of d(X), is defined inductively. As usual, we represent derivations by configurations from which the underlying trees and assignments of values to the nodes are clear.

(Dl) If f E X(A, B ) , then f : A + B E Der(d(X)). (D2) If AE ObX, then 1(A) : A + A E Der(d(X)).

(D3) a ( A , B,C) : A n ( B n C ) + ( A n B)n C E Der(d(X)).

(D4) a - ' ( A ,B, C ) : ( An B ) n C + A n ( B n C) E Der(d(X)). (D5) a ( A , B ) : A n B + B n A E Der(d(X)). (D6) h ( A ) : In A + A E Der(d(X)).

(D7) h - ' ( A ): A + I n A E Der(d(X)). (D8) p ( A ) : A NI+A E Der(&X)).

(D9) p - ' ( A ): A + A n I E Der(d(X)). (DlO) T ( A ): A + T E Der(d(X)). (Dl 1 ) T * ( A ): I+ A E Der(d(X)). (D12) r , i ( A ,B) : A

A

B + A E Der(d(X)).

(D13) r , ( A , B) : A

A

B + B E Der(d(X)).

222

THE LABELLED DEDUCTIVE SYSTEM

(814) vX(A, B ) : A + A v B

i\(x)

E Der(&X)).

(615) wZ(A, B ) : B + A v B E Der(&X)). P

(816) I f f : A< B and g : B+ C E Der(d(X)), then so does P 4 f:A+B g:B+C comp(g, f ) : A+ C

'

4

(617) Iff : A 5 B and g : C+ D E Der(&X)), then so does P 4 f : A + B g:C+D f n g :AnC+ BnD'

4 (018) JI f : A$ B and g : A+ C E Der(&X)), then so does P 4 f : A + B g:A+C (f,g) : A + B A C

*

4

(019) Iff : A 2 C and g : B+ C E Der(&X)), then so does P 4 f : A + C g:B+C [f,g]:AvB+C .

P

4

(620) I f f : A A B+ D and g : A A C+ D E Der(d(X)), then so does 4 f : A A B z D g:AAC+D ' (f,g ) : A A ( B v C ) + D

(D21) I f f : A+P B and

4

g : C+ D E Der(&X)), then so does

P

(622) I f f : A )x B+ C E Der(d(X)), then so does P

f : AnB+C

[A.4

A.41

THE CLASS

P

Der(d(X))

9

(823) If f : A + B and g : C+ D E Der(d(X)), then so does P f:A+B &f, g ) : ( A

4 g:C+D D)n C + B'

P

(D24) If f : A n B+ C E Der(&X)), then so does

f : A n B+P C ap(f) :A+ C B '

+

P 4 (D25) If f : A + B and g : C+ D E Der(d(X)), then so does P 4 f:A+B g:C+D eA(f, g) : A A ( B 3 C ) + D'

(D26) If f : A

A

B Z C E Der(d(X)), then so does P f:AhB+C aA(f): B + A + C '

(827) There are no other derivations of d(X).

223

APPENDIX B

THE UNLABELLED DEDUCTIVE SYSTEM A(X)

B.l. The axioms of A(X)

Relative to a fixed, but arbitrary small category X, the axioms of A(X) are defined inductively: ( A l ) Zf A, B E ObX and X(A, B) is non-empty, then the sequent A + B is an axiom. (A2) The sequent +I is an axiom. (A3) The sequent + T is an axiom. (A4) The sequent l+ is an axiom. (AS) There are no other axioms. B.2. The rules of inference of A(X) B.2.1. Structural rules

B.2.2. Operational rules

B.31

THE CLASS

Der(A(X))

225

B.2.3. REMARK.The restrictions to single formulas in the left, respectively right, premisses in (R14) and (R16) are proof-theoretically unnecessary, but facilitate the categorical interpretation of these rules and make the definition of normality in Appendix D somewhat easier. The corresponding restrictions in (R15) and (R17), however, are required in order to make the system A(X) intuitionistic. B.3. The class Der(A(X))

Relative to the category X determining the axioms of A(X), the class Der(A(X)), whose elements are the derivations of A(X), is defined inductively. As usual, we represent derivations by configurations from which the underlying trees and assignments of values to the nodes are clear.

f

(Dl) If f E X(A, B),then A + B E Der(A(X)). (D2) If a E ObX, then a --+ a E Der(A(X)). (D3)

+ I E Der(A(X)).

(D4)

+ T E Der(A(X)).

(D5) I-,E Der(A(X)). (D6) If

r+f @ y q and A y A s 0 E Der(A(X)), then f

g

AyA+0 AFA-+@09

r+@yq

'

so does

THE UNLABELLED DEDUCTIVE SYSTEM

226

(D7) If

rA+f

Q, E Der(A(X)), then

so does

rA+f

raA+

f

(D8) If raaA+

Q, E

A@)

Q, Q,'

Der(A(X)), then so does

f

raaA+ raA+

Q, Q,'

f

(D9) If rapA --* Q, E Der(A(X)), then so does

f

ra/3A+

rpaA+

(D10) If

Q, Q,'

r+f Q,Q E Der(A(X)), then so does f Q,Q r-+ r+ saw

(D11) If

r+f @aaQE Der(A(X)), then so does r+f OaaQ r+cpao .

(D12) If

r+f QapY EDer(A(X)), then so does f ~WBQ r+ r+ @paw*

(D13) If

f r+a

g

and A+p EDer(A(X)), then so does

f

F+a

g A+p

I'A+crnp (D14) If

rapA+f

Q, E Der(A(X)), then

'

so does

[B.3

THE CLASS Der(A(X))

B.31

r+aaApq

f

(D16) If rapA+ @ E Der(A(X)),then so does

rapA-,f @ raApA+@.'

f

(D17) If raA+ @ and

rpAZ @ E Der(A(X)),then so does f

raA+@ I'pAZ@ ra v P A + @

(D18) If

r+f @apq€Der(A(X)),then so does r+f oapq r+aaVpq\II'

f

g

(D19) If r + a and A p A + @€Der(A(X)),then so does

r +f a A B A Z ~ Ara+pA+@

.

f

(D20) If T a A + p E Der(A(X)),then so does

f

~

f

(D21) If A p A +

and

TaA+ B I'A+a+p'

r+g a E Der(A(X)),then so does f

MA+@

g r-ta

227

228

(D22) If

THE UNLABELLED DEDUCTIVE SYSTEM

rcu+f /3 E Der(A(X)),then so does f r+pecr*

Ta+ 6

(D23) There are no other derivations of A(X).

h(X)

tB.3

APPENDIX C

THE CUT ELIMINATION ALGORITHM

In this appendix, we describe a partial algorithm on Der(A(X)) which proves that for every f E Der(A(X)) which represents an arrow of one of the categories defined in Chapters 2-12, and which contains an instance of the cut rule (Rl), there exists a cut-free derivation g E Der(A(X)) deriving the same sequent. The algorithm is a generalization of the cut elimination process first introduced by Gentzen (cf. SZABO[1969] pp. 88-103). We enunciate the algorithm in terms of the transitive, monotone (i.e., if f > g , and k results from h by the replacement of the subderivation f by g, then h > k) relation > on Der(A(X)) generated by the following conditions: (Al, A l ) For all A, B, C E ObX, all f E X(A, B), and all g E X(B, C),

f

g

A+B B+C >A-C. A+C

comp(g, f )

(C.1)

(Al, R2) For all A, B E ObX, and all f E X(A, B),

f

I'AZ @

A+B I ' B A + @ > I'AZ@ I'AA+@ rAA+@

(C.2.1)

(C.2.2)

(C.2.3) 229

230

(R10, R2)

APPENDIX C

h AA+ 0

(R13, R2)

(R15, R2)

(R17, R2)

(C.7.1)

THE CUT ELIMINATION ALGORITHM

23 1

r+f (C.7.2)

(RS, R l l )

(C.6)

(RS, R12)

(RS, R14)

232

APPENDIX C

(RS, R16) n

r+f QVI

(C.11)

(C. 12)

(R10, R l l )

(C.14)

'THE C U T E L I M I NAT I O N ALGORITHM

233

(R15, R14)

(C.16) (R17, R16)

(C.17)

(C.18.l)

(C. 18.2)

(C. 19.1)

(C.19.2)

234

APPENDIX C

(C .20.3)

(C .20.4)

THE CUT ELIMINATION ALGORITHM

235

236

APPENDIX C

T H E C U T E L I M I NAT I O N ALGORITHM

137

238

APPENDIX C

(C.31.2)

(C.33)

(C.35)

T H E CUT ELIMINATION ALGORITHM

239

(C.37.1)

(C .37.2)

(C .38.2)

ArA+ @OVr

(C.38.3)

240

APPENDIX C

(C.39.3)

(C.39.4)

THE

cur

ELIMINATION AL GOR ITHM

24 1

242

APPENDIX C

A ~ A Zr fjCa ~ ~h ~

g h ApA+ @ y q E y n + 0 Ap+arA+@yq SyrI+@ EApAII+@@V r fj a > ~A~+~~AII-+@WP EA~+~~AII+~CYP (C.46) This completes the description of >. In order to be able to prove inductively that the relation > has the desired algorithmic properties, we require two measures of complexity, called degree and rank, which associate with each cut-free derivation f and each derivation g of the form

r+h

k

A r A + @@q

o

suitable natural numbers. The following two partial functions deg, rnk : Der(A(X))+ w , defined relative to f and g, are adequate for this purpose: (i) degCf) = 0. (ii) deg(g) = deg(y ) + 1. (iii) rnkCf) = 1 . (iv) rnk(g) = mkr(g) ( m + 1) + rnk,(g) (n + 1). (v) deg(y) = the total number of occurrences of the symbols N, A , v, and in y. (vi) rnkA(g)=the length of the longest branch of h ending with r + @ y q whose elements contain the cut formula y in the succedent. We call rnkr(g) the left rank of g. (vii) rnk,(g)= the length of the longest branch of k ending with AyA+ 0 whose elements contain the cut formula y in the antecedent. We call mkp(g) the right rank of g.

-

+,

+

THE

cur

E L I M I N A T I O N ALGORITHM

243

(viii) m = the number of consecutive applications of (R6) with which the derivation h ends. (ix) n = the number of consecutive applications of (R3) with which the derivation k ends. The pairs ( p , q ) in Conditions (C.1-46) are classified in such a way that in (C.1-17), rnk(p) = 2, in (C.18-33), rnk,(p)> 1, and in (C.34-46), rnk,(p)= 1 and r n k A ( p ) >1. An argument by cases shows that an induction on rank proves that > indeed determines the desired algorithm. The induction basis is established by a separate induction on degree. In each case, the instance or instances of (RI) in q has or have either lower rank or lower degree than the instance of ( R l ) in p. Gentzen’s original proof of the cut elimination theorem avoids the difficulties in the induction on rank caused by (R3) and (R6). Rule ( R l ) is replaced by a more general rule, called the mix, in which consecutive applications of (R3) and (R6) are collapsed into a single step. In the present context, this device is not available since the relation > is intended to be stable under various categorical interpretations of Der(A(X)). Hence Gentzen’s notion of rank must be replaced by that of weighted rank which counts the number of applications of (R3) and (R6) with which the premisses of an instance of (Rl) terminate. It is clear that any derivation with several cuts can be reduced to a derivation without cuts by applying the described algorithm to each cut separately from the top of the derivation down.

APPENDIX D

THE NORMALIZATION ALGORITHM

In this appendix, we extend the relation > of Appendix C to a global reducibility relation L which determines the choice and order of application of Rules (R2)-(R17) in cases of syntactic ambiguity. The relation L is defined on three subclasses of Der(A(X)) whose elements represent the arrows of (a) non-Cartesian, non-symmetric, (b) nonCartesian, symmetric, and (c) Cartesian categories, respectively. The adopted priorities may be summarized as follows: (a)

(R8) 5 (R14) 5 (R9) 5 (R2), (R15) I(R14) I(R9) I(R2), (R17) I(R14) I(R9) I(R2).

(b)

(R8) I(R14) I(R9) I(R2) I(R4), (R15) I(R14) I(R9) I(R2) I(R4).

(c) (R5) 5 (R13) 5 (R10) I(R12) 5 (R14) I(R3) I( R l l ) 5 (R2) 5 (R4), (R15) I(R12) I(R14) I(R3) 5 (R11) I(R2) I(R4), where I is transitive, and (Ri) and satisfying the following further conditions:

(R2-R2) f

I

244

T H E N O R M A L I Z A T I O N AI.GOR1THM

245

(R2-R3)

(D.2.1)

(D.2.2) h TaA+ @ (D.2.3)

(D.2.4)

(R2-R4)

(D.3.1)

(D.3.2)

(D.3.3)

(D.3.4)

246

APPENDIX D

(R2-R5

(R2-R8)

(D.5.1)

(D.5.2) (R2-R9)

(D.6.1)

(D.6.2) (R2-R 1 0)

T H E NORMAL.IZATION AI-GORITHM

247

(R2-Rll)

(D.8.1)

(R2-R 12)

(R2-R 13)

(D. 10)

248

APPENDIX D

(D.12.1)

(D. 12.2) (R2-R 16)

(D.13.1)

(D.13.2)

(R2-R 17)

(D.14.1)

provided that in (D.14.2),I is the only atomic subformula of a.

T H E NORMALIZATION ALGORITHM

249

(D.15.2) where n L 3, A = ( Y I . * an, ai = a ( 1 5 i 5 n ) , and (a)and ( 7 ) denote n - 1 instances of (R3), with ai active before aj in ( 7 ) if i < j.

(D.16.1)

(D.16.2)

(D.16.3) (R3-R5)

(R3-R10)

250

APPENDIX D

(R3-Rll)

(D. 19.1)

(D. 19.2)

(D. 19.3)

(R3-R 12)

25 1

T H E N O R M A L I Z A T I O N ALGORIT HM

(R3-R14)

(D.22.1)

(D.22.2)

(D.22.3)

(R3-Rl5)

(D.23.1)

(D.23.2) (R4-R4) For any permutation r of the integers 1 , . successions (cr) and ( 7 ) of instances of (R4),

r +f @

A+@

. . , n , and

any

(D.24.1)

provided that T = a1 * * an,A = PI * . . p., pi = aa(i)for 1 5 i 5 n, and ( 7 ) is the unique string of interchanges which first moves a,,(~) to PI, then a,(~)to 0 2 , etc. If r is the identity permutation, the right-hand side denotes f.

252

APPENDIX D

If I is the only atomic subformula of a,or if a = T, then

and

where g and k are the unique cut-free derivations obtained by applying the cut elimination algorithm of Appendix C to the derivations

where m is the derivation of -+ (Y described in 8.6.2, 9.6.2, and 10.6.4, and where ((T)involves no instances of (Rl) and (R4), and contains at most instances of (R8) or (R10) whose premisses have empty antecedents, and at most instances of (R14) whose left premisses have empty antecedents.

(R4-RS)

(D.25) (R4-R8)

(D.26.1)

(D.26.2)

T H E NORMALIZATION ALGORITHM

253

(R4-R9)

(D.27.1)

( D. 27.2)

(D.27.3)

(D.27.4)

(R4-RlO)

(R4-Rll)

(D.29.1)

254

APPENDIX D

(D.29.2)

(D.29.3)

(D.29.4)

(R4-R12)

(R4-R13)

(D.31)

T H E NORMALIZATION ALGORITHM

255

(D.32.4)

(D.32.5) (R4R15) (D.33.1)

(D.33.2)

(D.33.3)

256

APPENDlX D

(D.33.4) (R5-R5)

(D.34)

(R5-Rl0)

(D.35.2)

(D.35.3)

(D.35.4)

(D.35.5)

T H E NORMALIZATION ALGORITHM

257

(D.35.9)

258

APPENDIX D

(R5-R 1 1 )

(D.36)

(R5-RI2)

(R5-Rl3)

(D.38.1)

(D.38.2) (R5-RI4)

(D.39)

T H E NORMALIZATION ALGORITHM

259

(D.40.1)

n

(D.40.2)

(R8-R 14)

(R8-R 16)

(R9-R9)

(D.43)

260

APPENDIX D

(R9-R 14)

(R9-R 15)

(D.45.1)

(R9-R 16)

T H E NORMALIZATION ALGORIT HM

26 I

(R9-R 17)

(D.47) (R 10-R 10) c

262

APPENDIX D

(D.51.2)

(R 10-R 14)

(D.52)

(R1I-R11)

(D.53) (R11-R12)

T H E NORMALIZATION ALGORITHM

263

(R11-RI3)

(D.55) (Rll-R14)

(R11-Rl5)

264

APPENDIX D

(D.57.2)

THE NORMALIZATION ALGORITHM

265

266

APPENDIX D

(R14-R 15)

(D.65.3)

(D.65.4)

(D.65.5)

T H E NORMALIZATION ALGORITHM

-2

261

(D.65.7)

(R14-R 16)

(R14-R17)

(D.67) (R15-R 16)

268

APPENDIX D

(R16-R 16)

(R 16-R 17)

(D.70) (R2-R2-R 10)

(R2-R2-R 12)

I'HE N O K M A I I 7 A T I O N A I . ( i O R I T H M

(R2-R3-R 10)

(R2-R3-R 12)

269

270

(R2-R4-R 10)

(R2-R4-R 12)

APPENDIX D

T H E NORMALIZATION ALGORITH M

27 I

(D.76.9)

(D.76.10)

(D.76.I I )

(D.76.12)

(D.76.13)

(D.76.14)

(D.76.15)

(D.76.16)

THE NOKMAI 1 / 4 1 1 0 N

(R3-R3-R 10)

(R3-R3-R 12)

A I CnOKIIHM

273

214

APPENDIX D

(R4-R4-R 12)

n

r S y A a v PA-+ @

(D.80.

(D.81.1)

(D.8 1.2)

T H E NORMALIZATION ALGORITHM

rrAajph+ rra 3 P A A + r a 4 rra+pA+

Tct+PA-f2 If a

= p, and

ra+$A+

if I is the only atomic subformula of a,or if a

275

(D.81.3)

= T, then (D.81.4)

t

(D.82.2)

-

P

(D.82.4)

(R8) If I is the only atomic subformula of a and /? and of the terms of r and A, then r m n +a &(,) r+a A+P, - + a n p (D.83) TA+auP rA+anP

276

APPENDIX D

with r, s, and (a)involving at most instances of (R8), resp. (R14), whose premisses, resp. left premisses, have empty antecedents, as described in Lemma 8.6.2.

(R14) If I is the only atomic subformula of a and p and of the terms of A, and A, then

r,

(D.84) with r, s,(a), and (7) involving at most instances of (R8), resp. (R14), whose premisses, resp. left premisses, have empty antecedents, as described in Lemma 8.6.2.

(R15) If I is the only subformula of a,or if a =T, then

(D.85) where g is the unique cut-free derivation obtained by applying the cut elimination algorithm of Appendix C to the derivation

where h is the derivation of + a described in 8.6.2,9.6.2,and 10.6.4, and where (a)involves no instances of (Rl) and (R4), and contains at most instances of (R8) or (R10) whose premisses have empty antecedents, and at most instances of (R14) whose left premisses have empty antecedents. This completes the description of the global defining conditions of 2 . All additional local requirements ensuring the uniqueness of representation of unique arrows are listed in the main body of the text. The effectiveness of 2 requires that the semantic condition in (D.12.3), (D.12.4), (D.24.2), (D.81.4). and (D.85) is given a syntactic

T H E NORMALIZATION ALGORITHM

277

characterization. Corollaries 4.6.3, 5.6.6, 6.6.6, and Lemmas 9.6.3 and 10.6.2 achieve this purpose. We define a derivation f to be normal if it is cut-free and if f 2 g implies that f = g . We say that f reduces to g if there exists a finite sequence ( f l , . . . f n > of derivations of A(X) such that f = f l , fn = g, and f l 2 f z 2 . . . 2 fn, and say that f reduces immediately to g , written as f % g , if f = g or if f 2 g by virtue of precisely one of the defining conditions of 2 . An argument by cases shows that if f % g and f % h, then there exists a derivation p such that g 8 p and h ~ p and , an induction on n + m extends this result to show that if f % f l % . . . % fn and f % g l % . * * + gm, there exists a derivation q with the property that fn 2 q and gm 2 q. Hence any derivation fEDer(A(X)) reduces to at most one normal derivation g . Since the relation > is contained in 2 , every fEDer(A(X)) which represents an arrow of one of the categories constructed in Chapters 2-12, reduces to a cut-free derivation g , and an induction on the number of violations of Conditions (D.l-85) in g shows that f reduces to a normal, and hence unique normal, derivation h. The relation 2 thus defines an algorithm for normalizing any f E Der(A(X)) belonging to one of the subclasses Der(xA(X)) of Der(A(X)) mentioned at the beginning of this appendix. Let = be the equivalence relation on Der(A(X)) generated by 2 . Then an induction on the length of the proof that f - g shows that for all f, g E Der(A(X)) representing an arrow of one of the categories constructed in Chapters 2-12, there exists a derivation h such that f 2 h and g 2 h. This property is called the Church-Rosser property of 2 . The proofs of the normalization theorems for the various subsystems xA(X) of A(X) in the main body of the text consist of the appropriate verifications that the relation z Xis contained in where = x is the equivalence relation on Der(xA(X)) induced by the interpretation S : Der(xA(X))+ ArFx(X), and where Z x is the reducibility relation on Der(xA(X)) generated by the local refinement of the restriction of 2 to Der(xA(X)). The proofs of the Church-Rosser theorems for the various subsystems xA(X) of A(X) in the body of the text, on the other hand, consist of the appropriate verifications that distinct normal derivations f and g of a sequent r + @represent distinct arrows of Fx(X), i.e., that f = * g iff f - x g , where z Xis the equivalence relation on Der(xA(X)) generated by zx.

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[1976b] An addendum to my paper “A categorical equivalence of proofs”, Notre Dame Journal of Formal Logic 17, 78. [I9771 The logic of closed categories, Notre Dame Journal of Formal Logic 18, 441-457. TAIT, W. W. [1%8] Normal derivability in classical logic, in: BARWISE [1%8] pp. 204-236. TAYLOR,A. E. [1958] Functional Analysis (Wiley, New York). VOLGER,H. [I9711 Logical categories, Ph.D. thesis, Dalhousie, Halifax. [1975a] Completeness theorem for logical categories, in: LAWVEREet al. [I9751 pp. 5 1-86. [1975b] Logical categories, semantical categories and topoi, in: LAWVEREet al. [I9751 pp. 87-100. VOREADOU [1972] A coherence theorem for closed categories, Ph.D. thesis, Chicago. 119741 Some remarks on the subject of coherence, Cahiers de Topologie et Gbmetrie Differentielle 15, 399-417. [ 19751 Non-commutative diagrams in closed categories, Manuscript. [ 19771 Coherence and non-commutative diagrams in closed categories Memoirs of the American Mathematical Society 9, Number 182. WRAITH,G. C. [I9751 Lectures on elementary topoi, in: LAWVEREet al. [1975] pp. 14-206.

INDEX OF SYMBOLS

286

INDEX OF SYMBOLS

Axioms

17, 19, 24, 33,41,55,72,95, 108, 130, 152, 167, 181, 219 (A2) 17, 19, 24, 33,41, 55.72.95, 108, 130, 152, 167, 181, 206. 219 ( i 3 ) 17, 19, 24, 33, 9.5, 108, 167, 181, 219 (A4) 17, 19, 24, 33, 95, 108, 167, 181, 220 ( i 5 ) 17, 19, 33, 108, 220 (A6) 17, 19, 24, 33,95, 108, 181, 220 ( i 7 ) 17, 19, 24, 33, 95, 108, 181, 220 (A8) 17, 19, 24, 33, 95, 108, 181, 220 (h) 17, 19, 24, 33, 95, 108, 181, 220 (i10) 17, 19.41, 55.72, 130, 152, 206,219 (A1 I ) 17, 19, 5 5 , 72, 152, 206, 220 (A12) 17, 19, 41, 5 5 , 72, 130, 152, 206, 220 (i13) 17, 19, 41, 5 5 , 72, 130, 152, 206, 220 (i.14) 17, 19, 5 5 , 72. 152, 206, 220 ( i 1 5 ) 17, 19, 5 5 , 72, 152, 206, 220 ( i 1 7 ) 19, 207 ( i 1 8 ) 19, 207 (Al) 17, 19.26, 33,43,60,73, 96, 109, 131, 153, 168, 182, 204, 225 (A21 17, 19, 20, 26, 33, 96, 109, 182, 224 (A3) 17, 19,20,43,60,67,73,86, 131, 1.53, 159, 204, 224 (A4) 17, 19, 20, 60, 67, 73, 86, 153. 160, 204,224 (C1) 4 (C2) .5 (C3) 5 (F1) 6 (F2) 6 (F3) 6 (F4) 6 (F5) 15 (MI) 21, 164 (M2) 21, 164 (M3) 22, 164 (M4) 31 (M5) 31 (M6) 31 (il)

Categories P 6

7 Funct(C,D) 7 hom(P,Q) 7 P P 7 Cat 8, 23, 71, 72 CxD 8 Ens 8, 23, 66,71, 92, 93, 190, 197 Mon 11 prOrd I 1 Ens, 23, 71, 93, 129 Fm(X) 23 mCat 23 RMOdR 23, 32, 94, 107, 108, 164, 179 Fsm(X) 32 KMod 32 ModK 32 smCat 32 cCat 40 Fc(X) 40 bcCat 55 Fbc(X) 55 comRng 66 Ab 71,92 dbcCat 71 Fdbc(X) 71 L 92 Fmcl(X) 94 kTop 94, 202 mclcat 94 Fsmcl(X) 108 smclCat 108 V 116 B 119 cclCat 129 Fccl(X) 129 Funct(CoP,Ens) 129, 189, 199 H 149, 196 bcclCat 152 E 152, 189 Fbccl(X) 152 2’ 165, 180 cop

INDEX OF SYMBOLS

Fr(X) 166 rCat 166 Frnbcl(X) 181 rnbclCat 181 I 191. 192, 200, 203 Sub(A) 195 E(A,R) 196 Sh(C) 197. 199 Open(X) 197, 214 Funct(1, C ) 200 w 201 qcCat 206 Fqc(X) 207 Fqc(AtL*(X)) 207 Classes ObC 4 Arc 4 ArCx.,, Arc 4 C(A,B) 5 [ABI 5 0 (empty class) 5 I*} (singleton class) 5 horn(P,Q) 7 Funct(C,D) 7 Nat(C(A, -), F) 9, 129 w (finite ordinals) 12 A r c Xc,.v)ArC 12 ArCx,,,ArC 12 TL' 15,202 L*(X) 15, 204 L(X) 16, 20 FL(X) 16, 20 SeqdX) 16 LbSeq,(X) 16, 17 Lb(i\(X)) 17 N 17 N* 18 B: 18 Der(d(X)) 19,221 Der(A(X)) 19, 20, 243, 244 M(FL(X,, 20 Der(mA(X)) 23

Der(rnA(X)) 26 Der(sm&X)) 33 Der(srnA(X)) 34 Der(cd(X)) 41 Der(cA(X)) 44 Der(bc&X)) 55 Der(bcA(X)) 60 Der(dbci\(X)) 72 Der(dbcA(X)) 74 Der(rncli\(X)) 95 Der(mclA(X)) 97 Der(srnclA(X)) 108 Der(smdA(X)) 109 Der(ccl&X)) 130 Der(cclA(X)) 132 L(T) 135, 159 Der(bccl&X)) 153 Der(bcclA(X)) 157. 217 L(1) 159 Der(r&X)) 167 Der(rA(X)) 169 Der(mbcl&X)) 181 Der(rnbclA(X)) 183 Mon(A) 194 Sub(A) 194 E(A,R) 195, 196 Cov(A) 198 BV 202 EQ 202 FL.(X) 202-204 F " 202 F V 202, 213 IC 202 L*(X) 202-204 Ro 202 R" 202 sc 202 SeqL*(X) 202 UQ 202 AtFL.(X) 203 Fin FL*(X) 203 FL*(X) 203-204 FinL*(X) 204, 218 Seq,*(X) 204 Der(A*(X)) 206, 207, 211-213, 217

287

288

INDEX OF SYMBOLS

Der(h*(X)) 207 Ord 211 Lb(&X)) 219 Der(A(X)) 221 Der(A(X)) 225,229 Deductive systems

&X) 17, 19, 219-223 A(X) 17, 19, 22&228 m&X) 24 mA(X) 26 sm&X) 33 smA(X) 33-34 cA(X) 41 cA(X) 43,57 bc&X) 55 bcA(X) 57.60 dbc&X) 71-72 dbcA(X) 73-74 mcI&x) 95 mclA(X) 96-97 smcl&X) 108 smclA(X) 109 ccl&X) 130 cclA(X) 131 bccli\(X) 152 bcclA(X) 153-154, 156, 204 r&X) 166-167 rA(X) 168-169 mbclb(X) 181 mbclA(X) 182 6*(X) 206-207 Formulas

Functions dom 4 , 5 cod 4 , 5 comp 4, 5 F. 6 Fa 6 Y 9 subst 12 subst(p, u ) 13 T 18, 205 f, 18-19, 205-206 S(m) 26-27

(i) (binomial coefficient)

30, 118, 127,

177, 187 S(sm) 26-27, 34 n ! (factorial) 39, 52, 121, 127 S(c) 44-45 S(bc) 44-45,61 S(dbc) 44-45, 61, 76-77 S(mcl) 26-27, 98 S(smc1) 26-27, 98, 109-110 N ( p , q ) 117, 121, 173, 176, 177, 186, 187 r 121 Il(C,D)lI 127, 177, 187 S(ccl) 44-45, 132-133 S(bccl) 44-45, 61, 76-77, 132-133 S(r) 26-27,98, 170-171 S(mbcl) 26-27, 98, 170-171 k 103 con 195 dis 195 f 195 imp 195 t 195 S(L*(X)) 208

INDEX O F SYMBOLS

s ( q c ) 4 4 4 5 , 61, 76-77, 132-133, 208-211 deg 21 I , 212, 242 rnk 211, 212, 242 rnk, 212, 242 rnk, 212, 242 Functors

Const B 6

C(A,-) 8 C(-. A ) 8

F (pre-order) 11 M (monoid) I 1 F (free object) 12 U (forgetful) 12 (-)N(-) 21, 163 Um 23 Fm 23-25 Fm(H) 25 I-1 (m) 29 Usm 32 Fsm 32-33 Fsm(H) 33 U-1 (sm) 38 (-) A (-) 40 Uc 41 Fc 41-43 F c ( H ) 42-43 1-1(c) 52 Ubc 55 Fbc 55-57 Fbc(H) 42-43, 56 1-1(bc) 69 (-) v (-) 70 Udbc 71 Fdbc 71-73 Fdbc(H) 42-43.56.73 I-1 (dbc) 91 (-)*(-) 93, 128, 145, 163 Umcl 95 Fmcl 95-96 Fmcl(H) 25, 96 A J ( A M(-)) 97 A * ( A % - ) ) 97 8-1 (mcl) 106 Usmcl 108

Fsmcl 108-109 Fsmcl(H) 25, 33.96, 109 i-1 (smcl) 126 Uccl 130 Fccl 130-131 Fccl(H) 42-43, 96, 131 (ccl) 144 Ubccl 152 Fbccl 152-153 FbccKH) 42-43, 56, 96, 131, 153 [-I (bccl) 162 (-)H-)163 Ur 166 Fr 166-168 F r ( H ) 25, 96, 168 II-1 (r) 177 Umbcl 181 Fmbcl 181-182 FmbcKH) 25, 96, 168, 181-182 (mbcl) 187 F(1) 191 F ( I x J ) 191 Sub(-) 195 E(-,n) 195 F (sheaf) 197-198 v 200,201 3 200,201 u q c 206 Fqc 206-207 Fqc(H) 42-43, 56.96, 131, 153, 207 !-I (qc) 21 1, 218

u-11

u-n

Languages L*(X) 15, 213 mL(X) 24 smL(X) 33 cL(X) 41 bcL(X) 55 dbcL(X) 71 mclL(X) 95 smclL(X) 108 cclL(X) 130 bcclL(X) 152 rL(X) 166 mbclL(X) 95

289

290

INDEX OF SYMBOLS

FinL*(X) 218

L(X) 219

Natural transformations comp(p.u) 7

1(F) 7 u 9

a(-,-)10 q 10, 97-98, 107 10, 97-98, 107, 128 a 21, 22, 32,44, 163 A 21, 22, 32, 107, 178 p 21, 22, 32, 107, 178 u 31, 32, 52, 107 a, 40, 128 a, 40 a, 54, 70

a. 54, 70 70 aA 93, 128, 145, 164 (A,V,+) 150 ap 164 4 169-170 Z 169-170 ( L A ) 191 ( K , K ) 191 a v 201 a3 201 a8

Rules of inference

(R1) 17, 19, 23, 33,41, 55,72,95, 108, 130,

152, 167, 181 (R2) 17, 19, 24, 33, 95, 108, 167, 181 (R3) 17, 19, 41, 5 5 , 72, 130, 152 (R4) 17, 19, 55, 152 (R5) 17, 19, 72 (R6) 17, 19, 95, 108, 167, 181 (R7) 17, 19, 95, 108, 167, 181 (R8) 17, 19, 167, 181 (R9) 17, 19, 167, 181

( R l O ) 17, 19, 130, 152 ( f i l l ) 17, 19, 130, 152 (R12) 19, 207 (R13) 19, 207 (RI) 17, 19, 26, 27, 33, 43, 60, 73, 76-77, 96, 109, 131, 153, 156, 168, 182, 204, 213, 217, 224, 229 (R2) 19, 20, 26, 27, 33, 43, 44-45, 49, 60, 66,67, 73, 79, 85, 86, 96, 109, 131, 136, 154, 182, 204, 224 (R3) 17, 19, 20, 43, 44-45, 49, 60, 66, 73, 85, 131, 136, 154, 160, 204, 212. 224 (R4) 17, 19, 20, 33, 109, 113, 131, 132-133, 136, 154, 160, 204, 225 (R5) 17, 19, 20, 60, 61, 66, 67, 73, 85, 86, 154, 159, 160, 204, 224 (R6) 17, 19, 20,60,61,66,73,85, 154, 160, 204, 212, 224 (R7) 17, 19, 20, 154, 156, 224 (R8) 17, 19, 20, 26, 27, 34, 96, 109, 168, 182, 224 (R9) 17, 19, 20, 26, 27, 34, 96, 109, 168, 182, 224 (R10) 17, 19, 20, 43, 60, 67, 73, 74, 76-77, 86, 131, 154, 159, 205, 225 (RII) 17, 19, 20,43, 44-45, 67, 74, 86, 131, 154, 160, 204, 225 (R12) 17, 19, 20, 60, 67, 73, 74, 86, 154, 160, 204, 225 (R13) 17, 19, 20, 60, 61, 67, 74, 86, 154, 159, 204, 225 (R14) 17, 19, 20, 97, 98, 109, 131, 132-133, 136, 154, 156, 160, 168, 182, 204, 225 (R15) 17, 19, 20,97, 98, 109, 131, 132-133, 154, 156, 159, 169, 182, 204, 225 (R16) 17, 19, 20, 169, 170-171, 225 (R17) 17, 19, 20, 169, 170-171, 225 (R18) 19, 204-205, 213, 214 (R19) 19, 205 (R20) 19, 205 (R21) 19, 205, 213 (R22) 19, 205 (R23) 19, 205 (R24) 19, 205 (R25) 19, 205

INDEX OF SUBJECTS

Active formula, 20 Adjoint functor, 10 Adjunction, 10. 97 10.97 Counit of an Unit of an -, 10. 97 Admissible (rule of inference). 19 Algebra Boolean -, 148 Heyting -, 148 20 I Quantifier Algorithm Cut elimination -, 20. 229-243 Normalization -, 244-277 And, 2 Antecedent, 17 -symbol, 20 Application (of a rule of inference), Arrow, 4, 12 De Morgan -, 147-151 Finitarily representable -, 21 I Identity -, 4, 5 , 13, 20, 220 Arrowgram, 5 Atomic formula, 16, 203 Axiom, 19, 206-207.219-220, 224

Branch - of a tree, 18, 205 Length of a -, 18 Maximal -, 18

-.

Cartesian - category, 40 - closed -, 128 Category, I . 4, 5 Bicartesian -, 54 - closed -, 145 Cartesian -, 40 - closed -, 128 Closed -, 92, 107 Cocomplete -, 192 Complete -, 192 5 Discrete Distributive bicartesian -, 70, 145 Equivalent 9 I-cocomplete -, 192 I-complete -, 192 Large 5 Monoidal 20. 21, 22 - biclosed -, 178-179 - closed -, 93 - of finite sets. 199 Opposite -, 7 201 Quantifier-complete Residuated -, 163-164 Sequential -, 12-14 Simple -, 5 Small -, 5 Structured -, 1, 21, 40, 54, 70, 92, 93, 107, 127, 145, 163-164, 178-179, 190, 20 I Symmetric monoidal -, 31 - - closed -, 107

-.

-.

19

-.

-.

Bicartesian - category, 54 - closed -, 145 70. 145 Distributive Biclosed Monoidal - category, 178-179 Bifunctor, 8 Binary tree, 18 Boolean algebra, 148 w-complete -, 202 Bound variable, 202

-.

-.

-.

29 1

292

INDEX OF SUBJECTS

Church-Rosser - property, I , 3, 277 -theorem (cf. Theorem) Class. 5 Classifier Subobject 193, 195 Closed Cartesian 128 -category. 92. 107 Monoidal -, 93 Symmetric monoidal -, 107 Cocomplete category, 192 Codomain, 4 Coequalizer, 191 Coherence. 20, 22, 29, 32. 35, 36 Colimit, 191 Commutative diagram, 6 Complete - category, 192 Quantifier 189 Completeness theorem (cf. Theorem) Component (of a natural transformation), 7 Composition, 4 - of natural transformations, 7 Generalized 12 Internal -, 103 Computability theorem (cf. Theorem) Concatenation, I I Conclusion (of a rule of inference), 19 Conjunction. I , 195 -symbol, 202 Generalized -, 189. 204 Constant functor, 6 Contravariant functor. 7 Coproduct, 22, 192. 193. 200, 201 Finite -, 22, 192, 193 Counit (of an adjunction). 10, 97 Counter-examples, 32. 36, 40, 54, 71, 108, 118. 129, 139, 149, 152, 179, 214 Cut, 19 - elimination, 60 - - algorithm, 20, 229-243 - - theorem (cf. Theorem) -formula, 20

-. -.

-.

-.

Degree of a -, 211, 212 Cut-free derivation. 19. 229 Deductive system Labelled 17, 206-207. 219-223 Unlabelled -, 17, 204-205, 224-228 Degree - of a cut, 211, 212 - of a derivation, 212, 242, 243 - of a formula, 211 De Morgan arrows, 147-151 Derivable sequent, 19, 206 Derivation, 18-19, 205-206 Cut-free 19, 229 Degree of a -, 212, 242, 243 Height of a -, 19 Normal -, 29, 36, 50, 67-68, 86, 101. 114, 136-137. 160-161, 172, 184. 217218, 277 Width of a -, 19 Diagram, 6 Commutative -, 6, 7 Discrete category, 5 Disjunction, I , 195 -symbol, 202 Generalized 189, 204 Distributive bicartesian category, 70, 145 Distributivity, 57-60 Domain, 4 Dots, 35 Double negation, 156

-.

-.

-.

Eigenvariable, 213, 214 Elementary topos, 194 Embedding, 6 Faithful - (cf. Embedding) Equalizer, 191 Equivalence relation - on h r (d ( X ) ) , 41. 55. 95, 153 - - Der(A*(X)), 45, 61, 99, 157, 208-21 1 - - Der(bc&X)), 41, 55 - - Der(bcA(X)), 45, 61 - - Der(bccl&X)), 41, 55, 95, 153

293

INDEX OF SUBJECTS

Equivalence relation on (cont.) - - Der(bcclA(X)). 45. 61.99. 157 - - Der(cd(X)). 41 - - Der(cA(X)). 45 - - Der(ccl&X)). 41. 95. 130 - - Der(cclA(X)), 45. 99. 133 - - Der(dbc&X)). 41. 72 - - Der(dbcA(X)). 45, 77 - - Der(mi\(X)), 24 - - Der(mA(X)), 27 - - Der(mbcl&X)), 24. 95, 167. 181 - - Der(mbclA(X)). 27, 99. 171. 183 - - Der(mcl&X)), 24. 95 - - Der(mclA(X)), 27, 99 - - Der(r&X)), 24, 95, 167 - - Der(rA(X)), 27. 99. 171 - - Der(sm&X)), 24, 33 - - Der(smA(X)). 27. 34 - - Der(smcl&X)), 24. 33. 95, 108 - - Der(smclA(X)). 27. 34, 99, I10 - - L*(X), 208 - - Mon(A), 194 Equivalent categories, 9 Examples - of bicartesian categories. 54. 71, 149152, 189, 195-199. 201-202 - - - closed 149-152, 189. 190, 195199, 201-202 - - Boolean algebras, 150-151 - - Cartesian categories, 40,54, 71, 129, 149-152, 189, 190, 195-199, 201-202 - - - closed -, 129, 149-152, 189, 190, 195-199, 201-202 - - distributive bicartesian -, 71, 149152, 189, 190, 195-199, 201-202 --elementary topoi, 189, 190, 197-199 - - Grothendieck topoi, 189, 197-199 - - Heyting algebras, 149-150, 190, 195197 --monoidal categories, 22-23,40,54,71, 93-94, 107, 115-122, 129, 149-152, 179180, 189-190, 195-199, 201-202 - - - biclosed -, 107, 115-122, 129, 149152,179-180,1893 190,195-199,201-202 - - - closed -, 93-94, 107, 115-122, 129,

-.

149-152, 179-180, 189-190, 195-199, 201-202 - - quantifier-complete -, 189, 197-199, 201-202 - - residuated -, 107, 115-122, 129, 149152, 164-166, 179-180, 189-190, 195199,201-202 - - sequential -, 15-16,28,35,46,62,78, 99, 110, 133, 157, 172, 183, 211 - - sheaves, 197-198 - - symmetric monoidal categories, 32, 40,54, 71, 107, 115-122, 129, 149-152, 189, 190, 191-202, 195-199 - - - - closed -, 107, 115-122, 129, 149-152, 189, 190, 195-199, 201-202 Existential quantifier. 200 -symbol, 202 Export-import law, 92 External hom functor, 92 Faithful functor, 6 ’ False, 1, 20, 195 202 Symbol for Finitarily representable arrow. 21 1 Finitary formula, 203 Finite Category of - sets. 199 - coproduct, 22, 192, 193 - product, 22. 40, 54 - sequences, I 1 - tree, 18 Forgetful functor, I I Formal proof (cf. Derivation) Formula, 16, 202-204 Active -, 20 Atomic -, 16 c u t -, 20 Degree of a -, 21 1 Finitary -, 203 Passive 20 Sequence of -s. 20 Free - bicartesian category, 55-56 - - closed -, 152-153

-.

-.

294

INDEX OF SUBJECTS

Free Cartesian category, 40-42 - - closed -. 130-131 - distributive bicartesian 71-73 - monoid. I I - monoidal category, 24-25 - - biclosed -, 181-182 - object functor. 12 - pre-ordered set. 1I - quantifier-complete category. 206-207 - residuated -, 166-168 - symmetric monoidal 33 - - - closed -, 108-109 - variable. 202 Full -functor. 6 - subcategory, 15 Function symbols, 202 Functor, 4. 6. 14-15 Adjoint -s, 10 Bi-. 8 Constant -, 6 Contravariant 7 92 External horn Faithful -, 6 II Forgetful Free monoid -, 12 Free object -, 1 1 . 12 Full -, 6 Hom-. 8 Identity -, 8 Internal horn -, 92. 93 Set-valued -, 8

-.

-.

-. -.

-.

Generalized - composition, 12 - conjunction, 189 - disjunction, 189 - natural transformation, - transitivity. 12 Grothendieck topos, 199 Height - of a derivation, 19 - of a tree, 18, 205

97, 140

Heyting - algebra, 148 - - object, 196-197 External - algebra, 92 92, 93 Internal w-complete - algebra. 202 Hom functor, 8 Homomorphism - of categories (cf. Functor) - of pre-ordered sets, 6. 7

-.

Identity - arrow. 4. 5, 13. 20. 220 -functor. 8 I f . . . then, 2 Immediate reduction. 277 Implication. I , 195 -symbol. 202 Inference Operational rules of -, 3, 224, 225 Rules of 19. 220. 221, 224, 225 Structural rules of 3, 20. 224 Infinitary - conjunction symbol, 202 - disjunction symbol. 202 Infinite - coproduct functor (cf. Existential quantifier) - product functor (cf. Universal quantifier) Initial object, 22 Internal - composition, 103 - Heyting algebra, 196-197 - hom functor, 92 Interpretation - of Der(A*(X)), 44,61, 76, 132, 208 - - Der(bcA(X)), 44,61 - - Der(bcclA(X)), 44,61. 76, 132 - - Der(cA(X)), 44 - - Der(cclA(X)j, 44, 132 - - Der(dbcA(X)), 44,61, 76 - - Der(rnA(X)), 26 - - Der(mbclA(X)), 26.98, 170 - - Der(mclA(X)), 26, 98 - - Der(rA(X)), 26, 98, 170

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295

INDEX OF SUBJECTS

Interpretation of (cont.) - - Der(smA(X)), 26, 34 - - Der(smclA(X)), 26, 34, 98, 110 Isomorphic objects, 9 Isomorphism, 8 Natural -, 8 Label, 17, 206, 219 Labelled - deductive system, 17, 206-207, 219223 -sequent, 16 Language, 16, 202-204 Large category, 5 Left rank, 211, 212, 242, 243 Length of a branch, 18, 205 Limit, 191 Maximal branch of a tree, 18 Mix, 243 Monoid Free -, 11 Monoidal Biclosed - category, 178-179 Coherence in - categories, 20 - category, 20, 21, 22 - closed 93 Symmetric - -, 31 - - closed 107 Monotone relation, 229, 244 Monomorphism, 193

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Natural Component of a - transformation, 7 Generalized - transformation, 97, 140 - isomorphism, 8 - sink, 191 -source, 191 - transformation, 7 Negation operator, 147 Node, 18 Normal derivations - of bcA(X), 50,67-68 - - bcclA(X), 50, 67-68, 86, 136-137, 160-16 1 --cA(X), 50

- - CCIA(X), 50, 136-137 - - dbcA(X), 50.67-68,86 - - mA(X), 29 - - mbclA(X), 29, 101, 172, 184 - - mclA(X), 29, 101 - - rA(X), 29, 101, 172 - - smA(X), 29, 36 - - smclA(X), 29, 36, 101, 114 - - A*(X), 50, 67-68, 86, 136-137, 160161, 217-218 Normalization - algorithm, 244-277 - theorem (cf. Theorem) Object, 4, 12 Coproduct -, 23 196-197 Heyting-algebra Initial -, 22 Isomorphic -s, 9 Product -, 23 Terminal -, 22 Only if, i Operational (rule of inference), 3, 224 Opposite -category, 7 - pre-ordered set, 7 Or, 2 Ordered -tree, 18, 205 Pre- - set, 4, 6, 7, 11

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Passive formula, 20 Premiss (of a rule of inference), 19 Pre-ordered set, 4, 6, 7, 1 1 Free 11 Homomorphism of -s, 6, 7 Product, 22, 192, 193 Finite 22, 40, 54 Infinite - functor (cf. Universal quantifier) -object, 23 Proof Formal - (cf. Derivation) Pure variable 213 Proof theory of -I,M , 21.31

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296

INDEX OF SUBJECTS

Proof (cont.) - T . A . 32 - T. A . 1.v. 54. 70 -I.a . 3.92, 107 - T . A . 3. 128 - T, A . 1,v . 3. 145 - 1. 3. C. 163 178 - I, xx. 3. - T . A.I. v.j.V.3. 189 Pullback. 192, 193 Pure variable proof, 213 Pushout. 192. 193

e.

Quantifier. 189. 200 Existential 200 - algebra. 190, 201 - completeness. 189 - symbols. 202 Universal 200 Quantifier-complete category.

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20 I

Rank, 211, 212. 214. 217, 242. 243 242. 243 Left Right -, 242, 243 Weighted -, 243 Reducibility relation, I , 3, 244-277 - on Der(bcA(X)), 50, 68 - - Der(bcclA(X)), 50, 68, 87, 137 - - Der(cA(X)). 50 - - Der(cclA(X)). 50, 137 - - Der(dbcA(X)). 50, 68, 87 - - Der(mA(X)), 29 - - Der(mbclA(X)). 29, 101, 172. 184 - - Der(mclA(X)), 29, 101 - Der(r(A(X)), 29, 101, 172 - - Der(smA(X)), 36 - - Der(smclA(X)), 29. 36, 101, I14 - - Der(A*(X)), 50, 68, 87, 137. 217-218 Reflexive relation, 4 Relation Monotone 229. 244 244-277 Reducibility Reflexive -, 4 - symbols, 202 4, 229. 244 Transitive

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Residuated category. 163, 164 Restrictions - on Der(A*(X)). 212 - - eigenvariables. 213 - - (ki2). 220 - - (RI), 156-157. 213 - - (R14). 154-155. 225 - - (RIS). 225 --(Rl6). 225 --(Rl7). 225 - - (R18). 205. 208. 213 - - (R21). 20.5, 209. 213 Right rank, 211, 212, 242, 243 Rule of inference. 19, 220-221, 224-225 19 Admissible 19 Application of a Application of a structural -, 20 19 Conclusion of a Finitary -, 217 Instance of a -, 19 Operational -, 3, 224. 225 Premiss of a -, 19 Quantificational -, 204-205 Structural -, 3, 20, 224

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Sequence Finite -, 1 1 - of formulas, 20 Sequent, 16, 202-204, 219, 224, 229 Antecedent of a -, 17 19, 206 Derivable Labelled 16 Succedent of a 17 Unlabelled -, 16 Sequential category, 12-14 Set. 5 Pre-ordered 4, 6, 7, 1 I Set-valued -functor. 8, 10 -sheaf. 197 Sheaf, 197, 198-199 Simple category, 5 Sink, 191 Site, 198 Small category, 5 Source. 191

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297

INDEX OF SUBJECTS

Stability (under intersections), 198 Cut elimination -, 28,35,47,63,78, 100, Strictness (of an initial object), 85, 145 1 1 I , 134. 158, 172. 183, 21 I , 217, 243 Structural (rule of inference), 3, 20, 224 Normalization 29, 36. 50, 68, 87. 101. Structured category. I. 21, 40, 54. 70, 92, 114. 137, 161, 172, 184, 21 1, 218, 277 93, 107. 128, 145, 163-164. 178, 179. Topos 190,201 Elementary -, 194 Subcategory, 15 Grothendieck -, 199 Subderivation, 20 Transformation (cf. Natural transSubformula, 20 formation) Sublanguage, 20 Transitive relation, 4, 229 Subobject, 194 Transitivity - classifier, 193, 195 Generalized -, 12 Substitution. 12, 15, 16 Tree, 17, 18, 205 Subsystem Binary -, 18 Deductive 19. 20 Branch of a -, 18, 205 Succedent, 17 Finite 18 -symbol. 20 Height of a -, 18, 205 Symbols, 16, 202 Ordered -, 18. 205 Antecedent 20 Width of a -, 18, 205 Succedent -, 20 True, 1, 20, 195 Symmetric Symbol for 202 , - monoidal, 31 - - closed, 107 Unit System - of a tensor product, 23, 163 Labelled deductive 17. 206-207. 219- - an adjunction, 10, 97 223 Universal quantifier. 200 Unlabelled 17, 204-205. 224-228 -symbol, 202 Unlabelled Tensor product, 23, 92, 93 - deductive system, 17, 204-205. 224Terms, 15, 202-203 228 Terminal object, 22 -sequent, 16 Theorem Church-Rosser -, 3, 29, 36, 50, 69, 89, Variable 106, 110, 118, 122. 143, 162, 176, 186, Bound -, 202 2 18, 277 Free -, 202 Coherence - for mCat, 29 - - - smCat, 35, 36 Weighted rank, 243 Completeness 27, 34, 45, 61, 77, 99. Width 110. 133, 157, 171, 183, 211 - of a derivation. 19 Computability -, 29, 38, 52, 69, 91, 106, - _ -tree, 18, 205 127, 143. 164, 177, 205 Word problem, 11-12

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