Cut Elimination in Categories (Trends in Logic, 6)

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CUT ELIMINATION IN CATEGORIES

KOSTA DOŠEN University of Toulouse III and Mathematical Institute, Belgrade

revised version of January 2008 This book was published by Kluwer from Dordrecht (in the series Trends in Logic, vol. 6) in 1999. The addenda and corrigenda (in particular in §§ 4.10.1, 5.11, 6.4 and in the references) are here incorporated. The pagination differs from the published version.

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TABLE OF CONTENTS

PREFACE INTRODUCTION § 0.1. Aim and scope § 0.2. Summary of the work § 0.3. An introduction to cut elimination and adjointness § 0.3.1. Cut elimination § 0.3.2. Adjointness § 0.3.3. Cut elimination and adjointness § 0.3.4. Sequent systems and natural deduction § 0.3.5. Four types of sequent rules § 0.3.6. Identity atomization CHAPTER 1. CATEGORIES § 1.1. Foundations § 1.2. Morphisms and naturalness § 1.3. Graphs, graph-morphisms and transformations § 1.4. Deductive systems, functors, natural transformations and categories § 1.5. Equivalence of categories § 1.6. Free deductive systems § 1.7. Free categories § 1.8. Cut elimination in free categories § 1.8.1. Cut Disintegration in free categories § 1.8.2. Necessary conditions for Cut Disintegration in free categories § 1.8.3. Particular and Total Cut Elimination

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§ 1.9.Representing deductive systems and categories (Stone, Cayley and Yoneda) § 1.9.1. Cone graphs § 1.9.2. From graphs to deductive systems § 1.9.3. From deductive systems to categories § 1.9.4. The image of left compositional lifting § 1.9.5. Left-cone and right-cone graphs § 1.9.6. Deductive systems and categories in an alternative vocabulary § 1.9.7. Preorders and monoids § 1.9.8. The Yoneda Lemma for deductive systems CHAPTER 2. FUNCTORS § 2.1. Free functions and free graph-morphisms § 2.2. Free functors § 2.3. Cut elimination with free functors § 2.3.1. Cut Disintegration with free functors § 2.3.2. Necessary conditions for Cut Disintegration with free functors § 2.4. Functions redefined § 2.4.1. The standard definition of function § 2.4.2. The square of functions § 2.4.3. Cancellability of relations § 2.4.4. Functions and adjunction CHAPTER 3. NATURAL TRANSFORMATIONS § 3.1. Antecedental and consequential transformations § 3.2. Formations, formators and natural formations § 3.3. Free formations § 3.4. Free natural formations § 3.5. Cut elimination with free natural transformations § 3.5.1. Cut Disintegration in free natural formations § 3.5.2. Necessary conditions for Cut Disintegration in free natural formations

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CHAPTER 4. ADJUNCTIONS § 4.1.Definitions of adjunction § 4.1.1.Primitive notions in adjunction § 4.1.2. Hexagonal adjunction § 4.1.3. Rectangular || adjunction § 4.1.4. Rectangular \\ adjunction § 4.1.5. Rectangular // adjunction § 4.1.6. Triangular adjunction § 4.1.7. Seesaw adjunction § 4.2. Junctions, junctors and adjunctions § 4.3. Free junctions § 4.4. Free adjunctions § 4.5. Cut elimination in free adjunctions § 4.5.1. Cut Disintegration in free adjunctions § 4.5.2. Necessary conditions for Cut Disintegration in free adjunctions § 4.5.3. Constant-Cut Elimination § 4.5.4. Cut Molecularization § 4.5.5. Cut Disintegration with alternative notions of rectangular || adjunction § 4.6. Decidability in free adjunctions § 4.6.1. Decision problems in free adjunctions § 4.6.2. Free adjunctions between preorders § 4.6.3. The commuting problem in free adjunctions generated by arrowless graphs § 4.6.4. Decidability in free adjunctions generated by arbitrary graphs § 4.7. Rectangular \\ adjunctions § 4.7.1. Rectangular \\ junctions and adjunctions § 4.7.2. Cut elimination in free rectangular \\ adjunctions § 4.7.3. Decidability in free rectangular \\ adjunctions § 4.8. Triangular adjunctions § 4.8.1. Triangular junctions and adjunctions § 4.8.2. Cut elimination in free triangular adjunctions § 4.8.3. Decidability in free triangular adjunctions § 4.9. Cut elimination with other notions of adjunction

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§ 4.10. Model-theoretical normalization in adjunctions § 4.10.1. A simple decision procedure for commuting in adjunctions § 4.10.2. Normalizing via links and uniqueness of normal form § 4.11. The maximality of adjunction CHAPTER 5. COMONADS § 5.1. Definitions of comonad § 5.1.1. Standard definition of comonad § 5.1.2. The delta category § 5.1.3. Primitive notions in comonad § 5.1.4. Hexagonal comonads § 5.1.5. Triangular comonads § 5.1.6. The Kleisli category § 5.1.7. The Eilenberg-Moore category § 5.2. Adjunction between adjunctions and comonads § 5.2.1. The comonad of an adjunction § 5.2.2. Reflections and coreflections in comonads § 5.2.3. The adjunctions involving the categories of adjunctions and comonads § 5.2.4. The category of resolutions § 5.3. Comonographs, comonofunctors and comonads § 5.4. Free comonographs § 5.5. Connectives § 5.6. Free comonads § 5.7. Cut elimination in free comonads § 5.7.1. Cut Disintegration in free comonads § 5.7.2. Necessary conditions for Cut Disintegration in free comonads § 5.7.3. Cut Disintegration with alternative notions of comonad § 5.8. Decidability in free comonads § 5.8.1. Decision problems in free comonads § 5.8.2 Free comonads in preorders § 5.8.3. The commuting problem in free comonads generated by arrowless graphs

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§ 5.8.4. Decidability in free comonads generated by arbitrary graphs § 5.9. Model-theoretical normalization in comonads § 5.10. The maximality of comonad § 5.11. The links of adjunctions and the links of comonads CHAPTER 6. CARTESIAN CATEGORIES § 6.1. Rectangular || categories with binary product § 6.2. Triangular categories with binary product § 6.3. Cut elimination in free triangular categories with binary product § 6.4. Decidability in free triangular categories with binary product § 6.5. Alternative formulations of rectangular || categories with binary product § 6.6. Cut elimination in free rectangular || categories with binary product § 6.7. The terminal object § 6.8. Cut elimination in free cartesian categories § 6.9. Model-theoretical normalization in cartesian categories CONCLUSION REFERENCES INDEX

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PREFACE

The aim of this monograph is to study basic notions of category theory in a proof-theoretical spirit. This is not the first attempt of its kind. The general orientation to this subject was given some thirty years ago by Lambek, and I wish to believe this a kind of book he should have written. I would be glad if my book could attract the attention of categorists, and perhaps other mathematicians, as well as of logicians. But, though I have tried to shun away from logical jargon, I have forebodings about the reception of such a nonconformist work about categories, in which not a single commuting diagram of arrows has been drawn. I think diagrams serve mainly to ascertain that targets and sources of arrows agree in composition, and that the arrows on the two sides of an equality have the same source and target. I am usually unable to “see” that a diagram commutes just by looking at it. I must verify the fact somehow, either by deducing the corresponding equality of arrows, or by checking it in some other way, possibly by using another kind of drawing (see §§ 4.10 and 5.9). Apart from the disadvantages for typesetting of a notation with more than one dimension, the principal reason why I have not drawn the usual diagrams is that an alternative, and more provocative, title for this book could be “Diagram Elimination in Categories”. Diagrams lose their raison d’être if complex arrows are not built “horizontally” by composing, but “vertically” by applying unary operations to arrows (see § 0.3.3). The manuscript of this monograph was ready in February 1997, and was then submitted to the series where it appears. I made some

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additions to the text (§§ 4.10-11, 5.9-10 and 6.9) during the summer of 1997. I would like to express my gratitude to Jim Lambek for his comments and cordial encouragement. I am also very grateful to Peter Schroeder-Heister for providing by his unswerving hospitality two opportunities to present and discuss some parts of this book at the University of Tübingen: first, he sponsored a short course on categorial proof theory I gave at the Wilhelm Schickard Institute in July 1997, and, next, he invited me to the Workshop on ProofTheoretic Semantics at the Castle of Hohentübingen in January 1999. Giovanni Sambin was very kind to be my host during the First Workshop of the Paulus Venetus Group at the University of Padua in April 1998, where he enabled me to deliver a talk based on ideas propounded here. Next, I would like to thank Peter Aczel for showing an interest in my work, and for trying to help to make it appear sooner. I would also like to thank Krister Segerberg, who bore with much patience and tact the burden of editorship. During the preparation of this book I was supported by my home institutes, the Institut de Recherche en Informatique de Toulouse, at the University of Toulouse III, and the Mathematical Institute in Belgrade, where at the Logic Seminar I presented parts of the book. The Alexander von Humboldt Foundation financed my last stay in Tübingen. Finally, I am thankful to Mirjana Borisavljević for a diligent reading of the manuscript. And I owe many thanks to Zoran Petrić, the first reader of my text, and most helpful interlocutor in numerous and long conversations about matters treated in it. February 1999

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INTRODUCTION

§ 0.1. AIM AND SCOPE This is an introductory text in categorial proof theory. The aim is to explain fundamental notions of category theory, and in particular the notion of adjunction, from a proof-theoretical point of view. It will be shown that elementary notions, like the notions of category, functor, and natural transformation, and the more complex notion of adjunction, together with the related notion of comonad (we could as well deal with monads, also called triples), can all be formulated in such a manner that equalities between arrows tied to them are necessary and sufficient for eliminating composition in freely generated structures corresponding to these notions. This means, in particular, that for every arrow in categories involved in freely generated adjunctions and comonads we will have a term designating this arrow in which there is no composition. Moreover, such a term can be brought into a normal form unique for the arrow, so as to yield both syntactical and very simple model-theoretical, geometrical, decision procedures for the commuting of diagrams. This composition-free normal form serves also to demonstrate that strengthening the notions of adjunction and comonad with any new equality between arrows would trivialize these notions. The difference between these “composition-free” formulations of categorial notions and standard formulations is that natural transformations are not conceived as families of arrows, but as operations on arrows. Composition is a simple form of the cut rule of Gentzen’s sequent systems, and composition elimination is the form Gentzen’s cut elimination takes in categories. The techniques we shall use to achieve this categorial cut elimination are not exactly Gentzen’s, but they stem

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from his thesis [1935]. However, we will mention Gentzen’s results in logic only occasionally, and no particular knowledge of them is presupposed. An acquaintance with the basic ideas of general proof theory, as they are exposed in Gentzen’s thesis (or, eventually, other works like, for example, [Kleene 1952] or [Troelstra & Schwichtenberg 1996]), will be relied upon only for the sake of motivation. An acquaintance with the basic ideas of Prawitz from [1965, 1971], which develop Gentzen’s investigations of natural deduction, is not presupposed: these matters will be mentioned only in some side comments (especially those in §§ 0.3.4-6, 4.5.5 and 6.5). On the other hand, one part of our work presupposes a basic experience with normalization techniques in the lambda calculus, which, due to the success of this theory in theoretical computer science, are nowadays commonly taught. All we need may be found in [Barendregt 1981], but in many other, more elementary, texts, as well. The few basic notions of category theory that, in side comments, are presupposed and not defined may be found in Mac Lane’s book [1971], or other textbooks. In general, however, the treatment of categorial matters will be self-contained. Proof theory and category theory were first brought together by Lambek in [1968, 1969, 1972] (see also [Lambek & Scott 1986] and references therein; some additional references may be found in [D. 1997]). Out of Lambek’s work grows a branch of logic called categorial (or, more often, categorical) proof theory. In this theory, objects of categories correspond to formulae, i.e. propositions, arrows to deductions, the types of arrows to sequents, special arrows to axioms and operations on arrows to rules of inference; the symbol → in f : A → B stands for the turnstile ⏐⎯, and f is a code of a derivation of the sequent A → B (see [D. 1996, section 1]). Arrows in a graph freely generating a category correspond to nonlogical axiomatic sequents. In categorial proof theory one may either apply the apparatus of category theory to study logical matters—as this was attempted, for example, in [Szabo 1978]—or one may use proof-theoretical techniques to study problems in categories—as this was done, for example, in [Kelly & Mac Lane 1971]. In that paper, Kelly and Mac

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Lane apply Gentzen’s cut-elimination techniques to prove a coherence theorem for symmetric monoidal closed categories. (These categories entered into the domain of logic under the heading “linear logic” by the end of the 80s.) The present work is rather of the latter kind— applying proof theory in category theory—though its spirit and ultimate goal are more on the logical side. This monograph covers new stuff, but its style of exposition makes of that rather an introductory text. It covers also some familiar stuff, such as may be found in textbooks. However, the presentation is novel, and, we hope, simpler and clearer. We should not shun away from simplicity if the goal is to teach, rather than to impress. This style is adopted because the ambition is to lay the ground for a more extensive work on categorial proof theory. § 0.2. SUMMARY OF THE WORK In the remainder of the present introduction, we survey in an informal manner some general, motivating, aspects of cut elimination, adjointness, sequent systems and natural deduction. This is followed by six chapters. In Chapter 1 we deal with preliminary matters concerning cut elimination in free categories generated by arbitrary graphs. We show that the notion of category can be characterized by a form of cut elimination, which provides a proof-theoretical justification of this notion. There we give to our cut-elimination results the required form, which we call Cut Disintegration. In contradistinction to Gentzen’s cut elimination, where cuts are dealt with globally, Cut Disintegration concentrates on a particular cut and doesn’t try to eliminate others. Such a particular cut either disappears, or it is reduced to a cut between arrows inherited from the generating graph. Cut Disintegration entails Gentzen’s cut elimination. The assumptions made for categories will prove necessary and sufficient for proving Cut Disintegration in free categories generated by arbitrary graphs. This result is extremely easy to prove, but for pedagogical reasons it is better to deal with such matters first in a simpler context. Such a simpler context is also found in the following two chapters, where we deal with functors and natural transformations. In Chapter 2

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we show for the notion of functor—or, rather, semifunctor, i.e. functor that does not preserve identity arrows—that it can be characterized by Cut Disintegration. In Chapter 3 we do the same for a notion equivalent to the notion of natural transformation, where families of arrows indexed by objects are replaced by families of operations on arrows indexed by objects. So these notions have a proof-theoretical justification, too. Chapters 1 and 2 each have an appendix (§ 1.9 and § 2.4), which may be skipped by a reader who doesn’t want to lose trail of the main subject. In the first appendix we consider another proof-theoretical justification of the notion of category, which is also a commentary on the replacement of arrows by operations on arrows made in reformulating the notion of natural transformation. In the second appendix we explicate the adjointness present in the basic notion of function. In Chapter 4, which is the central part of the work, we deal with free adjunctions and in Chapter 5 with free comonads (we concentrate on comonads, rather than monads, because, from a logical point of view, comonads seem to come first, as the universal quantifier precedes the existential quantifier and the necessity operator precedes the possibility operator). We show that both of these notions can be characterized by Cut Disintegration, which, as before, provides a proof-theoretical justification for them. In these two chapters we also deal with the practical offshoots of our cut elimination: solutions of decision problems concerning free adjunctions and free comonads (these problems are not considered in the simpler context of Chapters 1-3, because, there, they are rather trivial). This involves both the problem whether for a pair of objects there is an arrow connecting them, which is the usual kind of decision problem solved with the help of cut elimination, and the problem whether a diagram of arrows commutes, which is a kind of decision problem considered in category theory in connection with coherence. We call the first kind of problem the theoremhood problem and the second kind the commuting problem. For the commuting problem, which is more interesting in this context, we obtain besides syntactical procedures quite handy model-theoretical procedures of a geometric kind. With the latter procedures, checking equalities of adjunctions and

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comonads is not more strenuous than calculating truth tables in propositional logic. Chapter 4 starts with a systematic survey of definitions of the notion of adjunction, and, similarly, Chapter 5 starts with a survey of definitions of the notion of comonad. The survey of comonads includes an exposition of the adjointness that involves the category of adjunctions and the category of comonads, an adjointness where the latter category is isomorphic to a full subcategory of the former. These surveys serve to locate the cut-free formulations of these notions, but they also provide some explanations, which may be unknown. We consider in these two chapters how alternative formulations behave with respect to cut elimination. In the concluding section of Chapter 4 we demonstrate as another application of our cut-elimination results that the notion of adjunction is maximal in the sense that strengthening this notion with any new equality between arrows would trivialize it. Chapter 5 has a similar concluding section, where we demonstrate the analogous maximality of the notion of comonad. In Chapter 6 we shall try to see how much of our cut-elimination characterization of adjointness is preserved for a special kind of adjunction, of particular interest to logic: namely, the adjunction involving product and the diagonal functor in cartesian categories. We take this as a sample case and don’t go further into such logical matters, which we leave for a separate work. It may seem that cut elimination in logic and related categorial structures should be better understood than cut elimination in such purely categorial structures like our free adjunctions and comonads, but, in fact, logic is rather more demanding. A thorough account of logical cut elimination should probably be rewritten, starting from foundations such as those we shall try to provide here. § 0.3. AN INTRODUCTION TO CUT ELIMINATION AND ADJOINTNESS Although adjunction is not the only notion of category theory for which we shall see that it is characterized by cut elimination, it is the central notion with which we deal, and the chapter on adjunction is the central piece of this work. The preceding chapters lead to that chapter,

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and the remaining two chapters, at the end, cover particular cases of adjunction. Therefore we should motivate, from a proof-theoretical point of view, our concern with this notion. We sketch in this introduction how cut elimination, and the related notion of normalization in natural deduction, lead to adjointness. § 0.3.1. Cut elimination Cut elimination or normalization is not a property of a logic, but of a formulation of the logic. The same logic can have a cut-free formulation and a formulation where cut is not eliminable. As a matter of fact, every logic has a trivial cut-free formulation: just postulate all provable sequents as axioms, without any rules of inference at all. Of course, such a formulation is not very useful. The point is usually to have a reduced, manageable, stock of axiomatic sequents, which together with some rules of inference yield the remaining provable sequents. And then, among all possible choices of axioms and rules, there may be some where cut is eliminable—i.e. can be dropped without losing any provable sequent—and others where it is not. The interesting thing is not to find any formulation where cut is eliminable, but one whose features are appealing, on aesthetic or technical grounds. Such a formulation may display symmetries among the connectives (symmetries of two sorts: within the framework of a particular connective or in between several connectives), or it may lead to a demonstration of normalization in natural deduction, the subformula property, decidability, interpolation, completeness, embeddability of one logic into another, or possibly something else. Without some such gain, cut elimination is of no consequence. It is trivial and uninteresting if we replace cut by rules that amount to a hidden form of cut, rules that, like some form of modus ponens, make cut derivable rather than just admissible. (A rule is admissible if it doesn’t increase the stock of theorems, without necessarily being derivable, i.e. explicitly definable, in terms of the primitive rules.) And even some contexts where cut is just admissible need not be interesting if the remaining primitive rules have disadvantages such as those we find in cut.

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So, when later in this work we look for cut-free formulations of categorial notions, we shall strive to obtain satisfactory formulations. They happen to be almost the same as standard formulations. The fact that cut elimination is a property of a formulation of a logic could perhaps be expressed by saying that it is an intensional, rather than an extensional, property. However, one must always be careful with such forms of speaking, which are encountered in different quarters of logic and philosophy, and whose meaning may therefore fluctuate. § 0.3.2. Adjointness The importance of cut elimination may be explained by its connection with the notion of adjunction, which category theory has taught us to recognize all over mathematics. To put it roughly, adjunction is half of an equivalence of categories, but taken wisely, in a “diagonal” way (cf. § 2.4.2). This means that, though the two categories need not be equivalent—one may be richer than the other—something essential is not lost in passing from the richer category to the poorer one: the two categories share a common core. A formal theorem concerning the equivalence of subcategories of categories in an adjoint situation reflects this fact (see [Lambek & Scott 1986, Part 0, sections 3-4] and [Lambek 1981], where rather “obscure” antecedents are found for this important principle). A typical adjunction is when we have, on the one hand, a category A whose objects are some algebras, like groups or vector spaces over a fixed field, with arrows being homomorphisms (in the case of vector spaces these are linear transformations), and on the other hand, the category B whose objects are sets, with functions as arrows. From A to B goes a forgetful functor G, which assigns to an algebra the underlying set of elements, and to a homomorphism the underlying function. This functor has a left-adjoint functor F from B to A that assigns to a set B the free algebra generated by B (with vector spaces, F(B) is the vector space with basis B). In passing with G from the richer category to the poorer one, not all information about the algebras is lost: something essential is preserved. The set G(A) still

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carries some information about the algebra A from the category A. When we apply next the adjoint functor F to G(A), the algebra F(G(A)) is not the same as the initial algebra A, but it is comparable to it: there is a homomorphism ϕA from F(G(A)) to A defined by mapping the free generators to themselves, which is a component of a natural transformation called the counit of the adjunction. Similarly, for a set B, the set G(F(B)) is comparable to B: there is a function γB from B to G(F(B)) amounting to inclusion, which is a component of a natural transformation called the unit of the adjunction. The categories A and B would be equivalent if F(G(A)) were isomorphic to A, and G(F(B)) were isomorphic to B. Of these isomorphisms, we have only halfs, chosen “diagonally”, in opposite directions: the homomorphism ϕA from F(G(A)) to A and the function γB from B to G(F(B)). Moreover, arrows derived from the unit composed with arrows derived from the counit give identity arrows: F(γB ) composed with ϕF(B) is the identity homomorphism on F(B), and γG(A) composed with G(ϕA ) is the identity function on G(A). This way, the forgetting of the forgetful functor is controlled. Some conclusions we may reach by reasoning in B can be transferred back to A. However, it seems that the point of describing an adjoint situation is not so much to provide a tool for proving new theorems, but rather to illuminate, clarify and systematize already known results. The ability to forget in a controlled manner is an important trait of rationality—perhaps the most important one. We should forget the unessential, so as not to be encumbered by it, and move more easily in our thoughts. But this forgetting should be controlled: what is essential shouldn’t be forgotten. There should be a way back: conclusions reached in the simpler context, where the unessential is forgotten, should be applicable to the original, more complicated, context. Controlled forgetting, such as we have in abstraction, is certainly a major characteristic of mathematical rationality, and an embodiment of it is found in the concept of adjunction. We can take it as a rule of thumb that behind theorems of the “if and only if” type we should look for adjunctions. In important

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theorems of this type, where in passing from one side to the other there is a gain, and where, typically, one direction of the theorem is easy to prove and the other difficult, there should be an adjunction that does not amount to equivalence of categories, but obtains between a richer and a poorer category. We should say, however, that not every adjunction not amounting to equivalence need hold between a richer and a poorer category. Two functors going from a category to this same category may be adjoints without the unit and counit giving isomorphisms. If we have a functor H from a category A to a category B that has both a left adjoint F from B to A and a right adjoint G from B to A, then the composite functors FH and GH from A to A are adjoints, FH being left-adjoint and GH right-adjoint (analogously, the composite functors HF and HG from B to B are adjoints, HF being left-adjoint and HG rightadjoint). Various examples of adjunction may be found in Mac Lane’s book [1971]. § 0.3.3. Cut elimination and adjointness Adjointness phenomena pervade logic. First, an essential ingredient of the spirit of logic is to investigate inductively defined notions, and inductive definitions engender free structures, which are tied to adjointness, as indicated in the preceding section (cf. § 5.5). We find also in logic the important model-theoretical adjointness between syntax and semantics, behind theorems of the “if and only if” type called semantical completeness theorems. Another kind of adjointness in logic is rather proof-theoretical: to this kind belongs the adjointness connected with cut elimination. This connection between cut elimination and adjointness appears when we realize that logical connectives (including quantifiers) are tied to adjoint functors—as Lawvere suggested in [1969]. A superficial aspect of these adjunctions is that logical connectives can be characterized within the framework of sequent systems by equivalences between, on the one hand, sequents with a single occurrence of a connective and, on the other hand, structural sequents, i.e. sequents from which connectives are absent (see [D. 1989];

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“structural” is here used as in the term “structural rules” of Gentzen’s proof theory). Following this pattern, in adjunctions tied to logical connectives one adjoint functor should involve an operation on the objects of a category—this operation corresponding to the connective—and the other should be a structural functor, not involving any such operation (here we shall not try to define the notion of structural functor more precisely). Conjunction corresponds to product in categories, which is tied to a bifunctor right-adjoint to the diagonal functor from a category to the product of this category with itself (see § 6.1), and disjunction corresponds to coproduct, which is tied to a bifunctor left-adjoint to the diagonal functor. The diagonal functor is structural. Another adjunction that characterizes conjunction is obtained by considering a cartesian category A, i.e. a category with finite products, and the polynomial cartesian category B generated by A with an indeterminate arrow (see [Lambek & Scott 1986, I.5] and [D. 1996]). From A to B there is a functor H (called the heritage functor in [D. 1996]), close to inclusion, which has a left adjoint F tied to conjunction, i.e. product (this adjunction is called functional completeness in [Lambek & Scott 1986, I.6] and deductive completeness in [D. 1996]). If A is a cartesian closed category, then H has also a right adjoint G tied to intuitionistic implication, i.e. exponentiation (the internal hom-functor of cartesian closed categories). The functor H is structural. The adjunction with the functors FH and GH from A to A is the well-known adjunction connecting conjunction and intuitionistic implication, i.e. product and exponentiation. (Here, strictly speaking, neither FH nor GH are structural functors, but FH is not far from such a functor.) The propositional constants true and absurd correspond respectively to the terminal and initial objects in a category, and these objects are characterized through adjunctions involving the trivial category that has a single object and a single arrow (see § 6.7). The constant functor into the trivial category, to which the functors tied to the true and absurd constants are respectively right and left adjoint, is also structural. (The quantifiers are characterized by adjunctions within the

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framework of Lawvere’s hyperdoctrines or fibered categories. The universal quantifier is tied to a functor right-adjoint and the existential quantifier to a functor left-adjoint to the substitution functor, which is a structural functor.) These adjunctions with structural functors that serve to characterize logical connectives do not amount to equivalences of categories. Behind definitions we usually have equivalences of categories, so that with adjunctions that don’t amount to equivalences of categories it doesn’t seem appropriate to speak of definitions. The introduction of the connective represents an enrichment; in particular, we need not be able to replace the connective in every context by something defining it. It is for such reasons that in [D. 1989] the term analysis of connective is used instead of definition of connective. The equalities between arrows of the adjunctions tied to logical connectives are related to equivalences between deductions, i.e. proofs, in a cut-elimination or normalization procedure (see [D. 1996, sections 2.3-2.4]). The situation is similar in other substructural logics, besides intuitionistic logic. (Classical logic does not lend itself easily to a categorial treatment; however, some classical connectives—in particular, conjunction—don’t differ from intuitionistic ones; see § 6.1.) In categories, cut is represented by its simple form—composition; so cut elimination is composition elimination. However, the matter is not clear cut. In some weaker logics, there may be connectives tied just to a functor, or to something not amounting to adjunction, which also permits cut elimination (cf. §§ 4.5.5, 4.7.2, 4.8.2). Besides that, not all of the equalities of the adjunctions tied to logical connectives need be related to cut elimination. This may be due to the complications of having several adjunctions mixed with one another. But (as we shall see in Chapter 6, which deals with cartesian categories) even with a single connective, the match between a particular sort of adjunction tied to the connective and cut elimination is not perfect. In this work we shall find out that this match becomes perfect if we don’t consider particular sorts of adjunction, but the pure, general, notion. Instead of explaining cut elimination in logic by adjointness, we shall turn the tables and show that adjointness may be explained by

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cut elimination, i.e. that it may be characterized through cut elimination. We shall show that the notion of adjunction may be formulated in such a manner that equalities between arrows tied to it are necessary and sufficient for eliminating composition. Although the usual formulations of the notion of adjunction are not such, the new formulation is, as a matter of fact, quite close to a usual formulation. The difference between the composition-free formulation of adjunction and a standard formulation is only that natural transformations are not conceived as families of arrows, but as operations on arrows. Our composition-free formulation is not trivial: we don’t just postulate primitive terms for every arrow; neither do we have primitive functions on arrows that make composition definable. The peculiarity, and in some context disadvantage, of composition is that it is a binary operation on arrows, whereas all our other primitive operations on arrows will be unary. (This means that with composition elimination we have practically eliminated the drawing of diagrams. It is binary operations on arrows like composition that make useful the replacement of ordinary equalities between arrows by commutative diagrams: without such operations diagrams have no role to play.) To state our cut-elimination result for adjointness precisely we need to introduce a certain notion of free adjunction. Cut elimination will be performed in free adjunctions. Before introducing this complex notion, we shall prepare the ground by considering the simpler notions of free category, free functor and free natural transformation, for which we have analogous, but simpler, cutelimination results. (These results, at the beginning, are very simple— practically trivial—but they help us to introduce our subject gradually, starting with most elementary matters.) So adjunction is not the only important notion of category theory that may be characterized through cut elimination. The notion of category itself is such, and a notion closely related to the notion of functor; namely, the generalization of the notion of functor that does not preserve identity arrows. The notion of natural transformation is also of that kind. As another example of a more complex notion characterized through cut elimination, of the calibre of adjunction, we shall consider the notion of comonad, which is tantamount to considering monads. (Monads and

21

comonads cover a particular kind of adjointness; see § 5.2.3.) If fundamental notions are such, it seems plausible that many other notions of category theory could be characterized in the same way. § 0.3.4. Sequent systems and natural deduction We concentrate in this work on cut elimination, and deal only implicitly with the related, but not identical, notion of normalization as found in natural deduction. Gentzen’s sequent systems, where cut elimination originates, differ from natural-deduction systems in several respects. First, structural rules, especially thinning and contraction, are explicit in sequent systems, whereas in natural-deduction systems they are implicit, usually hidden in the rule for introducing implication. Second, in sequent systems we have rules for introducing connectives on the left and on the right of the turnstile, whereas natural-deduction rules correspond to rules for introducing and eliminating connectives on the right of the turnstile. The sequent rules for introducing connectives on the left are equivalent to naturaldeduction rules for eliminating connectives on the right. With sequent systems we have instructions for building proofs from the middle, going upward towards the premises (i.e. hypotheses) and downward towards the conclusion. Moreover, we can paste together separately built proofs with the help of cut. In naturaldeduction systems we build proofs by going always downward from the premises to the conclusion. So the same proof can be built in more different ways in sequent systems than in natural-deduction systems: in sequent systems we can first go upwards, and then downwards, or the other way round, whereas in natural deduction we can proceed in a single manner, just downwards. Cut is trivially eliminable in natural-deduction systems (even in the absence of implication), but this cut elimination is of no consequence: it doesn’t guarantee that natural-deduction proofs will be in normal form—where, roughly speaking, eliminations are above introductions. It is cut elimination in sequent systems that corresponds to normalization in natural deduction, not cut elimination in natural deduction.

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Third, cut elimination in sequent systems is sufficient for obtaining proofs in normal form, but it is not necessary. It would be enough to eliminate cuts with nonatomic cut formulae. But even that is too much: it would suffice to eliminate cuts where the main connective of the cut formula is introduced on the right and on the left of the turnstile. (The innocuous cuts that may be left are analytic in the sense of [Smullyan 1968]: their cut formulae are subformulae of the premises and conclusion. However, not all analytic cuts can be left for proofs in normal form.) Phrased in terms of codes for natural-deduction proofs and codes for derivations in sequent systems, the first two kinds of difference are at the level of language, while the third, which should be considered after a translation between the natural-deduction language and the sequent language has been established, is at the level of equalities induced by normalization on the one hand and cut elimination on the other. After pointing where differences may lie, we shouldn’t forget about the similarities. It will become clear later on that much of what we say concerning characterizing categorial notions like adjunction through cut elimination could be phrased in terms of normalization in the style of natural deduction, rather than cut elimination. But to do one thing at a time, we prefer to stick here to the cut-elimination theme. § 0.3.5. Four types of sequent rules Gentzen’s sequent systems and natural-deduction systems don’t exhaust all the possibilities for formulating sequent rules. Let us take a brief look at the general situation. To simplify, let us suppose that we have sequent rules with a single premise. Then the rules of Gentzen’s sequent systems and naturaldeduction systems could be represented schematically as follows (“a” stands for “antecedent” and “c” for “consequent”): (ac)

→

* →

→ →*

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(cc)

→* →

→ →*

The star indicates the place where the connective of the rule occurs. The (ac) rules are sequent rules for introducing a connective on the left and on the right, and the (cc) rules correspond to naturaldeduction rules: the first one to an elimination rule and the second to an introduction rule. We put the elimination rule first because it is equivalent to the (ac) sequent rule immediately above. There are two more types of rules besides (ac) and (cc): (aa)

(ca)

→

* →

* →

→

→ * →

* → →

In the first column we have again rules equivalent to naturaldeduction elimination rules, and in the second column rules equivalent to natural-deduction introduction rules. Rules of the (aa) type are not usually given for connectives, but their pattern may be found with structural rules on the left in Gentzen’s sequent systems, provided we interpret the star as indicating where a comma, rather than a connective, occurs: the first rule is then like thinning and the second like contraction (see § 6.5). Rules of the (ca) type don’t seem to have a correlate in ordinary Gentzen systems. Without pasting proofs through cut, with the (ca) formulation we would have instructions for building natural-deduction proofs in a normal form where introductions are above eliminations, whereas in the usual natural-deduction normal form, eliminations are above introductions. We could describe this normal form as being “thick” in the middle part after the introductions and before the eliminations: there are more connectives in the middle than in the premises and conclusion. The usual normal form is

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premises

conclusion

premises

eliminations

introductions

introductions

eliminations

conclusion

Fig. 1. Thin normal form and thick normal form

“thin” in the middle: there are less connectives in the middle than in the premises and conclusion. Schematically, these two normal forms look as in Figure 1. The advantage of the thin normal form over the thick one is that we have a limit for its middle part—namely, zero: we eliminate connectives until we reach atomic formulae. With the thick normal form we need not have such a limit: we could continue introducing connectives up to infinity. However, perhaps such a limit can be imposed sometimes, and perhaps the thick normal may prove useful for some particular purposes (cf. the end of § 4.5.5). Without pasting proofs through cut, the directions for building proofs with the four types of sequents rules would be as in Figure 2. The first three directions are obtained by reading sequent derivations from top downwards. With this reading, the figure for (ca) would be the same as the figure of (ac). (The axiomatic sequents of the form A → A correspond in (ac) and (ca) to the middle part of the proof.) However, if we require that arrows go in the direction of adding connectives, as they do in (ac), then we obtain what we have for (ca) in Figure 2. (The arrows in (cc) and (aa) don’t follow this thickening

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strategy.) We should then read the (ca) sequent derivations from bottom upwards (as in tableau proofs). premises

(ac)

premises

middle (cc)

conclusion

conclusion

premises

(aa)

conclusion

premises

(ca)

middle

conclusion

Fig. 2. Directions for building proofs

The directions of (aa) and (ca) are found in real life when we are trying to find a proof: we often try to reach it by working backwards from the conclusion, or by starting from the premises and going backwards from the conclusion, trying to make the two ends meet. In this heuristical perspective, the direction of (ac) is the least natural, though in the perspective of analyzing proofs it is the most important. We have made this classification of sequent rules because we shall encounter later in this work rules of all the four types (see, in particular, § 4.5.5 and § 6.5). § 0.3.6. Identity atomization The identity deduction from A to A, which is recorded by the sequent A → A, is dual to cut, in a sense that will be made precise in § 1.9.2 (see also § 1.9.7). As there is a cut elimination, so there is an identity atomization, which consists in getting rid of axiomatic sequents A → A with complex A. We keep as axioms only A → A with atomic A, and derive the others. In the presence of nonlogical axioms (the only logical axioms in sequent systems are usually identity sequents A → A), cut elimination, too, does not enable us to get rid of all cuts: we must keep cuts

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between nonlogical axioms, where the cut formula should be atomic. The principal form of cut elimination of this work, which we call Cut Disintegration, takes this into account: it says that every cut can be either atomized or eliminated (see § 0.2, above, and § 1.8.1). In a context with only logical axioms, identity atomization usually takes the form described above, but in the presence of nonlogical axioms it may take a form analogous to Cut Disintegration, which would say that every identity can be either atomized or eliminated. With nonlogical axioms, there can be deductions where identity axioms are superfluous. Cut elimination, or rather cut atomization, leads to the thin normal form of the preceding section, and identity atomization leads to a refinement of this normal form where the middle part is atomic. (We shall encounter analogues of this refined, atomic, normal form in §§ 4.10-11 and 6.9.) It is normally required of a functor that it preserves identity arrows, as well as composition; i.e., we must have for a functor F the equality F(1A) = 1F(A), as well as F(g ° f ) = F(g) ° F( f ). We shall see that the first equality, which embodies identity atomization, is often superfluous for cut elimination in categorial structures. In particular, it is superfluous for the cut elimination with free functors of Chapter 2. However, in the main parts of Chapters 4 and 5, as well as in some other contexts mentioned there, we need this equality for cut elimination. So cut elimination may presuppose identity atomization, though they are in principle separate.

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CHAPTER 1

CATEGORIES

§ 1.1. FOUNDATIONS Category theory is sometimes taken as providing mathematics with foundations alternative to set theory. This point of view often leads into discussions about the size of classes, i.e. about the distinction between sets and proper classes. Such matters are, otherwise, rather foreign to the spirit of results about categories, which are more about structure than about size. (A presumably germane point is made when, using ancient philosophical terminology, categories are said to be about form, rather than substance; cf. [Lawvere 1964].) So these discussions are usually limited to a preamble of a typical work in category theory (such is the case, too, in the most widely cited text about categories—Mac Lane’s book [1971]). In general, they don’t leave much trace on the mathematics in the main body of the work, except a tendency to distinguish results that hold only for small categories, i.e. those whose objects and arrows, not being too numerous, can be collected into sets. These distinctions often don’t have much to do with the import of the results, and can be somewhat distracting. We are here approaching categories with a logical background and logical ambitions, but we shall neglect foundational matters. In fact, this neglect may be explained just by this background and these ambitions. If we were asked about foundations, we would rely on standard set-theoretical foundations, as they have become crystallized within logic. When on a few occasions we speak about the category of sets, the objects of this category should be taken as all the sets that are the elements of the domain of a given model of first-order axiomatic

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set theory. Since such a domain is itself a set, there is no problem in conceiving of the category of sets as being itself small. On the other hand, our ambition is to connect category theory with proof theory. In this area, objects are propositions and arrows are deductions, and there seems to be no reason to doubt that propositions and deductions may be collected into sets. Most often, they are taken to be countable. So we restrict our attention to small categories only. Categories bigger than these maybe exist, but they shall not be our concern. § 1.2. MORPHISMS AND NATURALNESS The dominant opinion is that the guiding principle of category theory is to look concerning every mathematical object for structurepreserving maps. When the object has no structure, when it is simply a set, then the maps are all functions from sets to sets. When the object has structure, then it may be an algebra, in which case the maps are homomorphisms, or it may be a set with a binary relation, in which case the maps are monotonic functions. Many other sorts of structure can be envisaged. In model theory, stress is often put on relational or functional structures with a single domain; i.e., relations are defined on a single set and functions are operations on a set. Category theory, on the other hand, is concerned much more with a plurality of domains. Let us consider the case of relations, and let us generalize monotonicity to relations between two sets. So let A and B be sets and let R ⊆ A × B. If we have another relation R' ⊆ A' × B', then a structurepreserving map from R to R' would be a pair of functions f : A → A' and g : B → B' such that for every a in A and every b in B if aRb, then f (a)R'g(b) (of course, aRb means (a , b) ∈ R). When A = B, A' = B' and f = g, then we obtain the ordinary monotonicity condition. The standard approach is to take a function as a special kind of relation, but we may also take the notion of function as being more primitive. Every relation R ⊆ A × B is associated to a function f from A to the power set of B such that aRb iff b ∈ f (a). To understand

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structure-preserving maps we shall then concentrate on the notion of function. Let a function pair from a pair of sets (A1, A2) to a pair of sets (A1', A2') be a pair of functions (g1, g2) such that g1: A1 → A1' and g2: A2 → A2'. A structure-preserving map from a function f : A1 → A2 to a function f ': A1' → A2' is a function pair (g1, g2) from (A1, A2) to (A1', A2') such that for every x in A1 and every y in A2 if f (x) = y, then f '(g1(x)) = g2(y). This implication is equivalent to requiring that for every x in A1 g2( f (x)) = f '(g1(x)),

which means that for the composite functions the following naturalness equality holds: g2 f = f 'g1.

We use the term morphism for function pairs that satisfy naturalness; so (g1, g2) is a morphism from f to f ' iff naturalness holds. This defines morphisms between functions. (Note that some authors use the term “morphism” for arrows in a category.) This terminology accords rather well with standard usage. For a binary operation f : A × A → A and another binary operation f ': A' × A' → A', the function pair that is an obvious candidate for a morphism from f to f ' is (g × g, g) where g : A → A' and (g × g)(x1, x2) is defined as (g(x1), g(x2)). Such a function pair (g × g, g) is a morphism from f to f ' iff g is a homomorphism in the ordinary sense. However, we shall speak of morphisms in other situations, too, where the structure mapped is not only that of a function, but something more complicated, involving several functions, which are moreover of a special kind. Then morphisms will not be simply function pairs, but something more involved, though analogous. In particular cases, we shall introduce special names for the morphisms in question. The guiding idea will always be to impose the naturalness condition for every function involved. Since many, if not all, important structures of mathematics can be expressed in terms of functions, and often gain in clarity by being expressed so, we shall

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find the notion of structure-preserving map appropriate to these structures by looking for naturalness conditions. § 1.3. GRAPHS, GRAPH-MORPHISMS AND TRANSFORMATIONS A graph is a function pair (S, T) from (X, X) to (Y, Y). So, S and T are both functions from X to Y. To help imagination, we call X the set of arrows, Y the set of objects, S the source function and T the target function. With that terminology, the denomination “graph” becomes justified. (In graph theory, the corresponding notion is sometimes called “directed multigraph with loops”.) A graph may be arrowless, i.e., its set of arrows may be empty, in which case the source and target functions are an empty function. An arrowless graph may be identified with a set of objects. The empty graph is the arrowless graph with the empty set of objects (the set of arrows must be empty if the set of objects is empty). For objects of graphs we use the letters A, B, C, …, and for arrows f, g, h, …, with indices if needed. We write f : A → B to indicate that the source of the arrow f is A and its target B; we say that A → B is the type of f. For graphs we use the script letters G, H, … A hom-set G(A, B) in a graph G is { f | f : A → B is an arrow of G}. An alternative way to define a graph is to identify it with a single function F from X to Y × Y. To pass from a graph (S, T) to a graph F, we have the definition

FS,T( f ) =def (S( f ), T( f )). Conversely, if we are given F, and p1 and p2 are, respectively, the first and second projection function, then we define S and T by SF( f ) =def p1(F( f )),

TF( f ) =def p2(F( f )).

It is clear that if we start from a graph (S, T), define FS,T, and then define SFS,T and TFS,T , we obtain that S is equal to SFS,T and T is equal to TFS,T . Analogously, FSF,TF is equal to F.

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We shall say that the two notions of graph, the (S, T) notion and the F notion, are equivalent. (This we do because there is an equivalence, actually an isomorphism, between the category of (S, T) graphs and the category of F graphs, as we shall see in § 1.5.) The equivalence of two notions does not always mean that the two notions are coextensive, i.e. that they cover exactly the same objects, as the notions of equilateral and equiangular triangles are coextensive. The (S, T) graphs and the F graphs are strictly speaking different objects, though they are in one-to-one correspondence. On the other hand, equivalence is more than just this one-to-one correspondence. The concept of equivalence of notions will be explained in detail in § 1.5 (after we have introduced the notion of equivalence of categories). A binary relation on Y may be identified with a graph F that is a one-one function. We can then forget about X, and consider just the image of F, i.e. a subset of Y × Y. If a binary relation is a set of ordered pairs, a graph is a family of ordered pairs indexed by the arrows, a family where the same ordered pair may occur several times with different indices. In other words, a graph is a multiset of ordered pairs. (Here is a watershed between proof theory and the rest of logic. Logicians are usually concerned with a consequence relation, if they are concerned with the business of consequence at all—there are so many things in logic—but proof theorists have to deal with a consequence graph, because there may be several different deductions with the same premise and the same conclusion.) If a graph is a function pair (S, T), then the appropriate notion of morphism is the following. Suppose S and T are functions from X to Y, while S' and T ' are functions from X' to Y '. Then as a morphism from G = (S, T) to H = (S', T ') we can take a function pair (MX, MY) from (X, Y) to (X', Y ') such that naturalness is satisfied, i.e. MY(S( f )) = S'(MX ( f )),

MY(T( f )) = T '(MX ( f )).

This means that arrows f : A → B of G are mapped to arrows MX( f ): MY(A) → MY(B) of H. As usual, we shall omit the subscripts from MX

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and MY , referring to both by M. We shall also find it handy to omit parentheses from M(A) and M( f ); instead we write MA and M f. So a graph-morphism M from G to H will be a pair of functions, both written M, assigning, respectively, to every object A of G an object MA of H, and to every arrow f : A → B of G an arrow M f : MA → MB of H. A graph-morphism M from G to H is faithful iff for every pair (A, B) of objects of G and for every pair ( f : A → B, g : A → B) of arrows of G if M f = Mg in H, then f = g in G; this means that M restricted to the hom-sets G(A, B) and H(MA, M B) is one-one. A graph-morphism M from G to H is full iff for every pair (A, B) of objects of G and for every arrow g : MA → MB of H there is an arrow f : A → B of G such that g = M f; this means that M restricted to the hom-sets G(A, B) and H(MA, M B) is onto. Note that if a graphmorphism is one-one on objects, then it is faithful iff it is one-one on arrows, and if it is a bijection on objects, then it is full iff it is onto on arrows. A graph-morphism is an embedding iff it is one-one both on objects and on arrows, and it is an isomorphism iff it is a bijection both on objects and on arrows. A graph G is a subgraph of a graph H iff there is a graphmorphism M from G to H that is the inclusion function both on objects and on arrows; M is called the inclusion graph-morphism from G to H. This means that the objects of G are included among the objects of H and the arrows of G among the arrows of H, and for every object A of G the object MA of H is A, while for every arrow f of G the arrow M f of H is f. Moreover, since M is a graph-morphism, the arrows of G have in H the same sources and targets as in G. The inclusion graph-morphism M is an embedding, and a fortiori it is faithful. A subgraph is full iff the inclusion graph-morphism is full. The identity graph-morphism IG from a graph G to G is the identity function both on objects and on arrows. If we have a graph-morphism

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M from a graph G to a graph H and a graph-morphism N from a graph H to a graph J, then we have the composite graph-morphism NM from G to J obtained by composing the functions M and N, on objects and on arrows. Let M and N be graph-morphisms from a graph G to a graph H. A transformation from M to N is a family τ of arrows τA : M A → NA of H, indexed by the objects A of G. More precisely, a transformation τ is a function from the set of objects of G to the set of arrows of H, with values τ(A), which is written τA , of type M A → NA. Note that a transformation need not be one-one (i.e., for different objects A and B of G, the arrows τA and τB may be equal, provided M A is M B and NA is NB). In § 3.1 we shall consider notions equivalent to the notion of transformation. A slightly more general notion than transformation is obtained by assuming that M and N are only functions from the objects of G to the objects of H, everything else being as for transformations. We shall have some occasions to rely on this notion of objectual transformation (see §§ 4.1.6, 4.8.1 and 5.1.5). § 1.4. DEDUCTIVE SYSTEMS, FUNCTORS, NATURAL TRANSFORMATIONS AND CATEGORIES An identity 1 in a graph G is a family of arrows 1A : A → A of G, indexed by the objects A of G. In other words, 1 is a transformation from IG to IG. The arrows 1A are called identity arrows. A composition ° in G is a function that to every pair ( f : A → B, g : B → C) of arrows of G assigns an arrow g ° f : A → C of G.

A deductive system is a triple 〈D, 1, ° ® where D is a graph, 1 is an identity in D and ° is a composition in D. The identity and composition of different deductive systems will always be denoted by the same symbols 1 and ° , assuming it is clear from the context to which deductive system they belong.

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A functor F from a deductive system 〈D, 1, ° 〉 to a deductive system 〈E, 1, ° 〉 is a graph-morphism from D to E such that the following equalities hold in E: (fun 1)

F1A = 1FA,

(fun 2)

F(g ° f ) = Fg ° F f.

These two conditions are just naturalness conditions for morphisms of identities (where identities are understood as functions) and morphisms of compositions. An embedding of deductive systems is a graph-morphism that is a functor and an embedding, and an isomorphism of deductive systems is a graph-morphism that is a functor and an isomorphism. A deductive system 〈D, 1, ° 〉 is a subsystem of a deductive system 〈E, 1, ° 〉 iff there is a functor from 〈D, 1, ° 〉 to 〈E, 1, ° 〉 that is an

inclusion graph-morphism from D to E. As for subgraphs in general, a subsystem is full iff the inclusion graph-morphism is full. It is clear that the identity graph-morphism ID on the graph D of a deductive system 〈D, 1, ° 〉 is a functor; it is called the identity functor. It is also clear that the composite graph-morphism GF is a functor when F and G are functors. Let M and N be graph-morphisms from a graph G to a graph H. If H has a composition ° , and, a fortiori, if H is the graph of a deductive system 〈H, 1, ° 〉, then a transformation τ from M to N is natural iff for every arrow f : A → B of G the following equality holds in H: (nat)

τB ° M f = Nf ° τA.

If M f, Nf, τA and τB are functions and ° is functional composition, then (nat) is the naturalness condition for the morphism (τA, τB) from M f to Nf. A deductive system is a category iff the following equalities hold between its arrows:

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(cat 1 right) (cat 1 left) (cat 2)

f ° 1A = f, 1B ° f = f,

(h ° g) ° f = h ° (g ° f ).

A subcategory is a subsystem of a category. We shall often denote later a deductive system 〈D, 1, ° 〉 simply by

D, taking the identity and composition for granted, provided it is clear from the context that we have in mind a deductive system, rather than simply a graph. We do the same for categories. If, however, we need to emphasize the difference between a deductive system and its graph, we shall use the notation 〈D, 1, ° 〉. Note that our notion of functor is slightly more general than the usual notion, which is given for categories only, whereas ours apply to arbitrary deductive systems. Note also that our notion of natural transformation is likewise more general than the usual notion, which is given for functors M and N from a category G to a category H. § 1.5. EQUIVALENCE OF CATEGORIES If a graph is a function pair (S, T), then a possible notion of morphism between graphs is not only our notion of graph-morphism, but also a more general notion, which we shall now introduce. Let ( f , h) be a function pair from (A1, B1) to (A2, B2) and ( f ', h') a function pair from (A1', B1') to (A2', B2'). A morphism from ( f , h) to ( f ', h') is then simply two function pairs, (g1, g2) from (A1, A2) to (A1', A2'), which is a morphism from f to f ', and (k1, k2) from (B1, B2) to (B1', B2'), which is a morphism from h to h'. If ( f , h) and ( f ', h') are graphs, then A1 = B1 = X, A2 = B2 = Y , A1' = B1' = X', A2' = B2' = Y ', but we could keep the same notion of morphism. Let us call these morphisms of graphs double morphisms. A graph-morphism as we have defined it in § 1.3 is a double morphism where g1 = k1 and g2 = k2. With double morphisms in general we would have a function pair (MX, MY) that in virtue of naturalness preserves sources, i.e. MY(S( f )) = S'(MX( f )), and another function pair

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(NX, NY) that in virtue of naturalness preserves targets, i.e. NY(T( f )) = T '(NX( f )). On the other hand, if a graph is a function F from X to Y×Y, then a possible notion of morphism is not only our notion of graphmorphism, but also another generalization of this notion. Namely, we would have a function pair (MX, MY × Y), where MX is, as before, a function from X to X', but MY × Y is a function from Y×Y to Y ' ×Y '. So pairs of objects are mapped to pairs of objects. The required naturalness condition is MY × Y(F( f )) = F '(MX( f )). Let us call these morphisms of graphs single morphisms. A graphmorphism is a single morphism where MY × Y is defined as MY × MY in terms of a function MY from Y to Y; for MY × MY we have (MY × MY)(A, B) = (MY(A), MY(B)). The notion of graph-morphism is a common denominator of double and single morphisms, which can serve for either notion of graph. An arrow f : A → B in a deductive system is an isomorphism iff there is an arrow g : B → A, called the inverse of f, such that g ° f = 1A and f ° g = 1B . Two objects A and B are isomorphic iff there is an isomorphism f : A → B. A natural transformation τ is a natural isomorphism iff τA is an isomorphism for every A. Two categories A and B are equivalent iff there is a functor F from B to A and a functor G from A to B such that there is in A a natural isomorphism from FG to IA and there is in B a natural isomorphism from GF to IB. An equivalence of categories where these natural isomorphisms are identities boils down to isomorphism of categories as we have defined it in the preceding section. It is easy to show that the category of graphs in the (S, T) sense (i.e. the category whose objects are these graphs) with graph-morphisms as arrows is isomorphic to the category of F graphs with graphmorphisms as arrows. Hence, these categories are also equivalent.

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This justifies our saying that the two notions of graph are equivalent. In general, two notions are to be called equivalent iff they cover objects of two categories that are equivalent. When two notions are equivalent, it is common to say that we have just two formulations of the same notion, or that the same notion is defined in alternative ways. Formulations are then called equivalent, rather than notions. We will often speak in this less formal way, too. Consider, now, the category of (S, T) graphs with double morphisms as arrows and the category of F graphs with single morphisms as arrows. These two categories are not equivalent, and neither of them is equivalent to the category of (S, T) graphs with graph-morphisms as arrows, or the category of F graphs with graphmorphisms as arrows. So, to determine whether two notions are equivalent, it is not enough to find a bijection between the objects that fall under these notions. We have to find also the appropriate morphisms, and prove an equivalence of categories. With the notions that will be found equivalent later in this work we will find mostly isomorphisms of categories, rather than simply equivalences. We stick, however, to the terminology of “equivalent notions”, because this way of speaking is more common (“isomorphic notions” would be a neologism), and because equivalence of categories catches well the intuitive idea of equivalence of notions. § 1.6. FREE DEDUCTIVE SYSTEMS To define the free deductive system 〈A, 1, ° 〉 generated by a graph G we proceed as follows. We first introduce atomic symbols in one-to-one correspondence with the objects of G; these symbols are called the object terms of A. We need not specify what exactly these symbols are: it is enough for our purposes to know that such symbols exist. We shall refer to them in a schematic way. (This is as in logic, where for most of the important purposes we need not specify the exact form of the symbols of an object language: we can refer to them with the help of schematic letters of the metalanguage.) We say that an object term that

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corresponds to an object according to the one-to-one correspondence stands for this object. Next we define the arrow terms of A. Every arrow term of A will have a unique type, which is a pair (A1, A2) of object terms of A. To indicate that an arrow term f of A has the type (A1, A2) we write f : A1 → A2. We introduce atomic symbols in one-to-one correspondence with the arrows of G; these symbols, which stand for the arrows of G, are called the generative arrow terms of A. As with object terms, we need not specify the exact form of these symbols, to which we shall refer only in a schematic way. If a generative arrow term stands for an arrow of G of type A1 → A2, then its type is made of the pair of object terms that stand for A1 and A2. Then we define inductively the arrow terms of A by the following clauses, using in addition to the generative arrow terms of A the new symbols 1A , where A is an object term of A, and the new symbol ° : (G) every generative arrow term of A is an arrow term of A; (1) if A is an object term of A, then 1A : A → A is an arrow term of A; (2) if f1: A1 → A2 and f2: A2 → A3 are arrow terms of A, then f2 ° f1: A1 → A3 is an arrow term of A. To be quite precise, we also need as auxiliary symbols the left and right parentheses, and we should write ( f2 ° f1) instead of f2 ° f1 in clause (2). But, as usual, we shall simplify our notation by not writing down the outermost parentheses of a term, taking them for granted. Later, in categories, when composition becomes associative, we can also omit other parentheses tied to composition. (Of course, we wouldn’t need any parentheses if we wrote terms in Polish notation.) As usual in inductive definitions, the set of arrow terms of A is meant to be the intersection of all sets satisfying clauses (G), (1) and (2). Conditions in clauses of an inductive definition such as this one are preserved by intersection.

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The object terms of A make the objects and the arrow terms of A the arrows of a graph A, with the obvious source and target functions. Moreover, 〈A, 1, ° 〉 is a deductive system, called the free deductive system generated by G. The objects and arrows of this free deductive system, as we have defined it, coincide with the object terms and arrow terms used to designate these objects and arrows. We could have had two sorts of things: on the one hand, objects and arrows, and, on the other hand, object terms and arrow terms different from objects and arrows. But in the case of free deductive systems we can identify the objects and arrows with the linguistic material used to designate them, because there is an isomorphism of deductive systems between these things. (We will have something analogous with free graph-morphisms in § 2.1, free formations in § 3.3, free junctions in § 4.3 and free comonographs in § 5.4.) It is a most important aspect of language that it is a free structure of some sort. Some free structures can actually not be better conceived than as a language. Arrow terms of A have been defined above as finite sequences of symbols. These sequences correspond in fact to finite binary trees with symbols at the leafs (topmost nodes), the same symbol possibly occurring at several different leafs. Parentheses serve just to give to the sequences a tree form, and are not needed if we define terms as trees. Instead of putting at the leafs of trees symbols for the arrows of G, we could put the arrows of G themselves. (However, this would mean behaving rather like Swift’s academicians of Lagado, who instead of words use things they carry around in sacks on their backs.) The free deductive system 〈A, 1, ° 〉 generated by G is free in the

following sense. Consider the generative graph-morphism H from G to A, which assigns to every object of G the corresponding object term of A and to every arrow of G the corresponding generative arrow term of A. So H amounts to identity on objects and inclusion on arrows. It

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is clear that H is faithful. The free deductive system 〈A, 1, ° 〉 and the generative graph-morphism H have the following universal property: For every deductive system 〈D, 1, ° 〉 and every graph-morphism M from G to D there is a unique functor N from 〈A, 1, ° 〉 to 〈D, 1, ° 〉 such that M = NH. This property characterizes 〈A, 1, ° 〉 up to isomorphism of deductive systems. (If this property holds for 〈A', 1, ° 〉, H' and N' as well as for 〈A, 1, ° 〉, H and N, then from H' = NH and H = N'H' we obtain H = N'NH; since H = IAH, by uniqueness we obtain N'N = IA. We derive NN' = IA' analogously.) We can express the universal property above by saying that the pair (A, H) is a universal arrow from G to the forgetful functor from the category of deductive systems (with functors as arrows) to the category of graphs (with graph-morphisms as arrows). Still another way to express the same thing is to say that the deductive system 〈A, 1, ° 〉 is the image of G under a left adjoint to the forgetful functor just mentioned (see [Mac Lane 1971, III.1, IV.1]). For the comment we have just made we assume an acquaintance with the notion of adjointness. However, this notion will also be introduced formally in § 4.1. (A functor analogous up to a point to the generative graph-morphism is called “the heritage functor” in [D. 1996]. It is unfortunate that we call here H the Generative graphmorphism, while the Heritage functor is called G in [D. 1996]. However, we reserve G for a right-adjoint functor in the main part of this work. And H does not correspond to a right adjoint in the adjunction mentioned above: there it corresponds to the unit of the adjunction, which is often denoted by a lower-case Greek H; we denote it by γ below. The heritage functor G is a right adjoint in another adjunction, called “deductive completeness” in [D. 1996].)

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§ 1.7. FREE CATEGORIES We shall first define the notion of free category 〈D*, 1, ° 〉 generated by a deductive system 〈D, 1, ° 〉.

Consider the equivalence relations ≡ on the arrows of D that satisfy the congruence law if f1 ≡ g1 and f2 ≡ g2, then f2 ° f1 ≡ g2 ° g1 (provided the types of the arrows on the two sides of ≡ are equal, and are such that the compositions are defined), as well as the following conditions, derived from (cat 1 right), (cat 1 left) and (cat 2) by replacing the equality sign by ≡: f ° 1A ≡ f, 1B ° f ≡ f, (h ° g) ° f ≡ h ° (g ° f ).

Let us call such equivalence relations categorial. It is straightforward to check that if we take the intersection ≡∩ of all categorial equivalence relations, we obtain again a categorial equivalence relation. (The assumptions for categories (cat 1 right), (cat 1 left) and (cat 2) are equalities; so the conditions on equivalences derived from them are preserved by intersection. These conditions are also preserved by intersection when they are derived from quasiequalities: i.e., implications of the form if f1 ≡ g1 and … and fn-1 ≡ gn1, then fn ≡ gn are preserved by intersection.) Then for every arrow f of D take the equivalence class [ f ] made of all the arrows f ' of D such that f ≡∩ f '. The objects of D* will be the same as the objects of D and its arrows will be the equivalence classes [ f ] we have just introduced. The type of [ f ] in D* is the same as the type of f in D (all arrows of D in the same equivalence class have the same type). The identity 1 and composition ° of 〈D*, 1, ° 〉 are defined by

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1A =def [1A ],

[g] ° [ f ] =def [ g ° f ], where on the right-hand side we have 1 and ° from D. It is clear that 〈D*, 1, ° 〉 is a category, which we call the free category generated by 〈D, 1, ° 〉.

Consider now the generative functor H from 〈D, 1, ° 〉 to 〈D*, 1, ° 〉, which is the identity function on objects and assigns to every arrow f of D the arrow [ f ] of D*. The free category 〈D*, 1, ° 〉 and the generative functor H have the following universal property: For every category 〈C, 1, ° 〉 and every functor M from 〈D, 1, ° 〉 to 〈C, 1, ° 〉 there is a unique functor N from 〈D*, 1, ° 〉 to 〈C, 1, ° 〉 such that M = NH. This property characterizes 〈D*, 1, ° 〉 up to isomorphism of deductive systems. The category 〈D*, 1, ° 〉 is the image of 〈D, 1, ° 〉 under a left adjoint to the forgetful functor from the category of categories (with functors as arrows) to the category of deductive systems (with functors as arrows). The free category 〈A*, 1, ° 〉 that is generated by the free deductive system 〈A, 1, ° 〉 generated by a graph G will be called the free category 〈A*, 1, ° 〉 generated by G. So we have constructed the free category generated by a graph in two stages. We first construct with linguistic material the free deductive system generated by the graph, and then we impose on this linguistic structure the categorial equalities. We shall have such a pattern also later in this work: we first construct a free structure of some sort with linguistic material, and then we impose on this structure appropriate equalities. Only the structures will be more complicated: they may involve not one, but two graphs, and also graph-morphisms between these graphs; moreover, the deductive systems will have additional operations on arrows (see §§ 2.1-2, 3.3-4, 4.3-4, 5.4-6).

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An arrow [ f ] of A* such that f is a generative arrow term of A is a generative arrow of A*. To economize on notation, we name the generative arrow [ f ] of A* by the generative arrow term f of A. Then we can designate the arrows of A* and A with the same arrow terms; i.e., the arrow [g] of A* is designated by any arrow term of A designating an arrow of A in [g]. We can also use the same terms for objects (which are anyway the same). We shall follow such a policy throughout the work in similar situations (see §§ 2.2, 3.4, 4.4, 5.6)—except when it is explicitly stated otherwise (as with F* in § 2.2, but even this only temporarily). Having separate languages for A and A*, though more precise, is not very practical. Besides that, avoiding this duplication is precise too, provided it is clear from the context to what our language refers. Consider the generative graph-morphism H* from G to A*, which assigns to every object of G the corresponding object of A* (i.e. the corresponding object term of A) and to every arrow of G the corresponding generative arrow of A*. So H* amounts to identity on objects, and since H* f is [ f ], which is designated by f in A*, we have that H* amounts to inclusion on arrows. It is clear that H* is faithful. The free category 〈A*, 1, ° 〉 and the generative graph-morphism H* from G to A* have the following universal property: For every category 〈C, 1, ° 〉 and every graph-morphism M from G to C there is a unique functor N from 〈A*, 1, ° 〉 to 〈C, 1, ° 〉 such that M = NH*. This property characterizes 〈A*, 1, ° 〉 up to isomorphism of deductive systems. The category 〈A*, 1, ° 〉 is the image of G under a left adjoint to the forgetful functor from the category of categories (with functors as arrows) to the category of graphs (with graph-morphisms as arrows). If HG is the generative graph-morphism from G to the graph A of the free deductive system 〈A, 1, ° 〉 generated by G, and HA is the

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generative functor from 〈A, 1, ° 〉 to 〈A*, 1, ° 〉, then it is clear that for the generative graph-morphism H* from G to A* we have H* = HAHG, and we could derive the universal property for H* from the universal properties for HG and HA. § 1.8. CUT ELIMINATION IN FREE CATEGORIES We shall now characterize the notion of category by a specific form of cut elimination, which we call Cut Disintegration. In contradistinction to Gentzen’s cut elimination, where all cuts are made to disappear, in Cut Disintegration we concentrate only on a particular cut—other cuts, or rather their descendants, will be kept. The particular cut on which we concentrate either disappears, or it is reduced to a cut between arrows inherited from the generating graph, which prooftheoretically correspond to nonlogical axioms. This Cut Disintegration entails Gentzen’s cut elimination, which we consider in § 1.8.3 under the name Total Cut Elimination. Gentzen’s cut elimination, too, can instead of simply eliminating cuts envisage reducing them to cuts between nonlogical axioms. Sometimes it is given this form (one finds that in the resolution method; cf. § 4.6.4). However, even in this form, by its global approach, it would differ from Cut Disintegration, which deals with cuts locally. We will consider this form of Gentzen’s cut elimination under the name Total Cut Molecularization in § 4.5.4, and we will compare it to Cut Disintegration. The assumptions made for categories will prove necessary and sufficient for having Cut Disintegration in free categories generated by arbitrary graphs. § 1.8.1. Cut Disintegration in free categories Let 〈A*, 1, ° 〉 be the free category generated by a graph G. Every equality f = g that holds for the arrow terms f and g of A* may be derived as follows. It may be axiomatic, which means that it is an instance of f = f, (cat 1 right), (cat 1 left) or (cat 2); else it is derived

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from axiomatic or previously derived equalities by the rules of symmetry or transitivity of equality, or the congruence rule (congr ° )

from f1 = g1 and f2 = g2, infer f2 ° f1 = g2 ° g1.

Let us call an occurrence of the composition symbol ° in an arrow term of A* a cut, and let f = g be an equality that holds in A* for the arrow terms f and g of A*. We define by induction on a derivation of f = g when a cut of the arrow term f is linked to a cut of the arrow term g in f = g. Officially, we define a set of unordered pairs {x, y}, called the links of f = g, where x is a cut in f and y a cut in g, and we say that x and y are linked in f = g iff {x, y} is in the set of links of f = g. These links are defined with respect to a particular derivation of f = g. The same equality derived in a different manner may have different links (see the example after the proof of Associativity Elimination in § 4.6.3). If f = g is the axiomatic equality f = f, then the n-th cut of f, counting from the left, is linked to the n-th cut of g. If f = g is an instance of (cat 1 right), which means that it is of the form g ° 1A = g, then the n-th cut of f is linked to the n-th cut of g, provided there are at least n cuts in g. The last cut of f, i.e. the main ° of f, displayed in g ° 1A, is not linked to any cut of g. If f = g is an instance of (cat 1 left), which means that it is of the form 1B ° g = g, then the n+1-th cut of f is linked to the n-th cut of g, provided there are at least n cuts in g. The first cut of f, i.e. the main ° of 1B ° g, is not linked to any cut of g. If f = g is an instance of (cat 2), then the n-th cut of f is linked to the n-th cut of g. The rule of symmetry of equality just preserves the links of the premise (linkage is a symmetric relation). If f = g is derived by applying the rule of transitivity of equality: from f = h and h = g, infer f = g, then a cut of f is linked to a cut of g iff there is a cut of h such that in f = h our cut of f is linked to this cut of h and in h = g this cut of h is linked to our cut of g. Finally, if f = g is derived by applying the rule (congr ° ), then the links between f1 and g1 in f1 = g1, and the links

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between f2 and g2 in f2 = g2, are preserved in f = g, which is of the form f2 ° f1 = g2 ° g1; moreover, the main ° of f2 ° f1 is linked to the main ° of g2 ° g1. This concludes the definition of links between cuts in an equality of * A . Note that in f = g every cut of f is linked either to a single cut of g or to no cut at all, and the same for f and g interchanged. Every cut in an arrow term f is the main ° of a subterm of f of the form f2 ° f1. We call this subterm the subterm of the cut. Let us call a cut atomic iff in its subterm f2 ° f1 both f1 and f2 are generative arrow terms. The degree of an arrow term is the number of occurrences of the symbols 1 and ° in it. If the subterm of a cut is f2 ° f1, then the degree of the cut is the degree of f2 ° f1. A cut is atomic iff its degree is the least possible, i.e. 1. Let us say that a cut x of an arrow term f can be atomized iff there is an arrow term f ' such that f = f ', there are as many cuts in f ' as in f, and in f = f ', for a certain derivation, every cut of f is linked to a cut of f ' so that x is linked to an atomic cut of f '. Let us say that a cut x of f can be eliminated iff there is an arrow term f ' such that f = f ', there is one cut less in f ' than in f, and in f = f ', for a certain derivation, every cut of f except x is linked to a cut of f '. We can now state and prove the following sort of cut-elimination theorem for the arrow terms of A*. Cut Disintegration. Every cut in an arrow term can be atomized or eliminated. Proof. It suffices to show that for every nonatomic cut in an arrow term f there is an arrow term f ' such that f = f ' in A*, in f = f ' every cut of f different from our nonatomic cut is linked to a cut of f ', every cut of f ' is linked to a cut of f, and our nonatomic cut of f is linked either to a cut of f ' of strictly smaller degree or to no cut at all. Once we have that, the theorem follows by induction on the degree of the nonatomic cut of f.

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Take a nonatomic cut in f whose subterm is f2 ° f1. Then we have the following cases. (1) f1 is 1A. Then f2 ° 1A = f2 by (cat 1 right), and we obtain f ' by replacing f2 ° 1A in f by f2. In f = f ', the main cut of f2 ° 1A in f is not linked to anything in f '. (2) f1 is f1" ° f1'. Then f2 ° ( f1" ° f1') = ( f2 ° f1") ° f1' by (cat 2) (with the rule of symmetry of equality), and the degree of the main cut of f2 ° f1" is strictly smaller than the degree of the main cut on the lefthand side linked to it. (G) f1 is a generative arrow term. Then we have the following subcases. (G.1) - (G.2) f2 is 1A or f2" ° f2'. These cases are treated like cases (1) and (2), using (cat 1 left) and (cat 2). (G.G) f2 is a generative arrow term. Then the main cut of f2 ° f1 is atomic. q.e.d. § 1.8.2. Necessary conditions for Cut Disintegration in free categories In the preceding section we have seen that the equalities (cat 1 right), (cat 1 left) and (cat 2), assumed for categories, are sufficient to demonstrate Cut Disintegration for the free category A* generated by a graph G. Our purpose is now to show for each of these equalities that in its absence Cut Disintegration would fail in A* for some choice of G. In the absence of the equality (cat 1 right) it is enough to suppose that we have a generative arrow term f, for an arrow of G of type A → B, to obtain the cut in f ° 1A, which can be neither atomized nor eliminated. In the absence of (cat 1 left) we have similarly 1B ° f. If G is arrowless, then one of the equalities (cat 1 right) and (cat 1 left) yields the other. A counterexample to Cut Disintegration can then be

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found in the absence of both of these equalities; this is 1A ° 1A, provided the graph G has at least one object A. In the absence of (cat 2) we have a counterexample with an arbitrary graph G, which can also be arrowless, provided it has at least one object A. The counterexample is (1A ° 1A) ° (1A ° 1A). The main cut of this arrow term cannot be eliminated so that the subsidiary cuts be linked to cuts. Nor can it be atomized. § 1.8.3. Particular and Total Cut Elimination It follows as a corollary of Cut Disintegration that if the graph G is arrowless, which means that the set of generative arrow terms is empty, then for the arrow terms of A* we can prove: Particular Cut Elimination. Every cut in an arrow term can be eliminated.

In the free category A* we then find only identity arrows. We can eliminate every cut in an arrow term of A* even if G has arrows provided 〈G, 1G, ° G〉 is a deductive system and we add to the equalities of A* the equalities g ° f = h, where f, g and h are generative arrow terms and h designates the arrow g ° G f of G. However, we are not then in the free category A*. The free category A* generated by an arrowless graph corresponds to a purely logical system. Among the generators we find only objects, i.e. formulae that make our language, and no arrows, i.e. nonlogical axioms. Gentzen’s cut elimination is not exactly the same as Particular Cut Elimination. He does not eliminate every cut as we did, but all cuts. Namely, he would prove: Total Cut Elimination. For every arrow term h there is a cut-free arrow term h' such that h = h'.

We can eliminate every particular cut and keep all others, whereas Gentzen gets rid of all of them. Of course, we can also eliminate all

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cuts one by one, and obtain Total Cut Elimination as a consequence of our Particular Cut Elimination. But if the goal is to eliminate all cuts, we can use a simpler procedure such as Gentzen’s, which eliminates cuts above which there are no cuts. We concentrate then on cuts in subterms h2 ° h1 where there are no cuts in h1 and h2. We will call such cuts topmost cuts. Concentrating on topmost cuts makes superfluous a case like (2) in the proof of Cut Disintegration. It is extremely simple to show Particular and Total Cut Elimination for free categories generated by arrowless graphs: these categories are discrete, i.e., all arrows in them are identity arrows. But it helps to grasp first what happens in this simplest of contexts, before tackling more involved matters, concerning which one can say something quite analogous. If G has arrows, then it might be impossible to atomize or eliminate all cuts in A*. More precisely, it might be impossible to find for every arrow term f an arrow term f ' in which every cut is atomic, such that f = f '. For example, the main, nonatomic, cut of f3 ° ( f2 ° f1), for f1, f2 and f3 generative arrow terms, is linked to an atomic cut of ( f3 ° f2) ° f1 in the equality (cat 2), but the atomic cut of f2 ° f1 is then linked to the main cut of ( f3 ° f2) ° f1, which is nonatomic. So there is no total version of Cut Disintegration, as there was a total version of Particular Cut Elimination (see, however, § 4.5.4). § 1.9. REPRESENTING DEDUCTIVE SYSTEMS AND CATEGORIES (STONE, CAYLEY AND YONEDA) In § 1.8 we have characterized the notion of category by a form of cut elimination: Cut Disintegration. This represents a proof-theoretical justification of this notion. In this part, which is an appendix to Chapter 1, we consider another proof-theoretical justification of the notion of category, which will be preceded by a similar justification of the notion of deductive system. These matters are not essential for the main topic of this work—characterization of categorial notions through composition elimination—and a reader who doesn’t want to lose the trail of the main course can skip the present part, and move

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immediately to Chapter 2. We shall, however, have a few occasions later to refer to matters treated in this part (see § 3.1 and § 4.1.1). They shed some light on the special role of composition in categories. In particular, the replacement of the standard formulation of the notion of natural transformation as a family of arrows by a formulation with operations on arrows, on which we rely for cut elimination in Chapters 3-6, may be illuminated by matters treated in the present appendix. The proof-theoretical justification of the notions of deductive system and category that we want to present now involves identifying objects with sets of arrows having them as source or as target. Prooftheoretically, this means identifying a proposition with a set of deductions. As we indicated in § 0.1, in categorial proof theory, objects of graphs, deductive systems and categories should be taken as formulae, i.e. propositions, and arrows as deductions. In intuitionism one sometimes hears that a proposition should be identified with the collection of proofs for it, or deductions leading to it. In the treatment adopted here, identifying a proposition with deductions where this proposition is a conclusion has no decisive primacy over identifying it with deductions where it is a premise. However, the former approach, which is also the one suggested by texts about intuitionism, is somewhat simpler to expose (it eschews a contravariance of the latter approach), and this is why we shall concentrate on it. Our topic is related to two well-known constructions in universal algebra: the Stone and Cayley representations. The proposition justifying the introduction of the notion of deductive system (Proposition 1 in § 1.9.2) is related to an elementary aspect of the Stone representation of lattice orders, while the propositions justifying the introduction of the notion of category (Proposition 2 in § 1.9.3 and Proposition 4 in § 1.9.5) are generalizations and variants of the Cayley representation of monoids. This leads to a representation of every category as a concrete category, i.e. a subcategory of the category of sets with functions (Proposition 3 in § 1.9.4; see [Freyd & Scedrov 1990, section 1.272]). The category of sets with functions is, of course, the category whose objects are sets and whose arrows are

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functions; identity arrows are identity functions and composition is composition of functions. This last representation is related to the well-known representation in category theory due to Yoneda. However, the connection with the Yoneda Lemma, which introduces some complications whose relevance to proof theory is not immediately clear, will not be considered before § 1.9.8, at the end of this part. It would be unwise to start immediately with more complex matters, which are anyway not essential for our exposition. The representation of categories by concrete categories is related to the topic of model-theoretical methods of normalization, which is within the realm of our main subject—cut elimination. We deal with normalization in Chapters 4-6, first in a more standard, syntactical, way, before considering also less standard techniques, of a modeltheoretical kind, in §§ 4.10, 5.9 and 6.9. In the case of free categories generated by graphs, with which we were concerned in this chapter, the matter is too simple to warrant consideration. However, to achieve in the syntactical approach the cut elimination that enables the normalization, we replace natural transformations conceived as families of arrows by natural transformations conceived as operations on arrows (see § 3.1). This is a move characteristic for the representation of categories by concrete categories, which we are going to consider in §§ 1.9.2-4 below. So it seems that there is an affinity between cut elimination and this representation. The model-theoretical methods of cut elimination and normalization bring some practical gain. However, what is wanting is not so much efficiency, but understanding. And one could perhaps look for such an understanding of cut elimination in representations of categories by concrete categories. (It is, of course, not enough to claim to possess such an understanding: provided one really has it, one must yet succeed in communicating it.) The results of the present part of our work will be quite simple to prove once they have been formulated. As it happens often in category theory, the point is in the formulation, not in the proof. The simplicity

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and directness of the proof should be one more reason to believe that the notions in question are natural. § 1.9.1. Cone graphs For every object A of a graph G let the left cone of A, denoted by V(A), be the set of arrows {g | for some C, g : C → A is an arrow of G}. So V(A) is ∪C G(C, A). A function ϕ from V(A) to V(B) assigning to every arrow g : C → A of G an arrow ϕ(g): C → B of G is called left-invariable (because the source of ϕ(g) is the same as the source of g). The left-cone graph V(G) of a graph G is a graph whose objects are all the left cones of G and whose arrows are all the left-invariable functions between such cones. The source of such a left-invariable function is its domain and its target is the codomain. For a graph G the graph Gop is obtained by taking the objects and arrows of G and making the source function of G the target function of Gop and the target function of G the source function of Gop. The right cone of an object A of a G is {g | for some C, g : A → C is an arrow of G}, which is equal to {g | for some C, g : C → A is an arrow of Gop}, i.e. the left cone of A in Gop. The right-cone graph of G is the left-cone graph of Gop. (We shall not talk about right-cone graphs until § 1.9.5.) Among the arrows of V(G) there is always an arrow IV(A): V(A) → V(A) that to every g in V(A) assigns g itself. So we always have an identity in V(G). For the arrows ϕ1: V(A) → V(B) and ϕ2: V(B) → V(D) of V(G) we define ϕ2 . ϕ1: V(A) → V(D) as composition of functions; i.e., for every g : C → A in V(A) (ϕ2 . ϕ1)(g) =def ϕ2(ϕ1(g)).

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So we always have a composition in V(G). It is clear that for every graph G the deductive system 〈V(G), I, . 〉 is a category. This category is a subcategory of the category of sets with functions. To adapt our notation to the subject matter, in the present part of our work (namely, in the sections within § 1.9), we shall revert to writing M(A) and M( f ) instead of MA and M f for M a graphmorphism. For succinctness, we shall also call embeddings of deductive systems deductive embeddings, and isomorphisms of deductive systems deductive isomorphisms. § 1.9.2. From graphs to deductive systems A graph-morphism F from G to V(G) is called lifting iff for every object A of G we have that F(A) is V(A). A graph-morphism G from V(G) to G is called grounding iff for every object V(A) of V(G) we have that G(V(A)) is A. Note that the function mapping every object A of G to V(A) is one-one, so that in a grounding graph-morphism G(V(A)) stands for a unique A. Then we have the following proposition. Proposition 1 1.1. The graph G has a composition iff there is a lifting graphmorphism from G to V(G). 1.2. The graph G has an identity iff there is a grounding graphmorphism from V(G) to G. Proof. 1.1. If G has a composition ° , then we take that L(A) is V(A) and for an arrow f : A → B of G we define the arrow L f : V(A) → V(B) of V(G), where Lf stands for L( f ), by

L f (g) =def f ° g. This defines a lifting graph-morphism L. Conversely, if we have a lifting graph-morphism F, then we define a composition ° in G by f ° g =def (F( f ))(g)

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(which, after replacing F by L, is exactly like the previous definition, only read in the other direction). 1.2. If G has an identity 1, then we take that G(V(A)) is A and for an arrow ϕ: V(A) → V(B) of V(G) we define the arrow G(ϕ): A → B of G by G(ϕ) =def ϕ(1A ). This defines a grounding graph-morphism. (As L is a compositional lifting, so this is an identity grounding.) Conversely, if we have a grounding graph-morphism G, then we define an identity 1 in G by 1A =def G( IV(A)). q.e.d.

Note that the proof of Proposition 1.1 would go through if V(A) were a hom-set G(C, A) for some fixed C, rather than a left cone, but the proof of Proposition 1.2 would then break down: we might be unable to define the grounding graph-morphism of the left-to-right direction, since 1A could fail to be in G(C, A) (restricting ourselves to G(A, A) would make undefinable the lifting graph-morphism L of Proposition 1.1). Note also that the left-invariability of ϕ is essential for proving Proposition 1.2 from left to right: otherwise, we would not know that ϕ(1A ) is of the type A → B. § 1.9.3. From deductive systems to categories For 〈G, 1, ° 〉 a deductive system, consider the lifting graph-morphism L from G to V(G) defined in the proof of Proposition 1.1. We call L the left compositional lifting of G. (Left compositional lifting maps the arrows X of G to partial operations from X to X; it is obtained by “currying” the partial operation ° from X × X to X.) Then we can prove the following.

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Lemma 1. The deductive system 〈G, 1, ° 〉 satisfies (cat 1 left) and

(cat 2) iff the left compositional lifting of G is a functor from 〈G, 1, ° 〉 to 〈V(G), I, . 〉. Proof. We have

L 1A(g) = 1A ° g, by definition IV(A)(g) = g, by definition.

So if the right-hand sides of these two equalities are equal by (cat 1 left), then the left-hand sides are equal, too. Conversely, if the left-hand sides are equal by (fun 1), then the right-hand sides are equal, too. We also have L f2 ° f1(g) = ( f2 ° f1) ° g, by definition (L f2 . L f1)(g) = f2 ° ( f1 ° g), by definition. Then we reason exactly as above, using (cat 2) and (fun 2). q.e.d. Proposition 2. The deductive system 〈G, 1, ° 〉 is a category iff the left compositional lifting of G is a deductive embedding of 〈G, 1, ° 〉 into 〈V(G), I, . 〉. Proof. Since L is one-one on objects, it will be an embedding iff it is one-one on arrows. Suppose 〈G, 1, ° 〉 satisfies (cat 1 right) and suppose L f1 = L f2. Then from L f1(1A ) = L f2(1A ) we obtain f1 ° 1A = f2 ° 1A , by definition f1 = f2, by (cat 1 right).

This, together with the left-to-right direction of Lemma 1, yields the proposition from left to right. Suppose now L is a deductive embedding of 〈G, 1, ° 〉 into 〈V(G), I, . 〉. Hence it is also a functor, and by the right-to-left direction of Lemma 1, we have that 〈G, 1, ° 〉 satisfies (cat 1 left) and

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(cat 2). It remains to show that it satisfies also (cat 1 right). For every arrow f : A → B of G and every arrow g in V(A) we have ( f ° 1A ) ° g = f ° (1A ° g), by (cat 2) = f ° g, by (cat 1 left). So L f ° 1A = L f , and since L is one-one on arrows, f ° 1A = f. q.e.d. Note that in the presence of (cat 1 left) and (cat 2) we have that L f1 = L f2 iff L f1(1A ) = L f2(1A ). This is related to the fact that in the definition of category the equality (cat 1 right) can be replaced by the implication if f1 ° 1A = f2 ° 1A , then f1 = f2 provided we keep (cat 1 left) and (cat 2). Similarly, (cat 1 left) can be replaced by the implication if 1B ° f1 = 1B ° f2 , then f1 = f2 provided we keep (cat 1 right) and (cat 2). And if we keep just (cat 2), then we can replace both (cat 1 right) and (cat 1 left) by these two implications provided we add the equality 1A ° 1A = 1A . The right-to left direction of Proposition 2 can be proved simply by appealing to the fact that 〈V(G), I, . 〉 is a category and that a deductive system that can be deductively embedded into a category must be a category. (To check (cat 1 right), for example, we would have L f ° 1A = L f . L 1A = L f . IV(A) = L f .) Our proof, which is not more involved, has the advantage of separating the derivation of (cat 1 left) and (cat 2), which does not depend on L being an embedding. Anyway, this renders the right-to-left direction of Proposition 2 more trivial than the left-to-right direction. The latter direction saves this proposition from being mistaken for the following really trivial assertion: A deductive system is a category iff there is a deductive embedding of it into a category.

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Proposition 2 is about a particular deductive embedding and a particular category built out of the original deductive system. Proposition 4 in § 1.9.5, which is related to Proposition 2, gives an alternative characterization of categories, which eschews the triviality of the right-to-left direction of Proposition 2. § 1.9.4. The image of left compositional lifting For 〈G, 1, ° 〉 a deductive system, an arrow ϕ: V(A) → V(B) of the leftcone graph V(G) will be called solidifiable iff for every g : C → A in V(A) we have (solid)

ϕ(g) = ϕ(1A ) ° g.

This terminology is explained by imagining that the operation ϕ has been “solidified” in the arrow ϕ(1A ), which together with composition can serve to define ϕ. Solidifiable functions of V(G) should be interpreted in logic as rules of inference that, in the presence of composition, i.e. cut, can be replaced by axioms. As, for example, the conjunction-elimination rule f :C→A∧B ϕ( f ): C → A

can be replaced by ϕ(1A ∧ B): A∧ B → A. The following two assumptions are related to (solid): (Lϕ) (Mϕ)

∃f ∀g (ϕ(g) = L f (g)), ϕ(h ° g) = ϕ(h) ° g.

It is easy to see that (solid) always implies (Lϕ), that (Lϕ) implies (Mϕ) in the presence of (cat 2), and that (Mϕ) implies (solid) in the presence of (cat 1 left). So in the presence of (cat 1 left) and (cat 2) all these assumptions are equivalent. We also have that (solid) is equivalent to (Lϕ) in the presence of (cat 1 right). The subgraphs of V(G) with the same objects whose arrows ϕ satisfy respectively (solid), (Lϕ) and (Mϕ) will be denoted by SV(G),

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LV(G) and MV(G). The graph LV(G) is the image of the left compositional lifting of G. We can prove the following lemmata concerning these graphs, Lemma 3 being a strengthening of Lemma 1 from left to right. Lemma 2. For every deductive system 〈G, 1, ° 〉 we have that 〈MV(G), I, . 〉 is a subcategory of 〈V(G), I, . 〉. Proof. We just have to check that (Mϕ) holds when we substitute IV(A) for ϕ, and that if (Mϕ) holds for ϕ1 and ϕ2, then it holds for ϕ2 . ϕ1. q.e.d. Lemma 3. The deductive system 〈G, 1, ° 〉 satisfies (cat 1 left) and

(cat 2) iff the left compositional lifting of G is a functor from 〈G, 1, ° 〉 to the category 〈SV(G), I, . 〉. Proof.

If 〈G, 1, ° 〉 satisfies (cat 1 left) and (cat 2), then (solid) is

equivalent to (Mϕ), i.e., the graphs SV(G) and MV(G) coincide, and by Lemma 2 we obtain that 〈SV(G), I, . 〉 is a category. In the presence of (cat 1 left) and (cat 2), the graphs SV(G) and LV(G) coincide, too, and it remains to apply the left-to-right direction of Lemma 1 to obtain the lemma from left to right. The other direction of the lemma is an immediate consequence of the right-to-left direction of Lemma 1. q.e.d. Lemma 4. If the deductive system 〈G, 1, ° 〉 satisfies (cat 1 right), then the left compositional lifting of G is an isomorphism between the graphs G and SV(G). Proof. Note first that in the presence of (cat 1 right) the graphs SV(G) and LV(G) coincide. We then define a graph-morphism G from SV(G) to G by taking that G(V(A)) is A, and that for ϕ: V(A) → V(B) an arrow of SV(G) the arrow G(ϕ): A → B of G is

G(ϕ) =def ϕ(1A ).

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This is the grounding graph-morphism G of the proof of the left-toright direction of Proposition 1.2 restricted to SV(G). We then have that G(L f ) = f, since the left-hand side is f ° 1A , by definition, and this is equal to f by (cat 1 right). We also have that LG(ϕ) = ϕ, since LG(ϕ)(g) is ϕ(1A ) ° g, by definition, and this is equal to ϕ(g) by (solid). q.e.d. Lemmata 3 and 4 yield the following version of Proposition 2 (whose right-to-left direction is again trivial). Proposition 3. The deductive system 〈G, 1, ° 〉 is a category iff the left compositional lifting of G is a deductive isomorphism between 〈G, 1, ° 〉 and the category 〈SV(G), I, . 〉.

It follows that for a category 〈G, 1, ° 〉 the graph-morphism G of the proof of Lemma 4, inverse to the isomorphism L, is a functor from 〈SV(G), I, . 〉 to 〈G, 1, ° 〉. When we want to check directly that G is a functor, we have that (fun 1) is satisfied by definition, while for (fun 2) we have to appeal to the solidifiability of ϕ2 to get ϕ2(ϕ1(1A )) = ϕ2(1A ) ° ϕ1(1A ).

This indicates that the graph-morphism G defined on the whole of V(G) need not be a functor from 〈V(G), I, . 〉 to 〈G, 1, ° 〉. § 1.9.5. Left-cone and right-cone graphs Analogous propositions, dual to those above, could be proved with right-cone graphs replacing left-cone graphs. There is not much interest in rehearsing what we obtained so far by turning things upside down. (The advantage of left lifting over right lifting is only that left lifting is covariant, whereas right lifting is contravariant.) However, it might be worth stating a consequence of Lemma 1 that combines leftcone and right-cone graphs.

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Let 〈G, 1, ° 〉 be a deductive system and let f : A → B and g : B → C be arrows of G. So g ° f : A → C is an arrow of G. For the arrows f : B → A and g : C → B of Gop we define the arrow f ° op g : C → A of

Gop to be the arrow g ° f : C → A of Gop. Then ° op is a composition in Gop, and since 1 is an identity in Gop as well as in G, we have that 〈Gop, 1, ° op 〉 is a deductive system. Proposition 4. The deductive system 〈G, 1, ° 〉 is a category iff

(1) the left compositional lifting of G is a functor from 〈G, 1, ° 〉 to 〈V(G), I, . 〉 and

(2) the left compositional lifting of Gop is a functor from 〈Gop, 1, ° op 〉 to 〈V(Gop), I, . 〉. Proof. If 〈G, 1, ° 〉 is a category, then 〈Gop, 1, ° op 〉 is a category, too, and by the left-to-right direction of Lemma 1, we obtain (1) and (2). Suppose now (1) and (2). Then, by the right-to-left direction of Lemma 1, it follows from (1) that (cat 1 left) holds in 〈G, 1, ° 〉, and

from (2) that (cat 1 left) holds in 〈Gop, 1, ° op 〉. But the arrow 1A ° op f : B → A of

Gop is the arrow f ° 1A : A → B of G; so (cat 1 right) holds in 〈G, 1, ° 〉. That (cat 2) holds in 〈G, 1, ° 〉 follows from the rightto-left direction of Lemma 1 and either (1) or (2). q.e.d.

Note that in the proof of this proposition from right to left it is sufficient to assume either for the graph-morphism of (1) or for the graph-morphism of (2) that it satisfies (fun 2): we need not assume that for both. But it follows from the proposition that if we assume (fun 2) for one of these graph-morphisms, then the other will satisfy (fun 2), too. The interest of Proposition 4 is that it gives an alternative characterization of categories without mentioning embedding into a category, as Proposition 2 does. The right-to-left direction of this proposition seems less trivial than the same direction of Proposition 2.

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§ 1.9.6. Deductive systems and categories in an alternative vocabulary The notions of deductive system and category need not be defined in terms of a binary operation of composition: instead they can be defined in terms of two kinds of unary operations on arrows, which amount to the images of left and right compositional lifting. Here are these alternative definitions. A deductive system is now a quadruple 〈G, 1, L, R〉 where

G is a graph and 1 is an identity in G, L is a function assigning to every arrow f : A → B of G a function L f that maps arrows g : C → A of G to arrows L f (g): C → B of G, R is a function assigning to every arrow f : A → B of G a function Rf that maps arrows g : B → C of G to arrows Rf (g): A → C of G, and the following equality holds: (L = R)

L f (g) = Rg( f ).

The notion of deductive system given by this definition is equivalent to the standard notion of § 1.4 by introducing ° in deductive systems 〈G, 1, L, R〉 by f ° g =def L f (g)

(° )

and by introducing L and R in deductive systems 〈G, 1, ° 〉 by (L)

L f (g) =def f ° g,

(R)

Rf (g) =def g ° f.

The category of these new deductive systems with appropriate functors as arrows is isomorphic, and hence equivalent, to the category of the old deductive systems with functors as arrows. It is obvious that in the new definition of deductive system we could omit either L or R, and define the omitted function in terms of

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the remaining one according to (L = R), but having both L and R enables us to state more clearly the new definition of category. A category is now a deductive system 〈G, 1, L, R〉 where the following equalities hold: (L1)

L f (1 A ) = f ,

(R1)

(LR)

Rf (L h(g)) = L h(Rf (g)).

Rf (1B ) = f,

Let us first note that (L = R) follows from these equalities, and hence need not be stipulated expressly: L f (g) = L f (Rg(1A )), by (R1) = Rg(L f (1A )), by (LR) = Rg( f ), by (L1). The new notion of category is equivalent to the old one, because the two corresponding categories of categories will be isomorphic. We shall now show how the new definition of category is related to Proposition 4. This proposition amounts to asserting that for a deductive system 〈G, 1, ° 〉, with the definitions (L) and (R), the equalities (L1)

L 1A(g) = g,

(L2)

L f2 ° f1(g) = L f2(L f1(g)),

(R1)

R 1C(g) = g,

(R2)

R f1 ° f2(g) = R f2(R f1(g)),

are interderivable with the equalities (cat 1 right), (cat 1 left) and (cat 2). It is quite straightforward to show that, in the presence of (L) and (R), these four equalities are interderivable with (L1), (R1) and (LR). Actually, either (L2) or (R2) is superfluous. So by starting from a deductive system 〈G, 1, L, R〉, with (L = R), and by adding the equalities (L1) and (R1), together with (L2) or (R2) where ° has been eliminated according to (L) or (R), we obtain still another alternative definition of category in the vocabulary with L and R. The characterization of categories of Proposition 2 doesn’t yield an equational formulation, in the style of a variety, as those suggested by Proposition 4, but a formulation in the style of a quasi-variety. We already mentioned such formulations in § 1.9.3. Without ° , a category

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could be defined as a deductive system 〈G, 1, L〉 that satisfies the equalities (L1) and (L2), where f2 ° f1 is short for L f2( f1), together with the implication if L f1(1A ) = L f2(1A ), then f1 = f2. There are other alternative formulations with R only, or both L and R, analogous to those mentioned in § 1.9.3. It may seem that with the formulations of the notion of category of this section we have also eliminated cut, i.e. composition. However, this cut elimination is spurious, since the unary operations on arrows that replace composition are just disguised binary operations. § 1.9.7. Preorders and monoids The notion of category is a common generalization of the notions of preorder (reflexive and transitive binary relation) and monoid (semigroup with identity). A preorder is a category whose graph is a binary relation, while a monoid is a category with a single object. Consider the following two important statements tied respectively to preorders and monoids. Stone Representation of Preorders. Consider a binary relation R ⊆ Y × Y. Then

R is transitive iff (∀A, B ∈ Y) ((A, B) ∈ R implies {C | (C, A) ∈ R } ⊆ {C | (C, B) ∈ R }); 2. R is reflexive iff (∀A, B ∈ Y) ({C | (C, A) ∈ R } ⊆ {C | (C, B) ∈ R } implies (A, B) ∈ R }).

1.

Cayley Representation of Monoids. An algebra 〈X, 1, ° 〉 is a monoid iff the map assigning to every f in X the function L f : g - f ° g is a monomorphism from 〈X, 1, ° 〉 to the algebra 〈XX, I, . 〉, where XX is the set of all functions from X to X, the element I is the identity function on X and the operation

.

is composition of functions.

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The first statement is of the same inspiration as Stone’s representation of distributive lattice orders in sets—it catches an elementary aspect of that representation. The second statement is best known in Cayley’s original version, where it applies to groups. The Stone Representation of Preorders is a specialization of Proposition 1. By that proposition, a relation R is transitive, i.e. has a composition, iff there is a lifting graph-morphism assigning to every pair (A, B) from R a left-invariable function from the left cone {(C, A) | (C, A) ∈ R } of A to the left cone {(C, B) | (C, B) ∈ R } of B. A relation R is reflexive, i.e. has an identity, iff there is a grounding graph-morphism assigning to a left-invariable function from {(C, A) | (C, A) ∈ R } to {(C, B) | (C, B) ∈ R } the pair (A, B) from R. To pass to the Stone Representation we need only remark that there is a leftinvariable function from {(C, A) | (C, A) ∈ R } to {(C, B) | (C, B) ∈ R } iff {C | (C, A) ∈ R } ⊆ {C | (C, B) ∈ R }. Note that {C | (C, A) ∈ R } is not the left cone {(C, A) | (C, A) ∈ R } of A, and the function that maps the objects A to the sets {C | (C, A) ∈ R } need not be one-one. It is one-one if R is a partial order, i.e., R is also antisymmetric, which in the categorial context corresponds to the deductive system being skeletal. (A deductive system is skeletal iff isomorphic objects in it are equal.) The analogue of the Stone Representation for equivalence relations is the assertion that R ⊆ Y × Y is an equivalence relation iff (∀A, B ∈ Y) ((A, B) ∈ R iff {C | (C, A) ∈ R } = {C | (C, B) ∈ R }). Then the equality of the equivalence classes {C | (C, A) ∈ R } and {C | (C, B) ∈ R } is matched by a bijection between the left cones of A and B. The Cayley Representation of Monoids is a specialization of Proposition 2. Now the left cone of the unique object of the monoid is simply the set of all arrows X of the monoid, and all functions from X to X are left-invariable (as well as right-invariable). When the monoid is a group, the functions from X to X in the image of left

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compositional lifting are bijections. In the categorial context, this corresponds to there being for every arrow in a graph an arrow of the converse type. Such are the graphs of groupoids, i.e. categories where all arrows have inverses, with whom they compose to give identity arrows. The specialization of Proposition 4 analogous to the Cayley Representation of Monoids reads as follows: 〈X, 1, ° 〉 is a monoid iff (1) the map assigning to every f in X the

function L f : g - f ° g is a homomorphism from 〈X, 1, ° 〉 to 〈XX, I, . 〉 and (2) the map assigning to every f in X the function Rf : g - g ° f is a homomorphism from 〈X, 1, ° op 〉 to 〈XX, I, . 〉, where f ° op g = def g ° f. § 1.9.8. The Yoneda Lemma for deductive systems We conclude this part of our work by considering briefly the connection between Proposition 3 and the Yoneda Lemma. Let 〈S, I, . 〉 be the category of sets with functions, and let S G be the graph of the category whose objects are graph-morphisms from G to S and whose arrows are natural transformations between these graphmorphisms. (Remember that our notion of natural transformation, defined in § 1.4, is more general than the usual one.) For 〈G, 1, ° 〉 a deductive system and C an object of G, consider the graph-morphism LC from G to S such that for every object A of G the object LC(A) is the hom-set G(C, A) and for every arrow f : A → B of G the arrow LCf : G(C, A) → G(C, B), where LCf stands for LC( f ), is defined by LCf (g) =def f ° g. The graph-morphism LC differs on objects from the left compositional lifting L of G: we now have hom-sets where we had left cones. Otherwise, on arrows, it is defined quite analogously.

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By reproducing the usual proof of the Yoneda Lemma, one can then establish the following generalization of this lemma. Yoneda Lemma. If F is a functor from the deductive system 〈G, 1, ° 〉 to 〈S, I, . 〉 and 〈G, 1, ° 〉 satisfies (cat 1 right), then for every object C of

G there is a bijection between F(C) and the hom-set S G(LC, F). Proof. For every element x of F(C), let the natural transformation τ x from LC to F be defined by τ xA(g) =def Fg(x)

where g ∈ G(C, A). That τ x is indeed a natural transformation is shown as follows: Ff (τ xA(g)) = Ff (Fg(x)), by definition τ xB(LCf (g)) = Ff ° g(x), by definition

and, since F is a functor, the right-hand sides are equal by (fun 2). For every natural transformation τ in S G(LC, F), let the object G(τ) of F(C) be defined by G(τ) =def τC (1C ). First we check that G(τ x) = x: G(τ x) = F1C(x), by definition = IF(C)(x), by (fun 1). It remains to check that τ G(τ) = τ: τ G(τ)A (g) = Fg(τC (1C )), by definition = τA (LCg(1C )), by τ being a natural transformation = τA (g), by definition and (cat 1 right). q.e.d.

Note that we didn’t assume for this version of the Yoneda Lemma that LC is a functor; otherwise, (cat 1 left) and (cat 2) would have to hold for 〈G, 1, ° 〉. However, the proof is exactly as for the usual

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Yoneda Lemma, where 〈G, 1, ° 〉 is assumed to be a category. (One can also show, without (cat 1 left) and (cat 2), that the bijection from F(C) to S G(LC, F) is natural in both C and F.) This Yoneda Lemma, where 〈G, 1, ° 〉 satisfies only (cat 1 right), is related to Lemma 4, which relies on the same assumption for 〈G, 1, ° 〉. In particular, the definition of the grounding graph-morphism G of the proof of Lemma 4 is quite analogous to the definition of the function G of the proof of the Yoneda Lemma. The usual Yoneda Lemma has the following well-known corollary, which may also be deduced from the version above. Corollary. If 〈G, 1, ° 〉 is a category, then for every object A and every object B of G there is a bijection between the hom-sets G(A, B) and op

S (G )(LA, LB). Proof. If 〈G, 1, ° 〉 is a category, then 〈Gop, 1, ° op 〉 is a category, too, and so by the Yoneda Lemma there is a bijection between op S (G )(LA, LB) and LB(A), which is by definition Gop(B, A), i.e. G(A, B). q.e.d. op

This yields an embedding of G into S (G ) with the Yoneda functor, which to A assigns LA and to f : A → B assigns the natural transformation τ f defined by τ fD (g) =def LBg( f ) = g ° op f = f ° g

for g ∈ Gop(A, D) = G(D, A). The Yoneda functor is parallel to the left compositional lifting L of G. Since, according to Proposition 3, the category 〈G, 1, ° 〉 is isomorphic to 〈SV(G), I, . 〉, we have an embedding of SV(G) into op

S (G ). This embedding is obtained by assigning to every left cone V(A) the functor LA from Gop to S. On arrows, we first apply to a

solidifiable left-invariable function ϕ: V(A) → V(B) the grounding graph-morphism G of the proof of Lemma 4, which yields the arrow

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ϕ(1A ): A → B of G; next, the Yoneda functor takes ϕ(1A ) to the

natural transformation τ ϕ(1A ), such that τ ϕ(1A )D (g), for g ∈ G(D, A) ⊆ V(A), is equal to ϕ(1A ) ° g. By (solid), this is ϕ(g). So the embedding op

of SV(G) into S (G

)

says that the solidifiable left-invariable functions op

of V(G) amount to natural transformations of S (G ).

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CHAPTER 2

FUNCTORS

§ 2.1. FREE FUNCTIONS AND FREE GRAPH-MORPHISMS We shall define first the notion of free function f generated by a pair of sets (A, B). Let B + f (A) be the set obtained by adding to B a new element f (a) for every a in A; the element f (a) is different from every element of B and from every element f (a ') where a ' is an element of A different from a. This defines a one-one function f from A to B + f (A), which is not onto if B is nonempty. So free functions are one-one functions. We shall now explain in what sense the function f is free. Consider the generative function pair (hA, hB) from (A, B) to (A, B + f (A)), where hA is the identity function on A and hB is the inclusion function of B into B + f (A). The free function f and (hA, hB) have the following universal property: For every function f ': A' → B ' and every function pair (µA, µB) from (A, B) to (A', B ') there is a unique morphism between functions (νA, νB + f (A)) from f : A → B + f (A) to f ': A' → B ' such that for every a in A we have µA(a) = νA(hA(a)) and for every b in B we have µB(b) = νB + f (A)(hB(b)). (The notion of morphism between functions was introduced and explained in § 1.2.) To define (νA, νB + f (A)) note that hA(a) is a, while hB(b) is b and νB + f (A)( f (a)) = f '(νA(a)).

This universal property characterizes f up to isomorphism of functions (defined via morphisms between functions that compose to

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give identity morphisms between functions). The function f is the image of (A, B) under a left adjoint to the forgetful functor from the category of functions (with morphisms between functions as arrows) to the category of pairs of sets (with function pairs as arrows). This forgetful functor assigns to a function from A to B the pair of sets (A, B). The notion of free graph-morphism between graphs generated by a pair of graphs (G, H), which we shall now define, is related to the notion of free function. First we define the graph H +F(G). Its objects are the objects of H together with a new object FA for every object A of G; the object FA is different from every object of H and from every object FA' where A' is an object of G different from A. The arrows of H +F(G) are the arrows of H plus a new arrow F f : FA → FB for every arrow f : A → B of G. Because of its type, the arrow F f must be different from all arrows of H; it should also be different from every arrow F f ' where f ' is an arrow of G different from f. This defines a graph-morphism F from G to H +F(G), such that F assigns the object FA to an object A of G and the arrow F f to an arrow f of G. This embedding of G into H +F(G) is called the free graph-morphism between graphs generated by the pair of graphs (G, H). A graph-morphism pair from a pair of graphs (J, K) to a pair of graphs (J ', K ') is a pair of graph-morphisms (MJ, MK) such that MJ maps J to J ' and MK maps K to K '. A morphism between graphmorphisms, which for short we call a morpho-morphism, from the graph-morphism G: J → K to the graph-morphism G': J ' → K ' is a graph-morphism pair (NJ, NK) from (J, K) to (J ', K ') such that the following naturalness equality holds: NK G = G'NJ. The notion of natural transformation is usually said to cover “morphisms between functors”. It is then instructive to compare this notion with the notion of morpho-morphism, because these two

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notions are not only different, but they may be correlated precisely only in particular situations. For some transformations from G: J → K to G': J → K, but not for all, the objects GA of K may be mapped to the objects G'A of K, and the arrows G f of K to the arrows G' f of K (we don’t have always such mappings because GA1 may be equal to GA2, while G'A1 is different from G'A2, and similarly for arrows). We can then take that these mappings are restrictions of the functions on objects and arrows of a graph-morphism NK from K to K, which on objects and arrows that are not of the form GA and G f are identity. This way a transformation may be correlated to a morpho-morphism (IJ, NK) from (J, K) to (J, K), where IJ is the identity graph-morphism on J. In our sense of the word “morphism”, the pairs (τA, τB) for a natural transformation τ from G to G' are rather like morphisms from G f : GA → GB to G' f : G'A → G'B. However, G f, G' f, τA and τB, which are arrows within the graph K, are not necessarily functions. The equality (nat) is a sort of “interiorization” of naturalness within a graph. Consider the generative graph-morphism pair (HG, HH) from (G, H) to (G, H +F(G)), where HG is the identity graph-morphism on G and HH is inclusion on the objects and arrows of H into H +F(G). The free graph-morphism F and (HG, HH) have the following universal property: For every graph-morphism F' from a graph G ' to a graph H ' and every graph-morphism pair (MG, MH) from (G, H) to (G ', H ') there from is a unique morpho-morphism (NG, NH + F(G)) F: G → H +F(G) to F': G ' → H ' such that MG = NGHG and MH = NH + F(G)HH. This universal property characterizes F up to isomorphism of graphmorphisms (defined via morpho-morphisms that compose to give identity morpho-morphisms). The graph-morphism F is the image of (G, H) under a left adjoint to the forgetful functor from the category of graph-morphisms (with morpho-morphisms as arrows) to the category

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of pairs of graphs (with graph-morphism pairs as arrows). This forgetful functor assigns to a graph-morphism from G to H the pair of graphs (G, H). However, we are not interested in graph-morphisms between any graphs, but in graph-morphisms between deductive systems, i.e. graph-morphisms between graphs of deductive systems. The appropriate notion of morphism between such graph-morphisms is the following. Let J, K, J ' and K ' be graphs of deductive systems. A morpho-morphism (NJ, NK) from the graph-morphism G: J → K to the graph-morphism G': J ' → K ' such that both NJ and NK are functors will be called a morpho-functor. We define now the free graph-morphism F between deductive systems generated by a pair of graphs (G, H). We first construct as in § 1.6 the free deductive system 〈A, 1, ° 〉 generated by G. Next we define the deductive system 〈B, 1, ° 〉. For that we introduce atomic symbols in one-to-one correspondence with the objects of H; these symbols are called the generative object terms of B. As before, we don’t specify what exactly these symbols are: it is enough to know that they exist. The object terms of B are defined inductively as follows: (G) every generative object term of B is an object term of B; (1) if A is an object term of A, then FA is an object term of B. Then we define the arrow terms of B. Every such arrow term will have a unique type, which is a pair (B1, B2) of object terms of B. To indicate that an arrow term g of B has the type (B1, B2) we write g : B1 → B2. We introduce atomic symbols in one-to-one correspondence with the arrows of H; these symbols are called the generative arrow terms of B. As with object terms, we don’t specify the exact form of these symbols. If a generative arrow term of B stands for an arrow of H of type B1 → B2, its type is made of the pair

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of generative object terms that stand for B1 and B2. Then we define inductively the arrow terms of B by the following clauses: (G) every generative arrow term of B is an arrow term of B; (1) if B is an object term of B, then 1B : B → B is an arrow term of B; (2) if g1: B1 → B2 and g2: B2 → B3 are arrow terms of B, then g2 ° g1: B1 → B3 is an arrow term of B; (3) if f : A1 → A2 is an arrow term of A, then F f : FA1 → FA2 is an arrow term of B; The object terms of B make the objects and the arrow terms of B the arrows of a graph B, with the obvious source and target functions. Moreover, 〈B, 1, ° 〉 is a deductive system, and F is a graph-morphism from A to B. This is the free graph-morphism F between deductive systems generated by (G, H). This graph-morphism is free in the following sense. Consider the generative graph-morphism pair (HG, HH) from (G, H) to (A, B ), where HG is the generative graph-morphism from G to A (defined as in § 1.6), while HH assigns to every object of H the corresponding generative object term of B and to every arrow of H the corresponding generative arrow term of B (so HH amounts to inclusion, both on objects and on arrows). The free graph-morphism F and the generative graph-morphism pair (HG, HH) have the following universal property: For every graph-morphism F' from the graph A ' of a deductive system to the graph B ' of a deductive system, and for every graphmorphism pair (MG, MH) from (G, H) to (A', B '), there is a unique morpho-functor (NA, NB ) from F: A → B to F': A ' → B ' such that MG = NAHG and MH = NB HH.

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This universal property characterizes F up to isomorphism of graphmorphisms between deductive systems (defined via morpho-functors that compose to give identity morpho-functors). The graph-morphism F is the image of (G, H) under a left adjoint to the forgetful functor from the category of graph-morphisms between deductive systems (with morpho-functors as arrows) to the category of pairs of graphs (with graph-morphism pairs as arrows). This forgetful functor assigns to a graph-morphism from C to D the pair of graphs (C, D). § 2.2. FREE FUNCTORS We shall first define the notion of free functor between categories generated by a graph-morphism between deductive systems. Let F be a graph-morphism from the graph A of a deductive system 〈A, 1, ° 〉 to the graph B of a deductive system 〈B, 1, ° 〉, and let 〈A*, 1, ° 〉 be the free category generated by the deductive system 〈A, 1, ° 〉. Then consider the categorial equivalence relations ≡ on the

arrows of B (in the sense of § 1.7), which in addition to the congruence law of categorial equivalence relations satisfy also the following congruence law: if [ f1] = [ f2] in A*, then F f1 ≡ F f2, as well as the following conditions, derived from (fun 1) and (fun 2) by replacing the equality sign by ≡: F1A ≡ 1FA, F( f2 ° f1) ≡ F f2 ° F f1. Let us call such equivalence relations functorial. It is straightforward to check that if we take the intersection ≡∩ of all the functorial equivalence relations, we obtain again a functorial equivalence relation. Then for every arrow g of B let [g] be the equivalence class {g '| g ≡∩ g '}, and let [ f ] be an arrow of A*. Then we define the following:

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1B =def [1B ],

F*A =def FA,

[g2] ° [g1] =def [g2 ° g1], F *[ f ] =def [ F f ].

The objects of B and the equivalence classes [g], together with these definitions, make the category 〈B*, 1, ° 〉. The F* we have defined is the free functor between categories generated by F. It is a functor from the category 〈A*, 1, ° 〉 to the category 〈B*, 1, ° 〉. Consider now the generative morpho-functor (HA, HB) from F: A → B to F*: A* → B*, where HA and HB are the identity functions on objects, HA assigns to every arrow f of A the arrow [ f ] of A*, and HB assigns to every arrow g of B the arrow [g] of B*. The free functor F* and the generative morpho-functor (HA, HB) have the following universal property: For every functor F' from a category A ' to a category B ' and every morpho-functor (MA, MB) from F: A → B to F': A ' → B ' there is a unique morpho-functor (NA, NB) from F*: A* → B* to F': A ' → B ' such that MA = NAHA and MB = NBHB. This property characterizes the free functor F* up to isomorphism of functors (defined via morpho-functors that compose to give identity morpho-functors between functors). The free functor F* is the image of F under a left adjoint to the forgetful functor from the category of functors between categories (with morpho-functors as arrows) to the category of graph-morphisms between deductive systems (with morpho-functors as arrows). The free functor F* between categories that is generated by the free graph-morphism F between deductive systems generated by a pair of graphs (G, H) will be called the free functor F* generated by (G, H). For the free functor F* generated by (G, H), the generative arrows of A* and B* are the arrows [ f ] and [g] such that f and g are generative arrow terms of, respectively, the graphs A and B of the deductive systems 〈A, 1, ° 〉 and 〈B, 1, ° 〉 associated to the free graphmorphism F from A to B generated by (G, H).

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Consider now the generative graph-morphism pair (HG*, HH*) from (G, H) to (A*, B*), where HG* is the generative graph-morphism from G to A* (defined as in § 1.7), while HH* assigns to every object of H the corresponding object of B* and to every arrow of H the corresponding generative arrow of B*. The graph-morphism HH* amounts to inclusion, both on objects and on arrows, and it is faithful. The free functor F* generated by (G, H) has the following universal property involving (HG*, HH*), quite analogous to the universal property of the free graph-morphism F between deductive systems generated by (G, H), which we have considered in the preceding section: For every functor F' from a category A ' to a category B ' and every graph-morphism pair (MG, MH) from (G, H) to (A', B ') there is a unique morpho-functor (NA, NB ) from F*: A* → B* to F': A ' → B ' such that MG = NAHG* and MH = NB HH*. This universal property characterizes F* up to isomorphism of functors (defined via morpho-functors that compose to give identity morpho-functors between functors). The functor F* is the image of (G, H) under a left adjoint to the forgetful functor from the category of functors between categories (with morpho-functors as arrows) to the category of pairs of graphs (with graph-morphism pairs as arrows). This forgetful functor assigns to a functor from C to D the pair of graphs (C, D). If (HG, HH) is the generative graph-morphism pair from (G, H) to (A, B), where A and B come from the free graph-morphism F between deductive systems generated by (G, H), and (HA, HB) is the generative morpho-functor from F to F*, then it is clear that for the generative graph-morphism pair (HG*, HH*) from (G, H) to (A*, B*) we have HG* = HAHG and HH* = HBHH, and we could derive the universal property for (HG*, HH*) from the universal properties for (HG, HH) and (HA, HB).

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The free functor F* generated by (G, H) is full and faithful. Since it is one-one on objects, it is also one-one on arrows. However, if H has some objects, then F is not onto on objects, and it need not be onto on arrows. We will designate the generative arrows [ f ] and [g] of A* and B* by the generative arrow terms f and g of A and B. Then we can use the object terms and arrow terms of A and B to designate the objects and arrows of A* and B*. In particular, we shall write F instead of F*. To make it clear what we are doing, we marked with a star superscript the difference between the F* of the free functor from A* to B* and the F of the free graph-morphism from A to B, but from now on we will omit this superscript. In doing that we follow the same policy we installed by denoting the identity and composition of 〈B*, 1, ° 〉 with the same symbols as the identity and composition of 〈B, 1, ° 〉. This economy makes our notation somewhat less precise, but rather more practical. § 2.3. CUT ELIMINATION WITH FREE FUNCTORS We shall now show that Cut Disintegration holds in the categories A* and B* associated to a free functor generated by an arbitrary pair of graphs. For A* this follows from the preceding chapter, and for B* the result is proved quite easily. It will appear that assumptions made for functors are not all necessary for Cut Disintegration in B*: the equality (fun 1) happens to be superfluous. However, it will not be superfluous in Chapters 4 and 5. The purpose of concentrating on Cut Disintegration with free functors, in spite of the easiness of the result and of the superfluity (fun 1), is to prepare the ground for later chapters.

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§ 2.3.1. Cut Disintegration with free functors Let F be the free functor from the category A* to the category B* generated by (G, H). Every equality g1 = g2 that holds for the arrow terms g1 and g2 of B* may be derived as follows. It may be axiomatic, which means that it is an instance of f = f, (cat 1 right), (cat 1 left), (cat 2), (fun 1) or (fun 2); else it is derived from axiomatic or previously derived equalities by the rules of symmetry or transitivity of equality, or the rule (congr ° ) of § 1.8.1, or the congruence rule (congr F)

from f1 = f2 in A*, infer F f1 = F f2 in B*.

We must now extend the definition of linked cuts from § 1.8.1 to equalities among arrow terms of B*. For that we have to take into account the following additional cases. If f = g is an instance of (fun 1), then no cut occurs in either f or g. If f = g is an instance of (fun 2), then the n-th cut of f is linked to the n-th cut of g. The rule (congr F) just preserves the links of the premise. The cuts linked in the equalities of the free category A* are defined as in § 1.8.1. Note that cuts of arrow terms of A* occurring as subterms of arrow terms of B*, which are hence cuts of A*, are linked through (fun 2) to cuts of B*. We just take over the other definitions of § 1.8.1, save that in the definition the degree of an arrow term of B* we also count, besides occurrences of 1 and ° , also occurrences of F applied to arrow terms (we don’t count F applied to object terms in the indices). In these definitions we take on an equal footing the cuts of A*and the cuts of B*. We can now prove Cut Disintegration for the arrow terms of B*, i.e. the codomain of the free functor F. Of course, according to § 1.8.1, Cut Disintegration holds for the arrow terms of the free category A*, the domain of the free functor F. Proof of Cut Disintegration for B*. We proceed as in the proof of § 1.8.1. So, in an arrow term of B*, take a nonatomic cut whose

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subterm is f2 ° f1. Then we have the following cases in addition to those considered in § 1.8.1. (3) f1 is Fg. Then we have the following subcases. (3.1) - (3.2) f2 is 1FB or f2" ° f2'. These cases are treated like cases (1) and (2) of the proof in § 1.8.1. (3.3) f2 is Fg '. Then Fg ' ° F g = F(g ' ° g) by (fun 2), and the degree of the main cut of g ' ° g is strictly smaller than the degree of the main cut on the left-hand side linked to it. (3.G) f2 is a generative arrow term of B*. This is excluded because f2 must have a source of the form FB. In case (G), where f1 is a generative arrow term of B*, we have the following additional subcase. (G.3) f2 is Fg. This is excluded because f1 would then have a target of the form FB. The present proof presupposes that the proof of Cut Disintegration for arrow terms of A* has been accomplished. q.e.d. § 2.3.2. Necessary conditions for Cut Disintegration with free functors Note that in the proof of Cut Disintegration in the preceding section we have used the equality (fun 2) of the definition of functor, but not the equality (fun 1). It is easy to see that (fun 1) is independent from the other equalities of functors between categories, i.e., it cannot be derived from (fun 2) and the categorial equalities (cat 1 right), (cat 1 left) and (cat 2). So, this equality is clearly not necessary for Cut Disintegration, whereas (fun 2) presumably is. That (fun 2) is indeed necessary is shown by the example where the graph G from the generating pair of graphs (G, H) has at least two arrows designated by the generative arrow terms f : A → B and g : B → C of A*, or G has at least one arrow designated by the generative arrow term f : A → A of A* (so G provides at least one possibility to compose some of its

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arrows). Then in the absence of (fun 2) we cannot atomize the cut of Fg ° F f , or F f ° F f . Of course, we can also not eliminate it. When G is arrowless, so that the category A* is discrete (i.e., all its arrows are identity arrows), then, as before, we obtain Cut Disintegration in B* with the help of (fun 2), but we could also use (fun 1) instead of (fun 2). The necessity of (cat 1 right), (cat 1 left) and (cat 2) is demonstrated as in § 1.8.2. So the notion characterized by the Cut Disintegration of the preceding section is not exactly the notion of functor between categories, but the closely related notion of graph-morphism between categories that must satisfy only (fun 2). This notion is known in the literature under the label semifunctor (see [Hoofman 1993] and references therein). The foregoing may be taken as a certain justification of this notion. However, in an artificial manner, we can find for the notion of functor an equivalent formulation where exactly this notion, including (fun 1), is characterized by Cut Disintegration. It is enough to make (fun 1) necessary for deriving (fun 2). For example, in the definition of functor we may replace (fun 2) by either of the following equalities: F(g ° f ) = (Fg ° F f ) ° F1A , F((g ° f ) ° h) = (Fg ° F f ) ° Fh; in both cases, we need (fun 1) to derive (fun 2). In Chapter 4 we consider likewise various equivalent formulations of the notion of adjunction, only one of which is exactly characterized by Cut Disintegration: in all the others some assumptions for adjunction are superfluous. However, the formulation of adjunction exactly characterized by Cut Disintegration is quite natural, and not artificial like the alternative formulations of functor we have just mentioned. The notion of functor enters into this formulation of adjunction in the standard formulation with (fun 1) and (fun 2) primitive. And there (fun 1) is necessary, too, for Cut Disintegration. We have there cuts in arrow terms of the form f ° F1B which we cannot dispose of without the help of (fun 1).

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§ 2.4. FUNCTIONS REDEFINED Chapter 1 is about characterizing the notion of category through cut elimination. In § 1.9 we had an appendix to this chapter concerning the notion of category and the more general notion of deductive system. To the present chapter, which deals with functors, we add this appendix about the notion of function, on which is built the notion of functor. The appendix may be skipped over as far as the cutelimination theme is concerned (the reader may go immediately to the next chapter). Its goal is only to show that understanding the simplest of all notions on which this, and practically every other, mathematical text is based involves up to a point the more complex notions to be considered later. The notion of adjunction, the main notion with which we are concerned in this work, presupposes the more elementary notion of function, whose importance and ubiquity in mathematics are of course not necessary to mention, let alone justify. We want to show, however, that underlying the notion of function there is an adjunction, and that this adjunction characterizes completely the notion of function. This will serve as another corroboration of the slogan that adjointness arises everywhere. The standard definitions of the general notions of function, onto function and one-one function don’t exhibit clearly the regularities and symmetries of these notions. It is not immediately clear from these definitions, without some deducing, that (1) the property of being a function is made of two components exactly dual to the onto and one-one properties (they go in the opposite direction), and that, taking duality in a different sense, (2) the onto and one-one properties are dual to each other. There are definitions of these notions that exhibit immediately (1) and (2), but these definitions are rarely and cryptically mentioned (the earliest reference for them I know of is [Riguet 1948, p. 127]). On their own, these definitions are quite simple. I believe that their ingredients belong to the folklore and sometimes crop up as exercises

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in textbooks. However, the general picture they provide seems to be missing in the standard textbook approach. Many students of mathematics probably stay pretty much in the dark about (2), or perhaps even (1); many are probably surprised when, after having known for some time about onto functions and one-one functions, they learn about (2) via the cancellation properties of epi and mono arrows in category theory. I don’t wish to suggest that these nonstandard definitions should supplant the standard ones—especially not for a first exposure to the defined notions. I suppose, however, that at some point in the study of mathematics one should get a systematic picture such as will occupy us here. § 2.4.1. The standard definition of function A binary relation is a set of ordered pairs R together with some specified domain D and codomain C such that R ⊆ D × C. We speak here only about “relations”, the epithet “binary” being tacitly presupposed, and, as usual, we write xRy for (x, y) ∈ R. A function from D to C is a relation R ⊆ D × C such that for every x in D there is exactly one y in C for which xRy. It is easy to deduce that R ⊆ D × C is a function iff (left-total) for every x in D there is at least one y in C such that xRy, (right-unique) for every x in D there is at most one y in C such that xRy. A function R ⊆ D × C is onto iff (right-total) xRy,

for every y in C there is at least one x in D such that

and it is one-one iff (left-unique) xRy.

for every y in C there is at most one x in D such that

For a relation R ⊆ D × C, the conjunction of (right-total) and (leftunique) is equivalent to asserting that for every y in C there is exactly

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one x in D such that xRy. So, after just a little bit of deducing, we obtained (1): the onto and one-one properties are the two components of functionality, but going from the codomain to the domain; functionality in the direction from the domain to the codomain is made of two completely analogous, dual, components. What is still not quite evident is (2); namely, that the onto and oneone properties are also, in a certain sense, dual to each other, though the expressions “at least one” and “at most one” hint at such a relationship. That “at least one” is dual to “at most one” may be gathered from the fact we can express that a set A is a singleton by the conjunction of “for some x1 and x2 in A, x1 = x2” (which amounts to “there is at least one member of A”) and “for every x1 and x2 in A, x1 = x2” (which amounts to “there is at most one member of A”). The existential quantifier is, of course, dual to the universal quantifier. When we deal specifically with functions, the duality between the onto and one-one properties is exhibited in category theory by showing that the first property amounts to cancellability on the right in functional composition, while the second property amounts to cancellability on the left. However, as we shall see in § 2.4.3, if we assume functionality neither for R ⊆ D × C nor for the converse set of ordered pairs, we will fail to exhibit in this manner the duality between (right-total) and (left-unique), or between (left-total) and (right-unique).

§ 2.4.2. The square of functions The definitions below will enable us to see the duality between the onto and one-one properties stated in (2), in a different, more basic, manner—without extra assumptions concerning R ⊆ D × C. They will also display clearly the duality of (1), namely, the connection between the onto and one-one properties and functionality. Let R ← ⊆ C × D be the relation converse to R ⊆ D × C, i.e., R← is {(y, x)| xRy}, and let R2 ° R1 be {(x, y)| for some z, xR1z and zR2y}. (For the composition of R1 with R2 we write R2 ° R1, rather than R1 ° R2, so as not to deviate from standard usage when we come to functional composition. This standard usage is unfortunate—it clashes with our

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inclination to read other things from left to right—but it is hard to fight against. Anyway, what we have to say about functions does not depend upon reforming the notation for functional composition.) Next, for every set A, let 1A be {(x, x)| x ∈ A}. Then consider the following properties a relation R ⊆ D × C might have: (left-total)

1D ⊆ R←° R

(left-unique) 1D ⊇ R←° R

(right-total)

R ° R ← ⊇ 1C

(right-unique) R ° R ← ⊆ 1C

We pass from left to right in this square by replacing R by its converse R ← (of course, R←← is equal to R). We pass from the upper row to the lower row by replacing an inclusion by the converse inclusion. The diagrams in Figure 3 illustrate the four properties in the square. Solid lines are in the antecedents and dotted lines in the consequents. For example, the upper left diagram is read as follows: “If x1 is equal to x2, then we have arrows going from them to the right towards a point y.” Since every point is equal to itself, this means that for every point in the domain we have at least one arrow going towards the codomain whose source is this point—the two dotted R arrows become one. The lower right diagram is read: “If we have arrows with the same source x, then their targets y1 and y2 are equal.” So at most one arrow can start from a point of the domain—the two solid R arrows become one. It is easy to check with the help of these diagrams that the properties in the square are equivalent to the previously introduced properties that bear the same names.

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x1

°

R

R

°y

= x 2°

x°

x1

°

R

x2°

R

°y

x °

° y1

=

R

R

(left-unique)

° y2

(right-total)

R

=

y1

=

R

(left-total)

°

° y2

(right-unique)

Fig. 3. The square of functions

Functions are defined by the properties in the upper left and lower right corners of our square. With onto functions we cover the upper row and the lower right corner, and with one-one functions the lower row and the upper left corner. A relation R ⊆ D × C satisfies the properties in the upper right and lower left corner iff the converse relation R ← ⊆ C × D is a function. Our square displays the duality between the onto and one-one properties, as well as the way how these properties are connected with functionality. Each corner of the square is “one quarter” of a bijection, i.e. one-to-one correspondence. The notion of function involves half of these corners in a diagonal way. An explanation for this judicious choice is given in § 2.4.4 below, when we talk of adjunction.

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§ 2.4.3. Cancellability of relations Let us now consider how the properties from the square are connected with cancellation properties for relations in relational composition. A relation R ⊆ D × C may satisfy the property (right-cancellable) S1 ⊆ S2,

for every S1 and S2, if S1 ° R ⊆ S2 ° R, then

where S1 ⊆ C × A and S2 ⊆ C × A for some set A, or the property (left-cancellable) S1 ⊆ S2,

for every S1 and S2, if R ° S1 ⊆ R ° S2, then

where S1 ⊆ A × D and S2 ⊆ A × D for some set A. Note that (rightcancellable) and (left-cancellable) are equivalent, respectively, to the properties obtained by replacing ⊆ in them by = (to show that, we may use (S1 ∪ S2) ° R = (S1 ° R) ∪ (S2 ° R) and R ° (S1 ∪ S2) = (R ° S1 ) ∪ (R ° S2 ); with ∪ replaced by ∩ we have the inclusions from left to right of these two distributions, but the converse inclusions may fail). Since for every relation R we have R ⊆ R ° R←° R , it is easy to verify that (right-cancellable) implies (right-total), but for the converse implication we only have that the conjunction of (rightunique) and (right-total) implies (right-cancellable); neither (rightunique) alone nor (right-total) alone does so. (Let D = {d}, C = {c1, c2} and A = {a}; then for R = {(d, c2)}, S1 = {(c1, a)} and S2 = Ø, we have that (right-unique) holds, while neither (right-total) nor (right-cancellable) does, and for R = {(d, c1), (d, c2)}, S1 = {(c1, a)} and S2 = {(c2, a)}, we have that (right-total) holds, while neither (right-unique) nor (right-cancellable) does hold.) We also have that the conjunction of (right-unique) and (left-cancellable) implies (leftunique), whereas (left-cancellable) alone does not (provided A is allowed to be empty). Of course, we obtain something quite analogous if in all these implications we replace everywhere “right” by “left” and “left” by “right”.

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So if R is a function, then (right-cancellable) is equivalent to (right-total) and (left-cancellable) is equivalent to (left-unique), but if R is not a function, these equivalences may fail. § 2.4.4. Function and adjunction Finally, let us try to justify the choice of properties from the square that enter into the definition of function. For R ⊆ D × C a relation, A a subset of D and B a subset of C, let R(A) be the set {y ∈ C | for some x ∈ A, xRy} and R←(B) the set {x ∈ D | for some y ∈ B, xRy}. If P(X) is the power set of a set X, then for every relation R ⊆ D × C, we have two functions R: P(D) → P(C) and R←: P(C) → P(D), monotonic with respect to ⊆ . We can easily verify that (left-total) is equivalent to

(γ)

for every A ⊆ D, A ⊆ R←(R(A)),

while (right-unique) is equivalent to (ϕ )

for every B ⊆ C, R(R←(B)) ⊆ B.

On the other hand, (γ) is equivalent to the left-to-right implication and (ϕ) to the right-to-left implication of the equivalence (∗)

for every A ⊆ D and every B ⊆ C, R(A) ⊆ B iff A ⊆ R←(B).

So, R and R← establish a covariant Galois connection between 〈P(D), ⊆ 〉 and 〈P(C), ⊆ 〉 iff R ⊆ D × C is a function. In more general terms, for the preorders 〈P(D), ⊆ 〉 and 〈P(C), ⊆ 〉 understood as categories (objects are subsets of D and C, and arrows exist between these objects whenever inclusion obtains), the functors R and R← together with the natural transformations induced by (γ) and (ϕ) make an adjunction, where R is left-adjoint and R← right-adjoint, the natural transformations of (γ) and (ϕ) being the unit and counit of the adjunction. We have this adjunction if and only if R ⊆ D × C is a function. (The if part of this equivalence is stated in [Mac Lane 1971, p. 94].) For every relation R ⊆ D × C we have that

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(∗∗) for every A ⊆ D and every B ⊆ C, R(A) ⊆ B iff A ⊆ D - R←(C B). (I am indebted to Aleksandar Lipkovski for having drawn my attention to (∗∗) by the equivalent form “R←(B) ⊆ A iff B ⊆ C - R(D A)”.) We also have that R ⊆ D × C is a function iff (∗∗∗)

for every B ⊆ C, R←(B) = D - R←(C - B).

So, underlying the Galois connection of (∗) there is a Galois connection of wider scope, but less pleasing. (The equivalence (∗∗) is implicitly present in temporal logic through the connection between future necessity and past possibility. The equality (∗∗∗) is also to be found in modal logic, when the functionality of the accessibility relation of Kripke models makes necessity and possibility coincide.) The equivalence “R ⊆ D × C is a function iff (∗)” can hardly serve as an alternative definition of the notion of function, since this notion is presupposed in the definitions of the mappings, or functors, R and R←. However, the adjunction in this equivalence may help to explain why the notion of function, rather than some other notion (for example, the notion of partial function, without left totality, or the notion of onto function, with right totality), is so important in mathematics. Conversely, if we are already convinced of the importance of the notion of function—as we should be—our equivalence may explain why Galois correspondence and adjointness are important.

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CHAPTER 3

NATURAL TRANSFORMATIONS

§ 3.1. ANTECEDENTAL AND CONSEQUENTIAL TRANSFORMATIONS We shall now introduce two notions that will replace the notion of natural transformation for our formulation of adjunction adapted to cut elimination. Let M and N be graph-morphisms from a graph G to a graph H. An antecedental transformation τa from M to N is a family of functions τaA, indexed by the objects A of G, that to an arrow f : NA → C of H assign the arrow τaA f : M A → C of H (note that, contravariantly, N is in the argument and M in the value). If H is the graph of a deductive system 〈H, 1, ° 〉, then the antecedental transformation τa is natural iff the following equalities hold in H for every arrow f : NA → C of H and every arrow h : A → B of G: (τ a ) (nat τa)

τaA f = f ° τaA1N A , τaANh = τaB1N B ° M h.

A consequential transformation τc from M to N is a family of functions τcA, indexed by the objects A of G, that to an arrow g : C → M A of H assign the arrow τcAg : C → NA of H. If H is the graph of a deductive system 〈H, 1, ° 〉, then the consequential transformation τc is natural iff the following equalities hold in H for every arrow g : C → M A of H and every arrow h : A → B of G:

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(τ c ) (nat τc)

τcAg = τcA1M A ° g, τcBM h = Nh ° τcA1M A.

(According to (τc), the function τcA is a solidifiable left-invariable function from the left cone V(M A) to the left cone V(NA); see § 1.9.4.) Every natural antecedental transformation τa from M to N with the definition τA =def τaA1N A

gives a natural transformation from M to N. Conversely, every natural transformation τ from M to N that satisfies the following instance of (cat 1 left): τA = 1N A ° τA

gives a natural antecedental transformation from M to N with the definition τaA f =def f ° τA.

These two notions can be shown equivalent. For that we would have to introduce the appropriate morphisms (cf. the next section) and demonstrate an equivalence of categories, which would in fact be an isomorphism of categories. The notions of natural consequential transformation and natural transformation that satisfies the following instance of (cat 1 right): τA = τA ° 1M A

are equivalent, according to the definitions τA =def τcA1M A , τcAg =def τA ° g.

(Note that the second of these definitions says that τcA is the image of τA under the left compositional lifting L, that is, τcA is LτA, while τA is the image of the solidifiable left-invariable function τcA under the

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grounding graph-morphism G, that is τA is G(τcA); see §§ 1.9.2-4. The presence of (cat 1 right) is explained by Lemma 4 of § 1.9.4. Likewise, τaA is the image of τA under right compositional lifting.) We also have that the notions of natural antecedental transformation that satisfies the following instance of (cat 1 right): τaA1N A = τaA1N A ° 1M A

and natural consequential transformation that satisfies the following instance of (cat 1 left): τcA1M A = 1N A ° τcA1M A

are equivalent, according to the definitions τcAg =def τaA1N A ° g, τaA f =def f ° τcA1M A.

Of course, when 〈H, 1, ° 〉 satisfies (cat 1 left) and (cat 1 right), and, a fortiori, when it is a category, we need not mention the instances of (cat 1 left) and (cat 1 right) we had above; then the notions of natural transformation, natural antecedental transformation and natural consequential transformation are all equivalent. It is easy to check that if 〈H, 1, ° 〉 is a category, then the equalities a (τ ) and (nat τa), which we used to define natural antecedental transformations, are equivalent to the equalities (τa 1)

τaA( f2 ° f1) = f2 ° τaA f1,

(τa 2)

τaA( f ° Nh) = τaB f ° M h,

while the equalities (τc) and (nat τc), which we used to define natural consequential transformations, are equivalent to the equalities (τc 1)

τcA(g2 ° g1) = τcAg2 ° g1,

(τc 2)

τcB( M h ° g) = Nh ° τcAg.

Mac Lane mentions in [1971, p. 19, Exercise 5] a formulation of natural transformation analogous to our natural antecedental and

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consequential transformations; however, in this formulation one finds operations on arrows that are simultaneously antecendental and consequential transformations. § 3.2. FORMATIONS, FORMATORS AND NATURAL FORMATIONS A structure made of two categories, two functors from the first to the second category and a natural transformation from one of the functors to the other will be called a natural formation. We shall define below (in § 3.4) the notion of free natural formation, and a free natural transformation will be the natural transformation of a free natural formation. We shall consider the notion of natural formation in its consequential variant: i.e., the natural transformation will be represented by a natural consequential transformation. It should be clear that equivalent notions can be obtained with a natural transformation or a natural antecedental transformation. Before introducing formally the notion of natural formation, we introduce a more general notion we call formation, which covers structures like natural formations, but without the required equalities between arrows (so a formation is to a natural formation what a deductive system is to a category, and what a graph-morphism is to a functor). Formations will also be considered in their consequential variant. Let 〈A, 1, ° 〉 and 〈B, 1, ° 〉 be deductive systems. Next, let F and G be graph-morphisms from A to B and let τc be a consequential transformation from F to G. Then we say that 〈A, B, F, G, τc〉 is a formation. To simplify the notation, we don’t mention the identities and compositions of 〈A, 1, ° 〉 and 〈B, 1, ° 〉, taking them for granted. (By replacing τc by a transformation τ or an antecedental transformation τa, we would obtain other, equivalent, notions of formation.) The appropriate morphisms between formations will be called formators. A formator from a formation 〈A, B, F, G, τc〉 to a formation 〈A', B ', F', G', τc'〉 is a pair (NA, NB) such that NA is a functor from the

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deductive system A to the deductive system A', and NB a functor from the deductive system B to the deductive system B '; moreover, the following naturalness equalities hold: NBF = F'NA, NBG = G'NA, c c NBτ Ag = τ 'NAANBg. A formator (NA, NB) is a formation isomorphism iff both NA and NB are isomorphisms of deductive systems. A notion slightly more general than our notion of formation is obtained by assuming for A that it is only a graph, and not necessarily the graph of a deductive system. The definition of formator has then to be adapted by requiring that NA be only a graph-morphism. However, we need the stricter notion we have introduced for our cutelimination results. A formation 〈A, B, F, G, τc〉 is a natural formation iff (i) 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories; (ii) F and G are functors; (iii) τc is a natural consequential transformation. § 3.3. FREE FORMATIONS In § 1.6 we defined the free deductive system generated by a graph with the help of the linguistic material, i.e. object terms and arrow terms, used for designating the objects and arrows of this deductive system. In § 2.1 we proceeded analogously to define the free graphmorphism between deductive systems generated by a pair of graphs. The notion of free formation P generated by a pair of graphs (G, H), which we shall now define, will also be linguistic in this sense. Let 〈A, 1, ° 〉 be the free deductive system generated by G. Next we define the deductive system 〈B, 1, ° 〉. We first introduce atomic symbols in one-to-one correspondence with the objects of H; these symbols are called the generative object terms of B. As before, we don’t specify what exactly these symbols are: it is enough to know

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that they exist. The object terms of B are defined inductively as follows: (G) every generative object term of B is an object term of B; (1) if A is an object term of A, then FA and GA are object terms of B. Then we define the arrow terms of B. Every such arrow term will have a unique type, which is a pair (B1, B2) of object terms of B. To indicate that an arrow term g of B has the type (B1, B2) we write g : B1 → B2. We introduce atomic symbols in one-to-one correspondence with the arrows of H; these symbols are called the generative arrow terms of B. As with object terms, we don’t specify the exact form of these symbols. If a generative arrow term of B stands for an arrow of H of type B1 → B2, its type is made of the pair of generative object terms that stand for B1 and B2. Then we define inductively the arrow terms of B by the following clauses: (G) every generative arrow term of B is an arrow term of B; (1) if B is an object term of B, then 1B : B → B is an arrow term of B; (2) if g1: B1 → B2 and g2: B2 → B3 are arrow terms of B, then g2 ° g1: B1 → B3 is an arrow term of B; (3) if f : A1 → A2 is an arrow term of A, then F f : FA1 → FA2 and G f : GA1 → GA2 are arrow terms of B; (4) if A is an object term of A and g : B → FA is an arrow term of B, then τcAg : B → GA is an arrow term of B; The object terms of B make the objects and the arrow terms of B the arrows of a graph B, with the obvious source and target functions. Moreover, 〈B, 1, ° 〉 is a deductive system, and 〈A, B, F, G, τc〉 is a formation. This is the free formation P generated by (G, H). (Without

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τc, what we have constructed would amount to a free pair of graphmorphisms between deductive systems generated by a pair of graphs.) The formation P is free in the following sense. Consider the generative graph-morphism pair (HG, HH) from (G, H) to (A, B ), where HG is the generative graph-morphism from G to A (defined as in § 1.6), while HH assigns to every object of H the corresponding generative object term of B and to every arrow of H the corresponding generative arrow term of B (so HH amounts to inclusion, both on objects and on arrows). The free formation P and the generative graph-morphism pair (HG, HH) have the following universal property:

For every formation P' = 〈A', B ', F', G', τc'〉 and every graphmorphism pair (MG, MH) from (G, H) to (A', B '), there is a unique formator (NA, NB ) from P to P' such that MG = NAHG and MH = NB HH. This universal property characterizes P up to formation isomorphism. The formation P is the image of (G, H) under a left adjoint to the forgetful functor from the category of formations (with formators as arrows) to the category of pairs of graphs (with graph-morphism pairs as arrows). This forgetful functor assigns to a formation 〈C, D, F, G, τc〉 the pair of graphs (C, D). § 3.4. FREE NATURAL FORMATIONS We shall first define the notion of free natural formation P* generated by a formation P. Let P be 〈A, B, F, G, τc〉 and let 〈A*, 1, ° 〉 be the free category generated by the deductive system 〈A, 1, ° 〉 of P. Then consider the categorial equivalence relations ≡ on the arrows of B (in the sense of § 1.7), which in addition to the congruence law of categorial equivalence relations satisfy also the following congruence laws:

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if [ f1] = [ f2] in A*, then F f1 ≡ F f2 and G f1 ≡ G f2, if g1 ≡ g2, then τcAg1 ≡ τcAg2, provided τcAg1 and τcAg2 are defined; these equivalence relations satisfy also the conditions derived from the equalities of (ii) and (iii) of the definition of natural formation in § 3.2 by replacing = with ≡ (these are the equalities (fun 1) and (fun 2) for F and G, plus (τc 1) and (τc 2); conditions derived from the categorial equalities of (i) are satisfied because we have categorial equivalence relations). Let us call such equivalence relations natural. It is straightforward to check that if we take the intersection ≡∩ of all the natural equivalence relations, we obtain again a natural equivalence relation. Then for every arrow g of B let [g] be the equivalence class {g '| g ≡∩ g '}, and let [ f ] be an arrow of A*. Then we define the following: 1B =def [1B ],

[g2] ° [g1] =def [g2 ° g1], F[ f ] =def [ F f ], G[ f ] =def [ G f ], τcA[g] =def [ τcAg ].

The objects of B and the equivalence classes [g], together with these definitions, make the category 〈B*, 1, ° 〉. It would be more precise if we distinguished notationally the F, G and τc of B* from those of B, but it is more practical if we don’t do so. Then 〈A*, B*, F, G, τc〉, where F, G and τc are not those of B, but those of B* we have just defined, is a natural formation. This is the free natural formation P* generated by P. Consider now the generative formator (HA, HB) from P to P*, where HA and HB are the identity functions on objects, HA assigns to every arrow f of A the arrow [ f ] of A*, and HB assigns to every arrow g of B the arrow [g] of B*. The free natural formation P* and the generative formator (HA, HB) have the following universal property:

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For every natural formation P' and every formator (MA, MB) from P to P' there is a unique formator (NA, NB) from P* to P' such that MA = NAHA and MB = NBHB. This property characterizes P* up to formation isomorphism. The natural formation P* is the image of P under a left adjoint to the forgetful functor from the category of natural formations (with formators as arrows) to the category of formations (with formators as arrows). The free natural formation P* that is generated by the free formation P generated by a pair of graphs (G, H) will be called the free natural formation P* generated by (G, H). (Without τc, what we have now constructed would amount to a free pair of functors between categories generated by a pair of graphs.) For the free natural formation P* generated by (G, H), the generative arrows of A* and B* are the arrows [ f ] and [g] such that f and g are generative arrow terms of, respectively, the graph A and the graph B of the free formation P generated by (G, H). We designate these generative arrows by the generative arrow terms f and g of A and B; then we can use the object terms and arrow terms of A and B to designate the objects and arrows of A* and B*. Consider now the generative graph-morphism pair (HG*, HH*) from (G, H) to (A*, B*), where HG* is the generative graph-morphism from G to A* (defined as in § 1.7), while HH* assigns to every object of H the corresponding object of B* and to every arrow of H the corresponding generative arrow of B*. The graph-morphism HH* amounts to inclusion, both on objects and on arrows, and it is faithful. The free natural formation P* generated by (G, H) has the following universal property involving (HG*, HH*), quite analogous to the universal property of the free formation P generated by (G, H), which we have considered in the preceding section:

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For every natural formation P' = 〈A', B ', F', G', τc'〉 and every graph-morphism pair (MG, MH) from (G, H) to (A', B '), there is a unique formator (NA, NB) from P* to P' such that MG = NAHG* and MH = NB HH*. This universal property characterizes P* up to formation isomorphism. The natural formation P* is the image of (G, H) under a left adjoint to the forgetful functor from the category of natural formations (with formators as arrows) to the category of pairs of graphs (with graph-morphism pairs as arrows). This forgetful functor assigns to a natural formation 〈C, D, F, G, τc〉 the pair of graphs (C, D). If (HG, HH) is the generative graph-morphism pair from (G, H) to (A, B), where A and B come from the free formation P generated by (G, H), and (HA, HB) is the generative formator from P to P*, then it is clear that for the generative graph-morphism pair (HG*, HH*) from (G, H) to (A*, B*) we have HG* = HAHG and HH* = HBHH, and we could derive the universal property for (HG*, HH*) from the universal properties for (HG, HH) and (HA, HB). § 3.5. CUT ELIMINATION WITH FREE NATURAL TRANSFORMATIONS We shall now show that for the free natural formation 〈A*, B*, F, G, τc〉 generated by an arbitrary pair of graphs Cut Disintegration holds in the category B*. That Cut Disintegration holds for A* follows from Chapter 1. The assumptions made for natural formations are all necessary, as well as sufficient, for having this Cut Disintegration, save for (fun 1), which happens to be superfluous, as in the preceding chapter. Cut Disintegration can also be proved for free natural formations generated by arbitrary graphs in the antecedental variant, where τc is replaced by τa. In this proof we would proceed in a manner quite analogous to what we have below.

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With the notion of free natural formation based on an ordinary natural transformation τ from F to G, Cut Disintegration is excluded because of cuts like those in τB ° F f and G f ° τA (these two arrow terms are equal by (nat)). We could deal with these cuts only when the graph G generating the free category A* is arrowless, so that this category is discrete (in that case (fun 1) would serve). § 3.5.1. Cut Disintegration in free natural formations Let 〈A*, B*, F, G, τc〉 be the free natural formation generated by (G, H). Every equality g1 = g2 that holds for the arrow terms g1 and g2 of B* may be derived as follows. It may be axiomatic, which means that it is an instance of g = g, (cat 1 right), (cat 1 left), (cat 2), (fun 1), (fun 2), (τc 1) or (τc 2); else it is derived from axiomatic or previously derived equalities by the rules of symmetry or transitivity of equality, or the rule (congr ° ) of § 1.8.1, or the rule (congr F) of § 2.3.1 for F and G, or the additional congruence rule (congr τc)

from g1 = g2 in B*, infer τcAg1 = τcAg2 in B*

provided τcAg1 and τcAg2 are defined. We must then extend the definition of linked cuts from § 1.8.1 and § 2.3.1 to equalities among arrow terms of B*. For that we have to take into account the following additional cases. If f = g is an instance of (τc 1) or (τc 2), then the n-th cut of f is linked to the n-th cut of g. The rule (congr τc) just preserves the links of the premise. The cuts linked in the equalities of the free category A* are defined as in § 1.8.1. Note that, as in § 2.3.1, cuts of arrow terms of A* are linked to cuts of arrow terms of B*. We just take over the other definitions of § 1.8.1, save that in the definition the degree of an arrow term of B* we also count, besides occurrences of 1 and ° , also occurrences of F (applied to arrow terms), G (applied to arrow terms) and τc. In these

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definitions we take on an equal footing the cuts of A*and the cuts of B*. We can now prove Cut Disintegration for the arrow terms of B*. Proof of Cut Disintegration for B*. We proceed as in the proof of § 1.8.1. So, in an arrow term of B*, take a nonatomic cut whose subterm is f2 ° f1. Then we have the following cases in addition to those considered in § 1.8.1 and § 2.3.1. In case (3), where f1 is Fg, we have the following additional subcase.

(3.4) f2 is τcA f. Then τcA f ° f1 = τcA( f ° f1) by (τc 1), and the degree of the main cut of f ° f1 is strictly smaller than the degree of the main cut on the left-hand side linked to it. The case where f2 is Gg ' is excluded. The case where f1 is Gg is treated quite analogously to case (3). We also have the following additional case. (4) f1 is τcA f. Then we have the following subcases. (4.1) - (4.2) f2 is 1GA or f2" ° f2'. These cases are treated like cases (1) and (2) of the proof in § 1.8.1, using (cat 1 left) and (cat 2). (4.3) f2 is Gg . Then Gg ° τcA f = τcB(Fg ° f ) by (τc 2), and the degree of the main cut of Fg ° f is strictly smaller than the degree of the main cut on the left-hand side linked to it. (4.4) f2 is τcA f. This case is treated analogously to (3.4) above. (4.G) f2 is a generative arrow term of B*. This is excluded because f2 must have a source of the form GA. In case (G), where f1 is a generative arrow term of B*, we have the additional subcase where f2 is τcA f, which is treated analogously to (3.4) above. The present proof presupposes that the proof of Cut Disintegration for arrow terms of A* has been accomplished. q.e.d.

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§ 3.5.2. Necessary conditions for Cut Disintegration in free natural formations In the proof of Cut Disintegration in the preceding section we have again not used the equality (fun 1) from the definition of natural formation: so this equality, which is independent from the other equalities of natural formations, is not necessary. The necessity of (cat 1 right), (cat 1 left) and (cat 2) is demonstrated as in § 1.8.2, while the necessity of (fun 2) is demonstrated as in § 2.3.2. The necessity of (τc 1) and (τc 2) is shown when the graph G has at least one arrow, so that in A* we have at least one generative arrow term f : A → B. Then we can neither atomize nor eliminate the cut of τcB 1FB ° F f in the absence of (τc 1). We can also neither atomize nor eliminate the cut of G f ° τcA 1FA in the absence of (τc 2). If G is arrowless, so that A* is discrete, then f in these examples would stand for an identity arrow, and neither (τc 1) nor (τc 2) would be needed to eliminate these two cuts: we could then use (fun 1), which is otherwise superfluous.

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CHAPTER 4

ADJUNCTIONS

§ 4.1. DEFINITIONS OF ADJUNCTION In this introductory part of the present chapter we survey the standard definitions of adjunction. However, rather than simply rehash familiar matters, we present also two presumably new definitions of this notion. One is a definition that does not economize on primitives. It takes as primitive notions the two adjoint functors, F and G, and both the natural transformations that are the counit and unit of the adjunction and the two bijections between the hom-sets A(FB, A) and B(B, GA). Usually, if the counit and unit are primitive, the bijections are defined, and vice versa. Having both kinds of notions primitive, together with the adjoint functors, enables us to formulate the specific equalities between arrows one finds in adjointness as a series of equalities defining one of these notions in terms of two remaining notions. These definitional equalities make a regular pattern, which may clarify standard definitions of adjunction. We shall compare this uneconomical, but regular and simple, definition to standard definitions of adjunction (like those that may be found in Mac Lane’s book [1971, IV]), and show that the notions defined are equivalent. Among the standard definitions we favour those that, like the uneconomical definition, are equationally presented. We also envisage defining adjunction in a more general kind of context—in particular, a context where F and G may fail to be functors because they don’t satisfy (fun 1), but only (fun 2). That is, F and G are only semifunctors (cf. § 2.3.2 and § 4.1.4 below). In § 4.1.7 we consider the other nonstandard definition of adjunction. This one is, on the contrary, an economical definition,

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where only the functions F and G on objects and the bijections between the hom-sets A(FB, A) and B(B, GA) are primitive. So neither of the adjoint functors F and G is taken as primitive. This economical definition simplifies one of the standard definitions. § 4.1.1. Primitive notions in adjunction Let A and B be two graphs. The objects of A will be designated by A, A1, A2, …, and the arrows of A by f, f1, f2, …, while the objects of B will be designated by B, B1, B2, …, and the arrows of B by g, g1, g2, … To refer neutrally to the arrows of either A or B we shall use later h, h1, h2, … Let F be a graph-morphism from B to A and G a graph-morphism from A to B. When we need it for emphasis, we shall write F a and Ga for the functions on arrows, and Fo and Go for the functions on objects, of the graph-morphisms F and G. However, in most cases we will, as usual, omit these superscripts. Let ϕ be a transformation from the composite graph-morphism FG to the identity graph-morphism IA and γ a transformation from the identity graph-morphism IB to the composite graph-morphism GF. (Remember that, as defined in § 1.3, a transformation is a family of arrows like a natural transformation for which we don’t assume (nat).) Finally, for every pair of objects (A, B) (where, according to our convention, A is from A and B is from B), let ΦB,A be a function assigning to an arrow g : B → GA of B the arrow ΦB,Ag: FB → A of A, and let ΓB,A be a function assigning to an arrow f : FB → A of A the arrow ΓB,A f : B → GA of B. We denote by Φ the family of all the functions ΦB,A and by Γ the family of all the functions ΓB,A; we call the functions in these families the seesaw functions. Consider now the following six notions we have just introduced: the functions on arrows F a and Ga, the transformations ϕ and γ, the families of seesaw functions Φ and Γ.

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If 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are deductive systems, each of these notions can be defined in terms of two other notions from the list (with the help of the identities and compositions of 〈A, 1, ° 〉 and 〈B, 1, ° 〉) by the following equalities: for g : B1 → B2 (F a) (ϕ )

for f : A1 → A2

Fg = ΦB1,FB2(γB2 ° g),

(Ga)

G f = ΓGA1,A2( f ° ϕA1),

ϕA = ΦGA,AG1A,

(γ)

γB = ΓB,FBF1B,

(Γ )

for f : FB → A ΓB,A f = G f ° γB.

for g : B → GA (Φ) ΦB,Ag = ϕA ° Fg,

The definitional dependences among these notions can be read off from the hexagon in Figure 4. F

a

γ

Φ

Γ

ϕ

Ga

Fig. 4. Definitional dependences in adjunction

The notion in each vertex is definable in terms of the two notions in the neighbouring vertices on the left and on the right. For example, F a is definable in terms of γ and Φ, while Φ is definable in terms of F a and ϕ, etc. On the left-hand side of the hexagon we have F and its Greek correlates, while on the right-hand side we have G with its Greek correlates. Vertices on the big, undrawn, diagonals have labels of the same type: (F a, Ga), (Φ, Γ) and (ϕ, γ).

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The small, dotted, diagonals are drawn to indicate possible choices of primitives, in terms of which all the six notions can be defined. In Table 1, the sign + indicates which notions are taken as primitive by the choice named in the leftmost column. Fa

Ga ϕ

γ

Γ

Φ

+

+

+

+

+

+

rectangular || +

+

+

+

rectangular \\ +

+

+

+

+

+

hexagonal

rectangular //

+

triangular ► +

+

triangular ◄

+

+

+ +

+

TABLE 1. Choices of primitives in adjunction

Besides these choices, there are six uneconomical pentagonal choices, with five primitives, and six more uneconomical choices with four primitives, obtained by adding a vertex to one of the triangular choices (so, altogether, we have 18 choices). What can be said about these additional uneconomical choices should be easy to infer from what is said below about the rectangular and triangular choices; so we shall not consider them separately. (In § 4.1.7 below, we shall find one more, very economical, choice with only Φ and Γ primitive; however, this choice is based on slightly different definitional equalities.) The hexagonal definitional pattern above becomes even more regular if we take into account the identities and compositions of the deductive systems A and B. Suppose that these deductive systems are formulated in the style of § 1.9.6, so that the first has 1A, LA and RA,

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while the second has 1B, LB and RB, as primitives. For the composition of A we have f2 ° f1 = LAf2( f1) = RAf1( f2)

and analogously for the composition of B. Then the definitional equalities above become (F a)

F g = ΦLBγ g,

(Ga)

G f = ΓRAϕ f,

(ϕ )

ϕ = ΦG1A,

(γ)

γ = ΓF1B,

(Φ )

Φg = LAϕF g,

(Γ )

Γ f = RBγG f,

where, to make matters clearer, we have omitted parentheses and subscripts referring to objects. Our hexagonal figure enriched with these additional notions involved in the definitions looks as in Figure 5. F

a

γ LB

Φ

RB

LA `A

ϕ

`B

Γ

RA Ga

Fig. 5. Definitional dependences in adjunction (revisited)

§ 4.1.2. Hexagonal adjunction The hexagonal choice of primitives of the preceding section is interesting because we can define adjunction as follows. The conditions

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〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F and G are functors, ϕ and γ are natural transformations, Φ and Γ are families of seesaw functions, (F a), (Ga), (ϕ), (γ), (Φ) and (Γ) hold

are satisfied iff the functors F and G are adjoint, F being left adjoint and G right adjoint. The natural transformations ϕ and γ are respectively the counit and unit of the adjunction (often written ε and η). In the next sections we shall verify that this notion of adjunction is indeed equivalent to the more usual ones, behind which stand more economical choices of primitives from the table above. Note that if we replace the equalities (ϕ) and (γ) by the equalities (ϕ°)

f ° ϕA1 = ΦGA1,A2G f,

(γ°)

γB2 ° g = ΓB1,FB2Fg,

for f : A1 → A2 and g : B1 → B2, then the condition that the transformations ϕ and γ are natural becomes redundant. The equalities (ϕ°) and (γ°) are an immediate consequence of (Φ), (Γ) and (nat) for ϕ and γ. On the other hand, these two equalities yield (ϕ) and (γ) in the presence of (cat 1 left) and (cat 1 right). However, (ϕ°) and (γ°) are not exactly definitions of the transformations ϕ and γ, but rather definitions of composing with ϕ on the right and γ on the left (i.e. of RAϕ and LBγ). § 4.1.3. Rectangular || adjunction Suppose 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F a and Ga satisfy (fun 2), (Φ) and (Γ) hold.

Then the equalities (F a), (Ga), (ϕ) and (γ) are interderivable with the equalities

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(ϕγF)

ϕFB ° FγB = F1B,

(ϕγG)

GϕA ° γGA = G1A,

(ϕ 1 )

ϕA ° FG1A = ϕA,

(γ1)

GF1B ° γB = γB,

from which Φ and Γ are absent. Let us first derive the latter equalities from the former. For (ϕγF) we have ϕFB ° FγB = ΦB,FBγB, by (Φ) = F1B, by (cat 1 right) and (F a).

For (ϕ1) we have ϕA ° FG1A = ΦGA,AG1A, by (Φ) = ϕA, by (ϕ).

We proceed analogously for (ϕγG) and (γ1). Conversely, we derive (F a) as follows: ΦB1,FB2(γB2 ° g) = (ϕFB2 ° FγB2) ° Fg, by (Φ), (fun 2) and

(cat 2) = F1B2 ° Fg, by (ϕγF) = Fg, by (fun 2) and (cat 1 left). For (ϕ) we use (Φ) and (ϕ1), and we proceed analogously for (Ga) and (γ). In the standard definition of adjunction with the rectangular || choice of primitives we have that 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F and G are functors, ϕ and γ are natural transformations, Φ and Γ may be defined by (Φ) and (Γ), (ϕγF) and (ϕγG) hold.

In fact, instead of the equalities (ϕγF) and (ϕγG) we usually have the equalities obtained from them by replacing the right-hand sides with 1FB and 1GA, respectively. These other equalities clearly amount to (ϕγF) and (ϕγG) in the presence of (fun 1) for F a and Ga.

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With this standard definition of adjunction, the equalities (ϕ1) and (γ1) follow either from (fun 1) for F a and Ga, or from the assumption that ϕ and γ are natural transformations (together with (cat 1 right) and (cat 1 left)). This is enough to conclude that the notion of adjunction of the preceding section is indeed equivalent to the standard notion with the rectangular || choice of primitives. § 4.1.4. Rectangular \\ adjunction Suppose 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F a and Ga satisfy (fun 2), (ϕ) and (γ) hold.

Then the equalities (F a), (Ga), (Φ), (Γ) and (nat) for ϕ and γ are interderivable with the following equalities (in which, since we have (cat 2), we don’t write parentheses in compositions, and the subscripts of Φ and Γ are omitted so as not to encumber notation excessively; these subscripts can be recovered from the context): (ΦΓ) (ΓΦ) (ΦΦ) (ΓΓ) (ΦF) (ΓG)

Φ(G f3 ° Γf2 ° g1) = f3 ° f2 ° Fg1, Γ( f3 ° Φg2 ° Fg1) = G f3 ° g2 ° g1, Φ(G f3 ° g2 ° g1) = f3 ° Φg2 ° Fg1, Γ( f3 ° f2 ° Fg1) = G f3 ° Γf2 ° g1, Φg ° F1B = Φg, G1A ° Γf = Γf.

In these equalities ϕ and γ don’t occur. Equalities like these were considered in [Hayashi 1985] and [Hoofman 1993], which deal with notions of adjoint semifunctors, i.e. graph-morphisms satisfying only (fun 2), and not necessarily also (fun 1). (Note that at the beginning of the preceding section we also didn’t assume (fun 1) to find equalities without Φ and Γ equivalent to (F a), (Ga), (ϕ) and (γ).)

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In the standard definition of adjunction with the rectangular \\ choice of primitives we have that 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F and G are functors, Φ and Γ are families of seesaw functions, ϕ and γ may be defined by (ϕ) and (γ), the following equalities hold:

(ΦΓ ') (ΓΦ') (ΦΦ')

ΦΓf = f, ΓΦg = g, Φ(g2 ° g1) = Φg2 ° Fg1,

(ΦΦ")

Φ(G f ° g) = f ° Φg.

The equalities (ΦΦ') and (ΦΦ") can be replaced by (ΓΓ ')

Γ( f2 ° f1 ) = G f2 ° Γf1,

(ΓΓ ")

Γ( f ° Fg) = Γf ° g.

It is easy to see that, due to the presence of (fun 1) for F a and Ga, the equalities (ΦΓ '), (ΓΦ'), (ΦΦ') and (ΦΦ") amount to (ΦΦ), (ΓΓ), (ΦΓ), (ΓΦ), (ΦF) and (ΓG). In this standard definition of rectangular \\ adjunction, (ΦΦ') can be replaced by Φg = Φ1GA ° Fg,

an equality that in the presence of (ϕ) and (fun 1) for Ga amounts to (Φ). Analogously, (ΓΓ ') can be replaced by an equality that in the presence of (γ) and (fun 1) for F a amounts to (Γ): Γf = G f ° Γ1FA.

The equalities (ΦΦ') and (ΦΦ") can be replaced by the implication if g2 ° g1 = G f ° g , then Φg2 ° Fg1 = f ° Φg (to show that we use (cat 1 right), (cat 1 left) and (fun 1) for F a and Ga). Analogously, (ΓΓ') and (ΓΓ") can be replaced by the implication

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if f2 ° f1 = f ° Fg, then G f2 ° Γf1 = Γf ° g. With these implications, which are involved in Lawvere’s definition of adjunction as an isomorphism of comma categories (see [Mac Lane 1971, p. 84, Exercise 2, and p. 53]), we abandon, however, the equational style of defining adjunction favoured here. § 4.1.5. Rectangular // adjunction If A and B are deductive systems that satisfy (cat 1 right) and (cat 1 left), and (F a) and (Ga) hold, then it is clear that the equalities (ϕ), (γ), (Φ) and (Γ) are interderivable with the equalities ϕA = ΦGA,AΓGA,AϕA,

γB = ΓB,FBΦB,FBγB,

ΦB,Ag = ϕA ° ΦB,FGA(γGA ° g),ΓB,A f = ΓGFB,A( f ° ϕFB) ° γB,

from which the functions F a and Ga are absent. (The equalities in the first line are instances of (ΦΓ ') and (ΓΦ'), respectively.) However, there doesn’t seem to be a standard definition of adjunction with the rectangular // choice of primitives, which would be based on equalities such as these. Standard definitions take the adjoint functors F and G, or at least one of them, as primitive. In § 4.1.7, we shall consider a definition of adjunction where neither of the functions F a and Ga is primitive. § 4.1.6. Triangular adjunction Suppose 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F a satisfies (fun 2), ϕ satisfies (nat), (Ga), (γ) and (Φ) hold.

Then the equalities (F a), (ϕ), (Γ), (fun 2) for Ga and (nat) for γ are interderivable with the equalities

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(β ) (ΓΓ ")

ϕA ° FΓB,A f = f ° F1B, ΓB1,A( f ° Fg) = ΓB2,A f ° g,

from which Ga, γ and Φ are absent. Note that again we have not assumed (fun 1) for F a (nor for Ga). The equality (β) could be replaced above by ϕA ° F(ΓB,A f ° g) = f ° Fg,

while in the presence of the assumptions that A is a category, that F a satisfies (fun 2) and that (Ga) and (β) hold, the equality (nat) for ϕ is replaceable by ϕA ° F1GA = ϕA.

This last equality follows, of course, from (cat 1 right) and (fun 1) for F a. In the standard definition of adjunction with the triangular ► choice of primitives we have that 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F is a functor and Go is a function on objects, ϕ is an objectual transformation, Γ is a family of seesaw functions, Ga, γ and Φ may be defined by (Ga), (γ) and (Φ), the following equalities hold:

(β')

ϕA ° FΓB,A f = f,

(η )

ΓB,A(ϕA ° Fg) = g.

Remember that an “objectual transformation”, as specified in § 1.3, is like a transformation between functions on objects, instead of graph-morphisms. We didn’t assume that Go belongs to a graphmorphism; so, to be precise, we can say only that ϕ is an objectual transformation from the composite function FG on the objects of A to the identity function on the objects of A.

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Note that in the presence of (Φ), the equalities (β') and (η) can be written as (ΦΓ ') and (ΓΦ'). (The labels “β” and “η” come from the adjunction of cartesian closed categories, where the corresponding equalities are related to β and η conversion in the typed lambda calculus.) It is clear that with (cat 1 right) and (fun 1) for F a the equality (β) amounts to (β'). On the other hand, (ΓΓ "), (cat 1 left) and (Γϕ)

ΓGA,AϕA = 1GA

yield (η), while, conversely, from (β'), (η), (cat 2) and (fun 2) for F a we obtain (ΓΓ "), and from (η), (cat 1 right) and (fun 1) for F a we obtain (Γϕ ). The equality (β') implies in the presence of (Ga) that ϕ satisfies (nat). The equality (η) is replaceable by the implication if ϕA ° Fg = f, then g = ΓB,A f, which together with (β') is tantamount to asserting that there is a unique g such that ϕA ° Fg = f. The definition of adjunction via a solution to a universal arrow problem is based on that (see [Mac Lane 1971, IV.1, p. 81, Theorem 2(iv)]). Since (β') is replaceable by the converse implication, and since we have (Φ), we could assume instead of (β') and (η) the equivalence g = ΓB,A f iff ΦB,Ag = f,

which is another way of assuming (ΦΓ ') and (ΓΦ'). However, with these implications and this equivalence we abandon the equational style of defining adjunction favoured here. For the definition of adjunction with the triangular ◄ choice of primitives we would have completely analogous considerations. § 4.1.7. Seesaw adjunction The rectangular \\ and rectangular // choices of primitives are not minimal for defining adjunction if we change slightly the defining

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equalities (F a), (Ga), (ϕ) and (γ). The transformations ϕ and γ may be defined as follows in terms of Φ and Γ without F a and Ga: (ϕ')

ϕA = ΦGA,A1GA,

(γ')

γB = ΓB,FB1FB,

which serves to transform (F a) and (Ga) into the following definitions of F a and Ga in terms of Φ and Γ without ϕ and γ: (F a ')

Fg = ΦB1,FB2(ΓB2,FB21FB2 ° g),

(Ga ')

G f = ΓGA1,A2( f ° ΦGA1,A11GA1).

We then have a definition of adjunction where 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, Fo and Go are functions on objects, Φ and Γ are families of seesaw functions, F a, Ga, ϕ and γ may be defined by (F a '), (Ga '), (ϕ') and (γ'), the following equalities hold:

(ΦΓ ') (ΓΦ') (ΦΦ'')

ΦΓf = f, ΓΦg = g, Φ(g2 ° g1) = Φg2 ° Φ(Γ1 ° g1)

(with the subscripts of Φ, Γ and 1 omitted). We could replace (ΦΦ'') by (ΓΓ '')

Γ( f2 ° f1 ) = Γ( f2 ° Φ1) ° Γf1.

To verify that this notion of adjunction is equivalent to the usual ones it suffices to show that it is equivalent to the notion with the rectangular \\ choice of primitives of § 4.1.4. For that we have first to check that F a and Ga defined by (F a ') and (Ga ') satisfy (fun 1) and (fun 2). Next, the equalities (ΦΦ'') and (ΓΓ '') amount to the equalities (ΦΦ') and (ΓΓ ') of § 4.1.4 in the presence of (F a ') and (Ga '), while equalities corresponding to (ΦΦ") and (ΓΓ ") are now derivable. Here is a derivation of (ΦΦ"):

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Φ(G f ° g) = ΦΓ( f ° Φ1) ° Φ(Γ1 ° g), by (Ga ') and (ΦΦ'') = f ° Φg, by (ΦΓ '), (ΦΦ''), (cat 1 left) and (cat 2)

(cf. [D. 1996, section 3.1]). This economical definition of adjunction is at the opposite end of the hexagonal definition of § 4.1.2, in which we did not economize on primitives. To prove strictly the equivalences of various notions of adjunction considered here, we would have to introduce the appropriate morphisms between adjunctions and demonstrate equivalences of categories, which would actually be isomorphisms of categories. We shall not do that, however, since this rather straightforward matter would take too much space. We define morphisms between adjunctions in the next section, § 4.2, and in § 5.2.1. § 4.2. JUNCTIONS, JUNCTORS AND ADJUNCTIONS We shall now define a notion of adjunction, equivalent to any of the standard notions, which will be adapted to cut elimination. This notion differs from the standard notion of rectangular || adjunction of § 4.1.3 only by replacing the natural transformation ϕ, i.e. the counit of the adjunction, by a natural antecedental transformation, and the natural transformation γ, i.e. the unit of the adjunction, by a natural consequential transformation. In the ordinary notion of rectangular || adjunction, with the natural transformations ϕ and γ primitive, we cannot eliminate cuts in arrow terms like ϕA ° ϕFGA and γGFB ° γB. Officially, our new notion of adjunction should be called the rectangular || notion of adjunction in the (ac) formulation (this “(ac)” will be explained in § 4.5.5). However, it is the main notion with which we shall work, and, for the time being, we call it simply “adjunction”. In § 4.5.5 we shall consider other notions of rectangular || adjunction, which will be qualified with an epithet, and in that context we shall use such an epithet for our main notion, too. But, if not stated otherwise, until § 4.7 and § 4.8, where we shall consider rectangular \\ and triangular notions, “adjunction” without qualification should be taken as referring to the notion of this section.

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Before introducing our new notion of adjunction, we introduce a more general notion we call junction, which covers structures like adjunction, but without the required equalities between arrows (so a junction is to an adjunction what a deductive system is to a category, what a graph-morphism is to a functor, or what a formation is to a natural formation). Let 〈A, 1, ° 〉 and 〈B, 1, ° 〉 be deductive systems. According to the conventions of § 4.1.1, we continue using the letters A, A1, A2, … for the objects of A, the letters f, f1, f2, … for the arrows of A, the letters B, B1, B2, … for the objects of B, and the letters g, g1, g2, … for the arrows of B. Next, let F be a graph-morphism from B to A and G a graph-morphism from A to B. Finally, let ϕa be an antecedental transformation from the composite graph-morphism FG to the identity graph-morphism IA and γc a consequential transformation from the identity graph-morphism IB to the composite graph-morphism GF. Note that here we can take that ϕa is a single operation on the arrows of A, assigning to an arrow f : A1 → A2 of A the arrow ϕa f : FGA1 → A2 of A (we cannot do that in general with antecedental transformations, because N need not be one-one on objects, as IA is). We can also take that γc is a single operation on the arrows of B, assigning to an arrow g : B1 → B2 of B the arrow γcg : B1 → GFB2 of B. The functions ϕaA and γcB are obtained by restricting the domains of the operations ϕa and γc. Then we say that 〈A, B, F, G, ϕa, γc〉 is a junction. We will sometimes say that this is a junction between A and B, in that order (this order is important). We use later the same form of speaking with adjunctions. To simplify the notation, we don’t mention the identities and compositions of 〈A, 1, ° 〉 and 〈B, 1, ° 〉, taking them for granted (as we did with formations in the preceding chapter). The appropriate morphisms between junctions will be called junctors. We take that a junctor from a junction 〈A, B, F, G, ϕa, γc〉 to a junction 〈A', B ', F', G', ϕa', γc'〉 is a pair (NA, NB) such that NA is a

117

functor from the deductive system A to the deductive system A', and NB a functor from the deductive system B to the deductive system B '; moreover, the following naturalness equalities hold: NAF = F'NB, NAϕa f = ϕa'NA f,

NBG = G'NA, NBγcg = γc'NBg.

A junctor (NA, NB) is a junction isomorphism iff both NA and NB are isomorphisms of deductive systems. A junction 〈A, B, F, G, ϕa, γc〉 is an adjunction iff (i) 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories; (ii) F and G are functors; (iii) ϕa is a natural antecedental transformation and γc a natural consequential transformation, which means that the following equalities hold in A and B: (ϕa 1) ϕa( f2 ° f1) = f2 ° ϕa f1,

(γc 1) γc(g2 ° g1) = γcg2 ° g1,

(ϕa 2) ϕa( f2 ° f1) = ϕa f2 ° FG f1,

(γc 2) γc(g2 ° g1) = GFg2 ° γcg1;

(iv) finally, the following rectangular equalities hold in A and B: (ϕaγcF) ϕa f ° Fγcg = f ° Fg,

(ϕaγcG) Gϕa f ° γcg = G f ° g.

These last two equalities are analogous to the equalities (ϕγF) and (ϕγG) of § 4.1.3 (which in [Mac Lane 1971, p. 83] are called “triangular”—some authors talk in the same context about “quasiinverses”; we don’t call (ϕaγcF) and (ϕaγcG) “triangular” because, first, the corresponding diagrams are not triangular any more, but rather rectangular, and, second, these equalities don’t come from the “triangular” definition of adjunction of § 4.1.6, but from the “rectangular” definition of § 4.1.3). It is straightforward to show that the notion of adjunction we have just introduced is equivalent to the rectangular || notion of adjunction of § 4.1.3, and hence to all other notions of adjunction we have considered in § 4.1. To achieve this precisely we have to check first that, taking ϕa and γc as defined in terms of ϕ and γ, our notion of

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junctor is appropriate for the rectangular || adjunctions of § 4.1.3 (cf. § 5.2.1). Next we should establish an equivalence of categories, which would in fact be an isomorphism of categories. § 4.3. FREE JUNCTIONS The notion of free junction J generated by a pair of graphs (G, H), which we shall now define, will be linguistic, as were the notions of free deductive system from § 1.6, free graph-morphism between deductive systems from § 2.1 and free formation from § 3.3. We first introduce atomic symbols for the objects of G and H, called the generative object terms of, respectively, A and B (these symbols are in one-to-one correspondence with the objects of G and H). As before, we don’t specify what exactly these symbols are: it is enough to know that they exist. Then we define inductively the object terms of A and B by the following clauses: (G) every generative object term of A is an object term of A; every generative object term of B is an object term of B; (1) if B is an object term of B, then FB is an object term of A; if A is an object term of A, then GA is an object term of B. Note that in clause (1) the letters A and B are crossed with each other. Nevertheless, the condition of this clause is such that it is preserved by intersection: namely, if we take all the pairs (Ai , Bi ) satisfying (G) and (1), then (∩i Ai , ∩i Bi ) satisfies (G) and (1) (that (G) is satisfied means that the generative object terms of A are members of Ai , and analogously with B). This is enough to guarantee the correctness of the inductive definition: the sets of object terms of A and B will be, respectively, ∩i Ai and ∩i Bi . Next we define the arrow terms of A and B. Every arrow term of A will have a unique type, which is a pair (A1, A2) of object terms of A. To indicate that an arrow term f of A has the type (A1, A2) we write

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f : A1 → A2, as before. Analogously, every arrow term g of B will have a unique type (B1, B2), where B1 and B2 are object terms of B; this is indicated by g : B1 → B2. We introduce atomic symbols for the arrows of G and H, called the generative arrow terms of, respectively, A and B (these symbols are in one-to-one correspondence with the arrows of G and H). As with object terms, we don’t specify the exact form of these symbols. If a generative arrow term stands for an arrow of G of type A1 → A2, its type is made of the pair of generative object terms that stand for A1 and A2, and similarly for H. Then we obtain the arrow terms of A and B by another inductive definition with a crossed clause:

(G) every generative arrow term of A is an arrow term of A; every generative arrow term of B is an arrow term of B; (1) if A is an object term of A, then 1A : A → A is an arrow term of A; if B is an object term of B, then 1B : B → B is an arrow term of B; (2) if f1: A1 → A2 and f2: A2 → A3 are arrow terms of A, then f2 ° f1: A1 → A3 is an arrow term of A; if g1: B1 → B2 and g2: B2 → B3 are arrow terms of B, then g2 ° g1: B1 → B3 is an arrow term of B; (3) if g : B1 → B2 is an arrow term of B, then Fg : FB1 → FB2 is an arrow term of A; if f : A1 → A2 is an arrow term of A, then G f : GA1 → GA2 is an arrow term of B; (4) if f : A1 → A2 is an arrow term of A, then ϕa f : FGA1 → A2 is an arrow term of A; if g : B1 → B2 is an arrow term of B, then γcg : B1 → GFB2 is an arrow term of B.

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The object terms of A make the objects and the arrow terms of A the arrows of a graph A, with the obvious source and target functions, and analogously with the graph B. Moreover, 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are deductive systems, and 〈A, B, F, G, ϕa, γc〉 is a junction. This is the free junction J generated by (G, H). This junction is free in the following sense. Consider the generative graph-morphism pair (HG, HH) from (G, H) to (A, B ), such that HG assigns to every object of G the corresponding generative object term of A and to every arrow of G the corresponding generative arrow term of A (so HG amounts to inclusion, both on objects and on arrows); HH is determined analogously with respect to H and B. The free junction J and the generative graph-morphism pair (HG, HH) have the following universal property: For every junction J' between A' and B ', and every graphmorphism pair (MG, MH) from (G, H) to (A', B '), there is a unique junctor (NA, NB ) from J to J' such that MG = NAHG and MH = NB HH. This universal property characterizes J up to junction isomorphism. The junction J is the image of (G, H) under a left adjoint to the forgetful functor from the category of junctions (with junctors as arrows) to the category of pairs of graphs (with graph-morphism pairs as arrows). This forgetful functor assigns to a junction between C and D the pair of graphs (C, D). § 4.4. FREE ADJUNCTIONS We shall first define the notion of free adjunction J* generated by a junction J. Let J be 〈A, B, F, G, ϕa, γc〉 and consider the pairs (≡A, ≡B ) such that ≡A is a categorial equivalence relation on the arrows of A (in the sense of § 1.7) and ≡B is a categorial equivalence relation on the arrows of

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B, which in addition to the congruence law of categorial equivalence relations: if f1 ≡A f1' and f2 ≡A f2', then f2 ° f1 ≡A f2' ° f1', if g1 ≡B g1' and g2 ≡B g2', then g2 ° g1 ≡B g2' ° g1', satisfy also the following congruence laws: if g1 ≡B g2, then Fg1 ≡A Fg2, if f1 ≡A f2, then ϕa f1 ≡A ϕa f2,

if f1 ≡A f2, then G f1 ≡B G f2, if g1 ≡B g2, then γcg1 ≡B γcg2;

these equivalence relations satisfy also the conditions derived from the equalities of (ii)-(iv) of the definition of adjunction in § 4.2 by replacing = with ≡A and ≡B , as appropriate (these are the equalities (fun 1) and (fun 2) for F and G, plus (ϕa 1), (ϕa 2), (γc 1), (γc 2), (ϕaγcF) and (ϕaγcG); conditions derived from the categorial equalities of (i) are satisfied because we have categorial equivalence relations). Let us call such pairs (≡A, ≡B ) pairs of adjunctional equivalence relations of J. It is straightforward to check that if we take all the pairs of adjunctional equivalence relations of J, and make the intersection ≡A∩ of all the relations ≡A and the intersection ≡B∩ of all the relations ≡B in these pairs, we obtain again a pair of adjunctional equivalence relations (≡A∩, ≡B ∩). Then for every arrow f of A and every arrow g of B take the equivalence classes [ f ] =def { f '| f ≡A∩ f '}, [g] =def {g '| g ≡B∩ g '}, and define on them the following: 1A =def [1A ],

[ f2] ° [ f1] =def [ f2 ° f1], F[g] =def [ Fg ], ϕa[ f ] =def [ ϕa f ],

1B =def [1B ],

[g2] ° [g1] =def [g2 ° g1], G[ f ] =def [ G f ], γc[g] =def [ γcg ].

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The objects of A and B together with the equivalence classes above make the categories 〈A*, 1, ° 〉 and 〈B*, 1, ° 〉. It would be more precise if we distinguished notationally the F, G, ϕa and γc of A* and B* from those of A and B, but it is more practical if we don’t do so. Then 〈A*, B*, F, G, ϕa, γc〉, where F, G, ϕa and γc are not those of A and B, but those of A* and B* we have just defined, is an adjunction. This is the free adjunction J* generated by J. Consider now the generative junctor (HA, HB) from J to J*, where HA and HB are the identity functions on objects, HA assigns to every arrow f of A the arrow [ f ] of A*, and HB assigns to every arrow g of B the arrow [g] of B*. The free adjunction J* and the generative junctor (HA, HB) have the following universal property: For every adjunction J' and every junctor (MA, MB) from J to J' there is a unique junctor (NA, NB) from J* to J' such that MA = NAHA and MB = NBHB. This property characterizes J* up to junction isomorphism. The adjunction J* is the image of J under a left adjoint to the forgetful functor from the category of adjunctions (with junctors as arrows) to the category of junctions (with junctors as arrows). The free adjunction J* that is generated by the free junction J generated by a pair of graphs (G, H) will be called the free adjunction J* generated by (G, H). For the free adjunction J* generated by (G, H), the generative arrows of A* and B* are the arrows [ f ] and [g] such that f and g are generative arrow terms of, respectively, the graph A and the graph B of the free junction J generated by (G, H). We designate these generative arrows by the generative arrow terms f and g of A and B; then we can use the object terms and arrow terms of A and B to designate the objects and arrows of A* and B*.

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Consider now the generative graph-morphism pair (HG*, HH*) from (G, H) to (A*, B*), where HG* assigns to every object of G the corresponding object of A* and to every arrow of G the corresponding generative arrow of A*; the graph-morphism HH* is defined analogously by replacing G by H and A* by B*. The graphmorphisms HG* and HH* amount to inclusion, both on objects and on arrows, and they are faithful. The free adjunction J* generated by (G, H) has the following universal property involving (HG*, HH*), quite analogous to the universal property of the free junction J generated by (G, H), which we have considered in the preceding section: For every adjunction J' between A' and B ', and every graphmorphism pair (MG, MH) from (G, H) to (A', B '), there is a unique junctor (NA, NB) from J* to J' such that MG = NAHG* and MH = NB HH*. This universal property characterizes J* up to junction isomorphism. The adjunction J* is the image of (G, H) under a left adjoint to the forgetful functor from the category of adjunctions (with junctors as arrows) to the category of pairs of graphs (with graph-morphism pairs as arrows). This forgetful functor assigns to an adjunction between C and D the pair of graphs (C, D). If (HG, HH) is the generative graph-morphism pair from (G, H) to (A, B), where A and B come from the free junction J generated by (G, H), and (HA, HB) is the generative junctor from J to J*, then it is clear that for the generative graph-morphism pair (HG*, HH*) from (G, H) to (A*, B*) we have HG* = HAHG and HH* = HBHH, and we could derive the universal property for (HG*, HH*) from the universal properties for (HG, HH) and (HA, HB).

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§ 4.5. CUT ELIMINATION IN FREE ADJUNCTIONS Let 〈A*, B*, F, G, ϕa, γc〉 be the free adjunction generated by (G, H). We shall now show that Cut Disintegration holds for the categories A* and B*. The assumptions made for adjunctions will prove necessary and sufficient for having this Cut Disintegration. For cuts where one of the arrow terms composed contains no generative arrow term, we shall demonstrate a strengthening of Cut Disintegration, called Constant-Cut Elimination, which we shall apply later. We shall also demonstrate a total version of Cut Disintegration, called Cut Molecularization, which corresponds to Gentzen’s cut elimination in a context where there are nonlogical axioms. Then we shall enquire whether this theorem could characterize adjunction as Cut Disintegration does. Finally, in the last section of this part (§ 4.5.5), we investigate how alternative formulations of rectangular || adjunction behave with respect to Cut Disintegration. We shall find that the assumptions for these formulations of adjunction are sufficient for Cut Disintegration, but they are not all necessary. Throughout this part (§ 4.5) the symbols A* and B* will refer to the categories of the free adjunction above. In § 4.5.5, where we talk of other formulations of adjunction, A* and B* would refer to categories of free adjunctions in these other formulations, but we don’t mention these categories explicitly. § 4.5.1. Cut Disintegration in free adjunctions Every equality f1 = f2 that holds for the arrow terms f1 and f2 of A* and every equality g1 = g2 that holds for the arrow terms g1 and g2 of B* may be derived as follows. It may be axiomatic, which means that it is an instance of h = h, (cat 1 right), (cat 1 left), (cat 2), (fun 1), (fun 2), (ϕa 1), (ϕa 2), (γc 1), (γc 2), (ϕaγcF) or (ϕaγcG); else it is derived from axiomatic or previously derived equalities by the rules of symmetry or transitivity of equality, or the rule (congr ° ) of § 1.8.1 (with f and g renamed appropriately), or the additional congruence rules

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from f1 = f2 in A*, infer G f1 = G f2 in B*, from g1 = g2 in B*, infer Fg1 = Fg2 in A*, from f1 = f2 in A*, infer ϕa f1 = ϕa f2 in A*, from g1 = g2 in B*, infer γcg1 = γcg2 in B*. The first two of these congruence rules are just variants of (congr F) of § 2.3.1, and the second two are variants of (congr τc) of § 3.5.1, with letters appropriately renamed. We must now extend the definition of linked cuts from § 1.8.1 to equalities among arrow terms of A* and B*. For that we have to take into account the following additional cases. If f1 = f2 or g1 = g2 is an instance of (fun 1), then no cut occurs on either side, and if it is an instance of (fun 2), (ϕa 1), (ϕa 2), (γc 1), (γc 2), (ϕaγcF) or (ϕaγcG), then the n-th cut on the left-hand side is linked to the n-th cut on the right-hand side. The additional congruence rules just preserve the links of the premises. Note that, as in § 2.3.1 and § 3.5.1, cuts of arrow terms of A* are linked to cuts of arrow terms of B*. We just take over the other definitions of § 1.8.1, save that in the definition of the degree of an arrow term we also count, besides occurrences of 1 and ° , also occurrences of F (applied to arrow terms), G (applied to arrow terms), ϕa and γc. In these definitions we take on an equal footing the cuts of A* and the cuts of B*. We can then prove Cut Disintegration for the arrow terms of A* and B*. Proof of Cut Disintegration for A* and B*. We proceed as in the proof of § 1.8.1. So, in an arrow term of A* or B*, take a nonatomic cut whose subterm is f2 ° f1. Then we have the following cases in addition to cases analogous to those considered in § 1.8.1 and § 2.3.1. In case (3), where f1 is Fg and it is an arrow term of A*, we replace B* by A* in (3.G), and we have the following additional subcases.

(3.4) f2 is ϕa f. Then we have the following subcases.

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(3.4.1) g is 1B. Then f2 ° F1B = f2 ° 1FB by (fun 1) for F, and the degree of the main cut of f2 ° 1FB is strictly smaller than the degree of the main cut on the left-hand side linked to it. (Alternatively, since B is GA, we can rely on (fun 1) for G and (ϕa 2). In both cases, we can eliminate the main cut of f2 ° f1 by using further (cat 1 right).) (3.4.2) g is g2 ° g1. Then f2 ° F(g2 ° g1) = ( f2 ° Fg2) ° Fg1 by (fun 2), (congr ° ) and (cat 2), and the degree of the main cut of f2 ° Fg2 is strictly smaller than the degree of the main cut on the left-hand side linked to it. (3.4.3) g is G f '. Then ϕa f ° FG f ' = ϕa( f ° f ') by (ϕa 2), and the degree of the main cut of f ° f ' is strictly smaller than the degree of the main cut on the left-hand side linked to it. (3.4.4) g is γcg '. Then ϕa f ° Fγcg ' = f ° Fg ' by (ϕaγcF), and the degree of the main cut of f ° Fg ' is strictly smaller than the degree of the main cut on the left-hand side linked to it. (3.4.G) g is a generative arrow term of B*. This is excluded because g must have a target of the form GA. (4) f1 is ϕa f. Then f2 ° ϕa f = ϕa( f2 ° f ) by (ϕa 1), and the degree of the main cut of f2 ° f is strictly smaller than the degree of the main cut on the left-hand side linked to it. In case (G), where f1 is a generative arrow term of A*, we have the following additional subcase. (G.4) f2 is ϕa f. This is excluded because f1 would then have a target of the form FGA. We treat analogously a nonatomic cut whose subterm is g2 ° g1, i.e. an arrow term of B*. q.e.d. As in § 1.8.3, Particular Cut Elimination and Total Cut Elimination for A* and B* follow from Cut Disintegration when G and H are arrowless graphs. However, if the goal is to prove Total Cut Elimination, we can imitate Gentzen’s procedure of eliminating

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topmost cuts (see § 1.8.3). This would make superfluous cases like (2) in the proof of § 1.8.1, (3.2) in the proof of § 2.3.1, (4.2) in the proof of § 3.5.1 and (3.4.2) in the proof of the present section. Free adjunctions generated by arrowless graphs are not trivial as were free categories generated by arrowless graphs. Categories in these adjunctions are not discrete and they are not preorders, provided at least one of the generating graphs is not empty, i.e., it has at least one object (see §§ 4.6.1-3). However, the graphs of these categories are disjoint unions of graphs of categories involved in free adjunctions generated by pairs of graphs (G, H) where one of the graphs G and H is arrowless with a single object and the other is the empty graph (i.e., it has neither arrows nor objects). In Gentzen’s cut-elimination procedure the degree (in German Grad, usually translated as degree) is the number of connectives in the cut formula (i.e. the formula A2 of f1: A1 → A2 and f2: A2 → A3 in the cut f2 ° f1: A1 → A3). Our degree is something else: it involves Gentzen’s degree, but it is rather closer to what Gentzen called rank (in German Rang), which is a measure of the complexity of the derivation above a cut. Gentzen’s degree is a secondary aspect of our degree, and it does not always decrease when our degree decreases. For example, it would decrease in cases like (3.4.3) and (3.4.4), but not in a case like (4), in the proof above; in (4) Gentzen would have only rank decreasing. § 4.5.2. Necessary conditions for Cut Disintegration in free adjunctions Note that in the complete proof of Cut Disintegration in the preceding section we have used all the equalities of (i)-(iv) of the definition of adjunction in § 4.2 (including (fun 1), which was not needed for the proofs in § 2.3.1 and § 3.5.1, but now appears in case (3.4.1); the equalities (γc 1), (γc 2) and (ϕaγcG) are used in unmentioned cases analogous to cases (4), (3.4.3) and (3.4.4)). Our purpose is now to show that all these equalities are also necessary. We shall show for each of these equalities that for the graphs A* and B* of the free adjunction-like structure generated by (G, H) that lacks this equality

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Cut Disintegration would fail for some choice of G and H. Whenever this is possible, we favour arrowless graphs G and H, because counterexamples to Cut Disintegration with such graphs can also be extended to other graphs, and because these counterexamples are also counterexamples for Particular and Total Cut Elimination. The necessity of (cat 1 right), (cat 1 left) and (cat 2) is demonstrated as in § 1.8.2. There, G and H cannot be arrowless if we want a counterexample for (cat 1 right) alone, or (cat 1 left) alone. A counterexample in the absence of both (cat 1 right) and (cat 1 left) can be found with arrowless graphs. Such graphs will also do for (cat 2). All the counterexamples to Cut Disintegration in the remainder of this section are with arrowless graphs G and H. In the absence of (fun 1) we have the counterexamples ϕa1A ° F1GA and G1FB ° γc1B. Here we assume that we lack (fun 1) for both F and G; i.e., we have neither F1B = 1FB nor G1A = 1GA. Because either of these equalities, together with the other equalities of adjunctions, would enable us to get rid of both counterexamples. Either of these equalities is sufficient for case (3.4.1) of the proof in the preceding section. In the absence of (fun 2) we have the counterexamples FγcGF1B ° Fγc1B or Gϕa1A ° GϕaFG1A, and others built analogously. (The counterexample in the absence of (fun 2) we had in § 2.3.2 was not for arrowless G, as those we have here.) In the absence of (ϕa 1) we have the counterexample a ϕ 1A ° ϕaFG1A, and in the absence of (γc 1) the counterexample γcGF1B ° γc1B. (These counterexamples correspond to the counterexamples to cut elimination with the natural transformations ϕ and γ primitive at the beginning of § 4.2.) In the absence of (ϕa 2) we have the counterexample a ϕ 1A ° FGϕa1A, and in the absence of (γc 2) the counterexample GFγc1B ° γc1B.

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In the absence of (ϕaγcF) we have the counterexample ϕaF1B ° Fγc1B, and, finally, in the absence of (ϕaγcG) we have the counterexample Gϕa1A ° γcG1A. § 4.5.3. Constant-Cut Elimination Let us call an arrow term of A* or B* constant iff no generative arrow term occurs in it, and let us call a cut whose subterm is h2 ° h1 constant iff either the arrow term h1 or the arrow term h2 is constant, where h stands for either f or g. Then we can prove the following version of Cut Disintegration for the arrow terms of A* and B*. Constant-Cut Elimination. Every constant cut in an arrow term can be eliminated. Proof. Note first that the cut-elimination procedure in the proof of Cut Disintegration in § 4.5.1 (including steps mentioned in previous chapters) either eliminates a constant cut of an arrow term or makes it linked to a constant cut of strictly smaller degree in an arrow term equal to the initial arrow term. By repeated applications of this procedure we obtain a sequence of cuts of ever decreasing degrees all linked to each other. This sequence must be finite. If the initial cut in the sequence was constant, the last cut in the sequence cannot be atomic. So this last cut can be eliminated. q.e.d.

Note that we have no version of Cut Disintegration that would assert that every nonconstant cut can be atomized. Because, by applying (cat 2) in cases (2), (3.2), (3.4.2) and (G.2) of the proof of Cut Disintegration, a nonconstant cut can become linked to a constant cut. For a nonconstant cut we can assert only, as in Cut Disintegration, that it can be atomized or eliminated. We can prove Constant-Cut Elimination also in the contexts of Chapters 1-3: for the free category A* generated by an arbitrary graph, the category B* that is the codomain of a free functor generated by an arbitrary pair of graphs and the category B* in a free natural formation generated by an arbitrary pair of graphs. We didn’t

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mention this result before because in the previous contexts it is more trivial, and because we didn’t need to apply it, as we shall do for free adjunctions in § 4.5.5. § 4.5.4. Cut Molecularization For reasons mentioned at the end § 1.8.3, it is impossible to prove a total form of Cut Disintegration for A* and B*, which would assert that for every arrow term h there is an arrow term h' in which every cut is atomic such that h = h'. Here, and also below, we use h to cover both the arrow terms f of A* and the arrow terms g of B*. However, if we replace “atomic” by weaker properties, we can prove such a version of Cut Disintegration. Let us first define these properties. Molecular arrow terms are defined inductively as follows: every generative arrow term is molecular; if h1 and h2 are molecular, then h2 ° h1 is molecular, provided h2 ° h1 is defined. Right-molecular arrow terms are defined as follows: every generative arrow term is right-molecular; if h1 is right-molecular and h2 is generative, then h2 ° h1 is right-molecular, provided h2 ° h1 is defined. So rightmolecular arrow terms are either generative or of the form hn ° (hn1 ° … ° (h2 ° h1)…), where n ≥ 2 and for every i in {1,…, n} the arrow term hi is generative. We define analogously left-molecular arrow terms, which are either generative or of the form (…(hn ° hn1) ° … ° h2) ° h1 for hi generative. A cut will be called molecular iff the subterm of this cut is molecular, and analogously with “rightmolecular” and “left-molecular”. An atomic cut, i.e. one whose subterm is h2 ° h1 for h1 and h2 generative, is both right-molecular and left-molecular, and, of course, it is molecular tout court. Let us define the molecularization of cuts analogously to atomization (see § 1.8.1); i.e., a cut x of h can be molecularized iff there is an arrow term h' such that h = h', there are as many cuts in h' as in h, and in h = h', for a certain derivation, every cut of h is linked to a cut of h' so that x is linked to a molecular cut of h'. Cut Disintegration trivially implies Particular Cut Molecularization, which says that every cut can be molecularized or eliminated. We also

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have Particular Cut Right-Molecularization and Particular Cut LeftMolecularization, because atomic cuts are both left-molecular and right-molecular. We don’t have a total form of Cut Disintegration, but we have the following total form of Particular Cut Molecularization for A* and B*. Total Cut Molecularization. For every arrow term h there is an arrow term h' in which every cut is molecular such that h = h'. Proof. We imitate Gentzen’s procedure of eliminating topmost cuts (see § 1.8.3). Only now, we eliminate topmost nonmolecular cuts; such a cut is a nonmolecular cut for whose subterm h2 ° h1 we have that all the cuts in h1 and h2 are molecular. We show that h2 ° h1 is either equal to an arrow term in which all cuts are molecular or it is equal to an arrow term with a single topmost nonmolecular cut such that this cut is of strictly smaller degree than h2 ° h1. The theorem then follows by induction on degree. (In the definition of degree, we could also omit counting molecular cuts, i.e. occurrences of ° in molecular arrow terms.) The possible cases we encounter in this proof are covered by cases considered in the proof of Cut Disintegration in § 4.5.1 (including cases considered in previous chapters). Note, however, that if f2 ° f1 is a topmost nonmolecular cut and f1 is f1" ° f1', then f2 can only be 1A (it can be neither f2" ° f2' nor generative, because f2 ° f1 would then be molecular, and it can be neither Fg2 nor ϕa f2' because the target of f1 would then be of the form FB, which is impossible since f1 is molecular). Note also that the case corresponding to case (3.4.2) of the proof in § 4.5.1 is excluded because in ϕa f ° F(g2 ° g1) the molecular arrow term g2 ° g1 cannot have a target of the form FGA. q.e.d.

It would not be a problem for the induction in this proof if h2 ° h1 were equal to an arrow term with more than one topmost nonmolecular cut, provided all these cuts are of strictly smaller degree than h2 ° h1. (We encounter such a situation in Gentzen’s proof of cut elimination in logic.)

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All the equalities of the definition of adjunction of § 4.2 are used in this proof, except for (cat 2). But we need also (cat 2) if in the statement of Total Cut Molecularization we replace “molecular” by “right-molecular” or “left-molecular”. These theorems, which we shall call Total Cut Right-Molecularization and Total Cut LeftMolecularization, easily follow from the ordinary Total Cut Molecularization above: just replace every molecular subterm h" of h' by a right-molecular or left-molecular h''' for which we have h" = h''' by, perhaps, repeatedly using (cat 2), or rather its instance (cat 2 mol)

(h3 ° h2) ° h1 = h3 ° (h2 ° h1)

where h1, h2 and h3 are molecular arrow terms. It is clear that (cat 2 mol) is necessary for these stronger forms of Total Cut Molecularization. The necessity of the other equalities of the definition of adjunction of § 4.2 can be shown along the lines of § 4.5.2. We shall see in § 4.6.4 that replacing (cat 2) by (cat 2 mol) is not a loss for free adjunctions generated by pairs of graphs: there, the former equality is derivable from the latter. So it might seem that Total Cut Right-Molecularization or Total Cut Left-Molecularization characterize adjunction as Cut Disintegration does. However, only Cut Disintegration does it in a straightforward manner. The equality (cat 2 mol) cannot do the work of (cat 2) for Cut Disintegration because of linked cuts (cf. the remark about linkage after the proof of Associativity Elimination in § 4.6.3). Besides that, (cat 2) is necessary for Cut Disintegration even with free adjunctions generated by arrowless graphs, whereas it is not necessary for Total Cut Molecularization in either form if we deal with such free adjunctions. We shall see in § 4.6.3 that in these free adjunctions (cat 2) is derivable from the remaining equalities, but, again, linked cuts in derived (cat 2) are not the same as linked cuts in primitive (cat 2). Total Cut Elimination for free adjunctions generated by arrowless graphs is a consequence of Cut Disintegration, but it is also a consequence of Total Cut Molecularization. Total Cut Molecularization corresponds to the form Gentzen’s cut elimination

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takes in a context where there are nonlogical axioms. These axioms correspond to generative arrow terms. Total Cut Molecularization, and its right and left stronger forms, are trivial to prove for free categories generated by an arbitrary graph. We just have to eliminate all arrow terms of the form 1A by using (cat 1 right) and (cat 1 left). What remains is a molecular arrow term, which can be transformed into a right-molecular or left-molecular arrow term by, perhaps, repeatedly using (cat 2 mol). Total Cut Molecularization can also be demonstrated in the contexts of Chapters 2 and 3: for the category B* that is the codomain of a free functor generated by an arbitrary pair of graphs and the category B* in a free natural formation generated by an arbitrary pair of graphs. § 4.5.5. Cut Disintegration with alternative notions of rectangular || adjunction If ϕa is the natural antecedental transformation and γc is the natural consequential transformation from the definition of adjunction in § 4.2, let ϕc be the natural consequential transformation equivalent to ϕa, and let γa be the natural antecedental transformation equivalent to γc (see § 3.1). More precisely, the natural consequential transformation ϕc, which should satisfy (ϕc 1) for f1: A1 → A2, f2: A2 → FGA3, (ϕc 2) for f1: A1 → FGA2, f2: A2 → A3, f2 ° ϕcA2 f1,

ϕcA3( f2 ° f1) = ϕcA3 f2 ° f1, ϕcA3( FG f2 ° f1) =

is defined in terms of ϕa by (ϕc) for f : A1 → FGA2,

ϕcA2 f =def ϕa1A2 ° f,

which is an instance of a definition mentioned in § 3.1. Conversely, we can define ϕa in terms of ϕc by (ϕa) for f : A1 → A2,

ϕa f =def ϕcA2FG f.

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By (cat 1 right) and (ϕc 2), the right-hand side of (ϕa) is equal to f ° ϕcA11FGA1, which is what we should have according to the corresponding definition in § 3.1. Analogously, the natural antecedental transformation γa, which should satisfy (γa 1) for g1: GFB1 → B2, g2: B2 → B3,

γaB1( g2 ° g1) = g2 ° γaB1g1,

(γa 2) for g1: B1 → B2, g2: GFB2 → B3,

γaB1( g2 ° GFg1) = γaB2g2 ° g1,

is defined in terms of γc by (γa) for g: GFB1 → B2,

γaB1g =def g ° γc1B1,

while γc can be defined in terms γa by (γc) for g: B1 → B2

γcg =def γaB1GFg.

Suppose that in the definition of adjunction in § 4.2 we replace (ac)

ϕa and γc

by either of the following (cc) (aa) (ca)

ϕc and γc, ϕa and γa, ϕc and γa.

This involves new sets of equalities, chosen among (ϕc 1), (ϕc 2), (γa 1) and (γa 2), for clause (iii) of the definition, and a rephrasing of the rectangular equalities of clause (iv) (see the equalities (ϕcγaF) and (ϕcγaG) below). The remainder of the definition would be unchanged. So we obtain four equivalent notions of rectangular || adjunction, which we shall call the (ac), (aa), (cc) and (ca) formulations. With all these alternative formulations of adjunction we can construct the free adjunction generated by a pair of graphs and prove Cut Disintegration. This proof can be obtained either directly or by reducing the alternative Cut Disintegration to the Cut Disintegration of § 4.5.1. Let us first follow the latter course. Suppose we have a free adjunction in the (aa) formulation generated by a pair of graphs. Then take an arrow term h in the new

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language with γa and replace every subterm γaBg in it by g ° γc1B, according to the definition (γa). This yields an arrow term h' in the old (ac) language. We can link every cut of h to its obvious descendant in h'. Moreover, we have new cuts in h', which have appeared through (γa). Then we apply Constant-Cut Elimination from § 4.5.3 to h' to eliminate all the new cuts. This yields an arrow term h", all of whose cuts are linked to cuts of h, establishing a one-to-one correspondence. Then we can atomize or eliminate any particular cut of h", according to Cut Disintegration for the (ac) formulation, which yields an arrow term h''. The arrow term h'', which is in the old (ac) language, can be transformed into an arrow term in the new (aa) language with the help of the definition (γc), which does not introduce any new cut. This is enough to show that Cut Disintegration can also be proved for the (aa) formulation, and we can proceed analogously with the (cc) and (ca) formulations, relying on the definitions (ϕc) and (ϕa). However, with the old (ac) formulation we have the advantage that all the equalities we have assumed in the definition of adjunction are necessary for Cut Disintegration, as was shown in § 4.5.2. With the new formulations and a direct demonstration of Cut Disintegration, this advantage is lost with the obvious equalities of adjunction of the new context. The situation is the clearest if we take as an example the (ca) formulation. We would then have instead of the rectangular equalities (ϕaγcF) and (ϕaγcG) the following two equalities: (ϕcγaF)

ϕcFB2FγaB1GFg = Fg,

(ϕcγaG)

γaGA1GϕcA2FG f = G f,

The cut-disintegration procedure looks now as follows. The degree of an arrow term h may now be taken to be a pair (n1, n2), where n1 is the number of symbols ϕc and γa occurring in h, and n2 the number of the remaining symbols 1, ° , F (applied to arrow terms) and G (applied to arrow terms) occurring in h. These degrees have a lexicographical order, i.e. an order of type ω2 with the definition (n1, n2) < (m1, m2) iff n1 < m1 or (n1 = m1 and n2 < m2). Then we can prove Cut Disintegration for (ca) arrow terms by proceeding as for the (ac)

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arrow terms, except in cases corresponding to (3.4), (4) and (G.4) in the proof of § 4.5.1. In the first and third case we apply (ϕc 1) or (γa 1), and in the second case (ϕc 2) or (γa 2), always ending up with a cut of strictly smaller degree. Case (3.4) is now much simpler than in § 4.5.1. In all that, the new rectangular equalities (ϕcγaF) and (ϕcγaG) have no role to play. Since these equalities are independent from the other equalities of the (ca) formulation, they are not necessary for Cut Disintegration in this formulation. The equality (fun 1) is now also not necessary. The situation is similar with the (aa) and (cc) formulations, but with only one of the new rectangular equalities being superfluous for Cut Disintegration. The case of the (ca) formulation shows that Cut Disintegration does not characterize only adjunction, but also a more general notion of junction, lacking the rectangular equalities, and also lacking (fun 1). Our standard (ac) formulation resembles a sequent system in which we have rules for introducing logical operations on the left and on the right of the turnstile: f : A1 → A2

g : B1 → B2

ϕaf : F GA 1 → A 2

γcg : B 1 → GFB 2

provided we take FG and GF as corresponding to a logical operation. One of F and G would be such an operation and the other would be something “structural” in the sense of Gentzen (and hence “invisible”; see § 0.3.5 and § 6.1). It is presumably not an accident that the notion of adjunction exactly characterized by cut elimination should be the closest in appearance to sequent systems for which cut elimination is usually proved. The (cc) formulation resembles a natural-deduction system with elimination and introduction rules: f : A 1 → F GA 2

g : B1 → B2

ϕcA 2 f : A 1 → A 2

γcg : B 1 → GFB 2

and the (aa) formulation resembles a sequent system with structural rules on the left of the turnstile:

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f : A1 → A2

f : GFB 1 → B 2

ϕaf : F GA 1 → A 2

γaB 1 f : B 1 → B 2

provided FG and GF correspond now to something structural (see § 0.3.5 and § 6.5). In the (ca) formulation, which does not seem to have a correlate in ordinary Gentzen systems, the corresponding rules are f : A 1 → F GA 2

f : GFB 1 → B 2

ϕcA 2 f : A 1 → A 2

γaB 1 f : B 1 → B 2

Cut-free arrow terms now lead to a “thick” normal form of naturaldeduction proofs, where introductions precede eliminations, whereas in the usual, “thin”, natural-deduction normal form eliminations precede introductions (see § 0.3.5). The (ca) formulation lacks the subformula property and is unadapted for solving the decision problem usually considered in logic—namely, whether there is an arrow of a given type. However, such a formulation may be as good as the others for solving other decision problems—in particular, the problem whether two arrow terms designate the same arrow. In the following part we deal with these decision problems for free adjunctions generated by pairs of graphs. § 4.6. DECIDABILITY IN FREE ADJUNCTIONS Cut elimination is often used for proving decidability in logic. It can be so used in the case of free adjunctions, too, as we shall demonstrate presently. (“Adjunction”, without qualification, refers throughout this part of the work to the standard (ac) formulation of § 4.2.) § 4.6.1. Decision problems in free adjunctions In the free adjunction between A* and B* generated by a pair (G, H) of arrowless graphs, we have Total Cut Elimination as a consequence of Cut Disintegration. Our standard (ac) formulation has an analogue of the subformula property of Gentzen’s sequent systems, as can be seen from the rules for ϕa and γc displayed at the end of § 4.5.5, as

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well as from rules for F and G. (The (cc), (aa) and (ca) formulations don’t have this property.) Combining Total Cut Elimination with this subformula property we obtain a procedure for deciding whether for a pair (A1, A2) of objects of A* there is an arrow of the type A1 → A2 in A*; there is a completely analogous procedure for pairs of objects and arrows of B*. This is the analogue of the decision procedure that cut elimination usually yields in logic. We call the sort of decision problem this procedure solves the theoremhood problem. Total Cut Elimination for free categories and free functors generated by arrowless graphs yields analogous decision procedures. However, the corresponding theoremhood problems are there quite trivial and don’t require considering such an involved matter as cut elimination. There we have discrete categories (all their arrows are identity arrows). The matter is not much more interesting with free natural formations generated by arrowless graphs (note that in that case the subformula property does not obtain). The theoremhood problem for free adjunctions generated by arrowless graphs is also rather easy to solve, and cut elimination is again not essential for that. We have arrows in A* only for types of the form A → A, FG…FGA → A and FB → FG…FGFB, and, analogously, we have arrows in B* only for types of the form B → B, B → GF…GFB and GF…GFGA → GA. We mention here the decision procedure for theoremhood based on cut elimination only because of its relationship to one of the main applications of cut elimination in logic. Moreover, this procedure is interesting because it is constructive, which means that it actually yields an arrow for a type having an arrow. Sometimes it yields a unique arrow: for example, 1FB , which is equal to F1B , is the only arrow of A* with the type FB → FB for B a generative object term (cf. the proof of the Proposition in the next section). Another decision problem, which arises in the context of category theory, is less trivial than these theoremhood problems. This is the problem whether two arrow terms of the same type designate the same arrow, or to put it in standard categorial terminology, the problem

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whether a diagram of arrows commutes. So we shall call it the commuting problem. This problem is again uninteresting for free categories and free functors generated by arrowless graphs, which yield preorders, i.e. categories where for every pair of objects, not necessarily distinct, there is at most one arrow between these objects. (As we said above, with free categories and free functors, all arrows are identity arrows.) However, the commuting problem is significant for free adjunctions between A* and B* generated by arrowless graphs with at least some objects. In such adjunctions we have in A* and B* different arrows with the same source and target. For example, in A* we have ϕaFγc1B : FGFB → FGFB,

which is different from 1FGFB , and in B* we have γcGϕa1A : GFGA → GFGA,

which is different from 1GFGA. (We shall see below how these inequalities can be established.) Actually, if these arrows were equated, then A* and B* would be preorders. We shall show this in the next section. After that we shall return to the commuting problem. § 4.6.2. Free adjunctions between preorders Note first that in any adjunction we have one of the following four equalities only if we have all of the remaining ones: ϕaFγcg = FGFg, ϕaFG f = FGϕa f,

γcGϕa f = GFG f, γcGFg = GFγcg

(see the proof of the Maximality of Adjunction in § 4.11 below; cf. [Lambek & Scott 1986, 0.4, Lemma 4.3]). These equalities are assumed to be satisfied for every g and f. Instead of these four equalities we can take the equivalent equalities obtained from them by replacing g by 1B and f by 1A . Anticipating matters, let us call any of these equalities a preordering equality. (As a matter of fact, any

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equality of junctions not satisfied in the free adjunction generated by a pair of arrowless graphs could replace the preordering equalities; see § 4.11.) Let an adjunction that satisfies the preordering equalities be a special adjunction, and consider the free special adjunction between A** and B** generated by a pair of arrowless graphs. (This free special adjunction is constructed quite analogously to the free adjunction: we only have to take into account an additional condition, derived from a preordering equality, for the adjunctional equivalence relations of the free junction.) Then we can show the following. Proposition. The categories A** and B** are preorders. Proof. For every object B of B**, the objects FB and FGFB are isomorphic in the category A**, with the isomorphisms

Fγc1B : FB → FGFB, ϕaF1B : FGFB → FB (to show that we use (ϕaγcF) besides the first preordering equality above). Then we can infer that any two objects FB and FG…FGFB are isomorphic in A**. (It is clear that we don’t have this isomorphism in A*: just take an adjunction where G is a forgetful functor and FB is a free algebra of some sort generated by a set of free generators B.) Now suppose we have in A** the arrows f1: A1 → A2 and f2: A1 → A2. If A1 and A2 are both generative object terms, then A1 must be the same as A2, and f1 and f2 are the same identity arrow. If only one of A1 and A2 is a generative object term, this can only be if A2 is generative and A1 is not. Then both f1 and f2 must be ϕa…ϕa1A2: FG…FGA2 → A2.

If neither of A1 and A2 is a generative object term, then A1 and A2 are of the form FG…FGFB or FB where B is a generative object term or B is GA for A a generative object term. The same generative object term B or A must occur in both A1 and A2, because otherwise there would be no arrow between A1 and A2 (take the trivial adjunction

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〈A, A, IA, IA, ϕa, γc〉 where ϕa f and γc f are equal to f; this is clearly a special adjunction, i.e., the preordering equalities are satisfied). Consider the isomorphisms i : FB → A1 and j : A2 → FB, which, as we have said above, must exist. Then we have that j ° ( f1 ° i): FB → FB and j ° ( f2 ° i): FB → FB must both be equal to 1FB ; because, when we eliminate all the cuts of j ° ( f1 ° i) and j ° ( f2 ° i) by Total Cut Elimination (which holds in A**, as well as in A*), we obtain a cutfree arrow term of type FB → FB, which can only be 1FB , or F1B , or FG1A . If i ' is the inverse of i and j ' the inverse of j, then both f1 and f2 are equal to j ' ° i ', and so f1 = f2. We show something completely analogous in B**. q.e.d.

(Another way of proving that A** is a preorder might be to reduce f1 and f2 to the same normal form with the help of the four preordering equalities displayed at the beginning of the section. For this normal form, redexes are at the left-hand sides of the equalities and contracta on the right-hand sides; see the proof of the Maximality of Adjunction in § 4.11.) A path from an object A to an object A' in a graph, A and A' not being necessarily distinct, is a finite sequence of arrows of this graph f1: A0 → A1, f2: A1 → A2, … , fn: An-1 → An, where n ≥ 1, the object A0 is A, the object An is A' and the arrows in the sequence f1,…, fn are not necessarily all mutually distinct. A path f1: A0 → A1, … , fn: An-1 → An is acyclic iff the objects A0,…, An are all mutually distinct. A graph is treelike iff for every pair of objects (A, A') in it, where A and A' are not necessarily distinct, there is at most one path from A to A'. It is easy to conclude that a treelike graph is an irreflexive relation, and that every path in it is acyclic. The Proposition we have demonstrated in this section can be generalized to free special adjunctions between A** and B** generated by a pair of treelike graphs. (Arrowless graphs are a particular kind of treelike graphs.) So any of the preordering equalities is sufficient for turning into preorders the categories of special adjunctions satisfying these

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equalities that are freely generated by treelike graphs. Of course, conversely, if we have an adjunction between preorders, then the preordering equalities must hold. § 4.6.3. The commuting problem in free adjunctions generated by arrowless graphs In the free adjunction between A* and B* generated by a pair of arrowless graphs, cut-free arrow terms of A* and B* in which there are no subterms of the form F1B or G1A will be considered to be in normal form. So arrow terms in normal form have the shape Q1…Qn1C, where n ≥ 0, every Qi is either F, or G, or ϕa, or γc, and Qn is neither F nor G. With the help of Cut Disintegration, and the ensuing Total Cut Elimination, together with (fun 1), we easily obtain an ordinary normal form theorem: namely, the assertion that for every arrow term h of A* or B* there is an arrow term h' in normal form such that h = h'. We shall introduce a collection of reductions that eliminate cuts from arrow terms, or replace them by cuts of strictly smaller degree, or eliminate subterms of the form F1B or G1A. With respect to these reductions, which are obtained by orienting equalities postulated for adjunction according to the proof of Cut Disintegration, we have that irreducible arrow terms are in normal form. For these reductions we can also prove a strong normalization theorem: namely, the assertion that every sequence of reduction steps is finite. Moreover, we can establish the Church-Rosser property for this collection of reductions, which yields a unique normal form for every arrow term of A* or B*. Strong normalization and the Church-Rosser property, together with a lemma guaranteeing that every arrow of A* or B* is designated by a unique term in normal form, yield a decision procedure for the commuting problem for A* and B*. Let us now introduce our collection of reductions. Here are the redexes and contracta of A*, where the redexes are either subterms of

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topmost cuts, i.e., they are of the form f2 ° f1 with no cuts in f1 and f2, or they are of the form F1B: redexes f ° 1A 1A ° f F1B Fg2 ° Fg1 f2 ° ϕa f1 ϕa f2 ° FG f1 ϕa f ° Fγcg

contracta f f 1FB

F(g2 ° g1) ϕa( f2 ° f1) ϕa( f2 ° f1) f ° Fg

(cat 1 right) (cat 1 left) (fun 1) (fun 2) (ϕa 1) (ϕa 2) (ϕaγcF)

And here are the redexes and contracta of B*, where the redexes are either subterms of topmost cuts, i.e., they are of the form g2 ° g1 with no cuts in g1 and g2, or they are of the form G1A: redexes g ° 1B 1B ° g G1A G f2 ° G f1 γcg2 ° g1 GFg2 ° γcg1 Gϕa f ° γcg

contracta g g 1GA

G( f2 ° f1) γc(g2 ° g1) γc(g2 ° g1) Gf ° g

(cat 1 right) (cat 1 left) (fun 1) (fun 2) (γc 1) (γc 2) (ϕaγcG)

In the rightmost column we mention the corresponding equality. Note that with these reductions we have covered all the equalities of the definition of adjunction except the equality (cat 2). It is clear that the redexes cover all the possible forms of topmost cuts. This entails that an irreducible arrow term, i.e. one that has no redex as a subterm, will have no topmost cuts, and hence no cuts at all. Since it also cannot have subterms of the form F1B or G1A, it will be in normal form.

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An arrow term reduces to another arrow term, i.e., it is in the reduction relation with this arrow term, iff the latter is obtained by replacing a subterm in the former that is a redex by the corresponding contractum. The strong normalization theorem for our reductions says that there is no infinite sequence of reduction steps, i.e. infinite sequence of arrow terms where every member reduces to its successor. This is a consequence of the fact that cuts in contracta either disappear or are of strictly smaller degree than cuts in the redexes, and of the fact that the degree of 1FB and 1GA is strictly smaller than the degree of F1B and G1A. Also the degree of a contractum is never greater than the degree of the corresponding redex: it is always strictly smaller, except in the reductions corresponding to (ϕa 1) and (γc 1), but then the degree of the cut is strictly smaller. So we can take as the complexity measure of an arrow term, which decreases in reduction steps, the sum of the degrees of all cuts plus the degree of the arrow term. As usual, we say that an arrow term h has a normal form h' iff h and h' are in the reflexive and transitive closure of the reduction relation (i.e., there is a sequence, possibly empty, of reduction steps leading from h to h') and h' is in normal form. The Church-Rosser property for reductions says that the reflexive and transitive closure of the reduction relation has the diamond property (namely, that branching is confluent). This property for our collection of reductions is established by straightforward considerations of all possible cases (since we have strong normalization, we can use Newman’s Lemma to infer the ChurchRosser property from the weak Church-Rosser property; see [Barendregt 1981, 3.1.25]). It follows that every arrow term of A* or B* has a unique normal form. It remains to establish that every arrow of A* and B* is designated by a unique arrow term in normal form. Namely, we have to show that two arrow terms are equal in A* and B* iff they can be reduced to the same normal form. From right to left, this follows immediately from the fact that all reductions are covered by equalities that hold between the arrow terms

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of A* and B*. For the other direction we need the following lemma, which says that for our particular categories A* and B*, between which there is a free adjunction generated by a pair of arrowless graphs, assuming the associativity of composition is superfluous. We call this lemma Associativity Elimination. (Note that the elimination in question is not of the same kind as the elimination of cut: with cut we eliminate an operation, whereas here we eliminate an equality.) Associativity Elimination. The equality (cat 2) is derivable from the remaining equalities assumed for the arrow terms of A* and B*. Proof. We have to derive

(cat 2)

( f3 ° f2) ° f1 = f3 ° ( f2 ° f1),

where f1, f2 and f3 are arrow terms of A*, from the remaining equalities of A* and B*. We can assume that f1, f2 and f3 are in normal form. For if they are not, then they are equal respectively to f1', f2' and f3' in normal form without using (cat 2). Establishing (cat 2) for f1', f2' and f3' will then establish (cat 2) for f1, f2 and f3. To derive (cat 2) we make an induction on the sum of the degrees of f1, f2 and f3. The minimal sum is 3, when all of f1, f2 and f3 are identity arrow terms, i.e. when they are all of the form 1A. For the basis of the induction we have that all cases where either of f1, f2 and f3 is an identity arrow term are easily taken care of by (cat 1 right) and (cat 1 left). We need not consider such cases in the induction step either. The following cases remain to be considered in the induction step. (1) f1 is Fg1. Then we have the following subcases. (1.1) f2 is Fg2. Then we have the following subcases. (1.1.1) f3 is Fg3. Then we have (Fg3 ° Fg2) ° Fg1 = F((g3 ° g2) ° g1), by (fun 2) = F(g3 ° (g2 ° g1)), by the induction hypothesis = Fg3 ° (Fg2 ° Fg1), by (fun 2). (1.1.2) f3 is ϕa f3'. Then we have the following subcases.

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(1.1.2.1) g2 is G f2'. Then we have the following subcases. (1.1.2.1.1) g1 is G f1'. Then we have (ϕa f3' ° FG f2') ° FG f1' = ϕa(( f3' ° f2') ° f1'), by (ϕa 2) by the induction = ϕa( f3' ° ( f2' ° f1')), hypothesis = ϕa f3' ° (FG f2' ° FG f1'), by (ϕa 2) and (fun 2). (1.1.2.1.2) g1 is γcg1'. Then we have the following subcases. (1.1.2.1.2.1) f2' is Fg2'. Then we have (ϕa f3' ° FGFg2') ° Fγcg1' = ( f3' ° Fg2') ° Fg1', by (ϕa 2) and (ϕaγcF) = f3' ° ( Fg2' ° Fg1'), by the induction hypothesis = ϕa f3' ° Fγc(g2' ° g1'), by (fun 2) and (ϕaγcF) by (γc 2) and = ϕa f3' ° (FGFg2' ° Fγcg1'), (fun 2). (1.1.2.1.2.2) f2' is ϕa f2". Then we have (ϕa f3' ° FGϕa f2") ° Fγcg1' = ( f3' ° ϕa f2") ° Fg1', by (ϕa 2) and (ϕaγcF) = ( ϕa f3' ° FGf2") ° Fg1', by (ϕa 1) and (ϕa 2) by the induction = ϕa f3' ° ( FGf2" ° Fg1'), hypothesis by (fun 2) and = ϕa f3' ° ( FGϕa f2" ° Fγcg1'), a c (ϕ γ G). (1.1.2.2) g2 is γcg2'. Then we have (ϕa f3' ° Fγcg2') ° Fg1 = ( f3' ° Fg2') ° Fg1, by (ϕaγcF) = f3' ° ( Fg2' ° Fg1), by the induction hypothesis = ϕa f3' ° Fγc(g2' ° g1), by (fun 2) and (ϕaγcF) = ϕa f3' ° (Fγcg2' ° Fg1), by (γc 1) and (fun 2). (1.2) f2 is ϕa f2'. Then we have the following subcases. (1.2.1) g1 is G f1'. Then we have

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( f3 ° ϕa f2') ° FG f1' = ϕa(( f3 ° f2') ° f1'), by (ϕa 1) and (ϕa 2) = ϕa( f3 ° ( f2' ° f1')), by the induction hypothesis = f3 ° (ϕa f2' ° FG f1'), by (ϕa 1) and (ϕa 2). (1.2.2) g1 is γcg1'. Then we have ( f3 ° ϕa f2') ° Fγcg1' = ( f3 ° f2') ° Fg1', by (ϕa 1) and (ϕaγcF) = f3 ° ( f2' ° Fg1'), by the induction hypothesis = f3 ° (ϕa f2' ° Fγcg1'), by (ϕaγcF). (2) f1 is ϕa f1'. Then we have ( f3 ° f2) ° ϕa f1' = ϕa(( f3 ° f2) ° f1'), by (ϕa 1) = ϕa( f3 ° ( f2 ° f1')), by the induction hypothesis = f3 ° ( f2 ° ϕaf1'), by (ϕa 1). We proceed analogously to derive (cat 2) when f1, f2 and f3 are replaced by arrow terms of B*. q.e.d. (Besides this syntactical proof of Associativity Elimination, there is a shorter, model-theoretical, proof, which we shall present in § 4.10.2.) Note that the derived equality (cat 2) induces different linked cuts than the primitive equality (cat 2). For example, in ( f3 ° 1A ) ° f1 = f3 ° f1, by (cat 1 right) = f3 ° (1A ° f1), by (cat 1 left) the right ° of ( f3 ° 1A ) ° f1 is linked to the left ° of f3 ° (1A ° f1), and the left ° of ( f3 ° 1A ) ° f1 and right ° of f3 ° (1A ° f1) are not linked to anything (cf. the comments on Total Cut Right-Molecularization and Total Cut Left-Molecularization in § 4.5.4). Once we have eliminated (cat 2), the Church-Rosser property of our reductions implies that if two arrow terms of A* or B* are equal, then they have the same normal form. This is because for every equality f = f ' of A* obtained without (cat 2), either f is the same arrow term as f ' or there is a sequence of arrow terms f1,…, fn, n ≥ 2, such that f1 is f while fn is f ', and for every i in {1,…, n-1} either fi

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reduces to fi+1 or fi+1 reduces to fi. Of course, the same holds for equalities of B*. Hence every arrow of A* or B* is designated by a unique arrow term in normal form. (An alternative proof of that, eschewing the Church-Rosser property and Associativity Elimination, is given in § 4.10.2.) This yields a procedure for deciding equality of arrow terms of A* or B*, i.e. a decision procedure for the commuting problem. Just reduce the two arrow terms to normal form (according to strong normalization, reductions may be applied randomly). If we end up with the same normal form, then the arrow terms are equal, and if we end up with different normal forms, then they are not equal. For example, the arrow terms ϕaFγc1B and 1FGFB , considered at the end of § 4.6.1, which are both in normal form, must designate different arrows of A*, and, for the same reason, γcGϕa1A and 1GFGA are different in B*. (Another way to establish these inequalities is as in the preceding section, by appealing to the nonisomorphism of FB with FGFB, and of GA with GFGA, in some particular adjunctions.) Strong normalization, the Church-Rosser property and decidability of commuting for free adjunctions generated by arrowless graphs need not be essentially tied to our standard (ac) formulation. Analogous results should be obtainable with the (cc), (aa) and (ca) formulation. However, we would then have to take into account other reductions in order to obtain uniqueness of normal form. For example, with the (ca) formulation, arrow terms in normal form would not be simply those without cuts and without subterms F1B and G1A, but redexes corresponding to the left-hand sides of the rectangular equalities (ϕcγaF) and (ϕcγaG) of § 4.5.5 would also need to be eliminated. (The proof of Associativity Elimination is much simplified with the (ca) formulation.) Solving the theoremhood or commuting problem in one formulation of adjunction solves it, of course, for all equivalent formulations. To decide the question in any formulation it suffices to translate it into the formulation for which decidability has been shown.

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§ 4.6.4. Decidability in free adjunctions generated by arbitrary graphs Up to now we have considered decision procedures for the theoremhood and commuting problem only for free adjunctions generated by a pair of arrowless graphs. We shall now consider these problems for free adjunctions generated by a pair of arbitrary graphs. (For that we rely on matters treated in § 4.5.4.) We shall be able to find a decision procedure for the theoremhood problem in the free adjunction between A* and B* generated by (G, H) if we have a decision procedure for the following problem in the graph G: Given a pair (A, A') of objects of G, is there a path in G from A to A'?

and a decision procedure for the analogous problem in the graph H (for the notion of path see § 4.6.2). By combining these decision procedures with Total Cut Molecularization and the subformula property of the (ac) formulation of adjunction, we obtain a decision procedure for the theoremhood problem in A* and B*. (This decision procedure is analogous to the resolution method of automatic theorem proving, where cut is applied only to axiomatic sequents.) We also have a decision procedure for the commuting problem in A* and B*, independently of any presuppositions concerning G and H. Let us describe this procedure. Arrow terms of A* and B* will now be considered to be in normal form iff every cut in them is molecular and they have no subterms of the form F1B or G1A. So arrow terms in normal form have the shape Q1…Qnh, where n ≥ 0, every Qi is either F, or G, or ϕa, or γc, and h is either 1C, in which case Qn is neither F nor G, or it is a molecular arrow term. With the help of Total Cut Molecularization and (fun 1) we easily obtain an ordinary normal form theorem. We can also introduce a collection of reductions with respect to which we can prove strong normalization, too. In this collection of reductions the redexes are

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either subterms of topmost nonmolecular cuts (defined as in the proof in § 4.5.4; i.e., the subterms of these cuts are nonmolecular arrow terms h2 ° h1 such that all the cuts in h1 and h2 are molecular), or they are of the form F1B or G1A. The form of the redexes and contracta is otherwise exactly as in the preceding section. Strong normalization and the Church-Rosser property are established as in that section. It follows that every arrow term of A* and B* has a unique normal form. But we shall not be able to establish that every arrow of A* and B* is designated by a unique arrow term in normal form, because different molecular cuts can be equal by (cat 2). To get that every arrow is designated by a unique arrow term in normal form we have to strengthen the notion of normal form to right-normal form: in rightnormal form we require that every cut be right-molecular, besides requiring as before that there be no subterms of the form F1B or G1A. (We could as well put “left” instead of “right”, and work throughout with left-normal forms.) We reduce an arrow term to right-normal form by enlarging our collection of reductions with instances of the following one: redex (h3 ° h2) ° h1

contractum h3 ° (h2 ° h1)

(cat 2 mol)

provided that h1, h2 and h3 are molecular arrow terms. We can again obtain strong normalization and the Church-Rosser property. (For strong normalization, we have to add to the complexity measure something that would make this measure decrease after applying (cat 2 mol) reductions. One way to do it is the following. We associate with every right parenthesis in a molecular arrow term h the number of cuts in h on the right of this parenthesis. Then we take the sum of all these numbers and add that to the complexity measure. This sum is zero for a right-molecular arrow term.) We can then prove that two arrow terms are equal in A* or B* iff they have the same right-normal form. From right to left this follows immediately from the fact that all reductions are covered by equalities that hold between the arrow terms of A* and B*. For the other

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direction we need the following generalization of Associativity Elimination. Associativity Molecularization. The equality (cat 2) is derivable from (cat 2 mol) and the remaining equalities assumed for the arrow terms of A* and B*. Proof. We follow the general pattern of the proof of Associativity Elimination in the preceding section. We have to derive

(cat 2)

( f3 ° f2) ° f1 = f3 ° ( f2 ° f1),

from (cat 2 mol) and the remaining equalities of A* and B*. We assume that f1, f2 and f3 are in normal form and make an induction on the sum of their degrees, where in these degrees we don’t count molecular cuts. In the basis of the induction, the case where all of f1, f2 and f3 are molecular (and where the sum of their degrees is hence 0) is taken care of by (cat 2 mol). The cases that have to be considered in the induction step are exactly those of the proof of Associativity Elimination. We just have to check that no additional case can arise with molecular arrow terms because of incompatibilities in the targets and sources in composition. (This checking is not short, but it is quite straightforward.) We proceed analogously to derive (cat 2) when f1, f2 and f3 are replaced by arrow terms of B*. q.e.d. Once we have replaced (cat 2) by (cat 2 mol), by imitating what we had in the preceding section, we can establish that two arrow terms of A* or B* are equal iff they have the same right-normal form. This yields a procedure for deciding equality of these arrow terms. Associativity Molecularization is a generalization of Associativity Elimination, because the latter follows from the former: when there are no generative arrow terms, there is no opportunity to apply (cat 2 mol). Altogether, the treatment of decidability, and in particular of the commuting problem, for free categories generated by arrowless graphs is just a particular case of the treatment that in this section we have accorded to this matter for free categories generated by arbitrary graphs. However, it seems preferable to treat first separately the

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simpler case, and then consider additions for the more general case. This is why we followed that course. (In particular, the statement of Associativity Molecularization is less sharp than the statement of Associativity Elimination, which is its consequence.) We shall approach decidability in such a gradual manner also later (in § 5.8 and § 6.4). § 4.7. RECTANGULAR \\ ADJUNCTIONS Among the notions of adjunction of § 4.1, we shall now consider the rectangular \\ notion, to see whether it behaves with respect to cut elimination as the rectangular || notion. We shall find that, though Cut Disintegration can be proved, it does not characterize exactly rectangular \\ adjunction, but a more general kind of rectangular \\ junction. In § 4.8 we shall then consider in the same way the triangular notion of adjunction. We select these two notions because adjunction is often defined in terms of them. In § 4.9 we will say something about cut elimination with other notions of adjunction from § 4.1. Throughout this part (§ 4.7), “junction” and “adjunction” should be understood in the sense of the definitions we are now going to give. § 4.7.1. Rectangular \\ junctions and adjunctions Let 〈A, 1, ° 〉 and 〈B, 1, ° 〉 be deductive systems. We continue applying the conventions of § 4.1.1 concerning letters for the objects and arrows of A and B. Next, let F be a graph-morphism from B to A and G a graph-morphism from A to B. For every object A of A let there be a function ΦA assigning to an arrow g : B → GA of B the arrow ΦAg: FB → A of A. Let Φ be the family of all the functions ΦA, or, more precisely, the function that assigns to A the function ΦA. Similarly, for every object B of B let there be a function ΓB assigning to an arrow f : FB → A of A the arrow ΓB f : B → GA of B. Let Γ be the family of all the functions ΓB, or,

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more precisely, the function that assigns to B the function ΓB. The double-indexed seesaw functions ΦB,A and ΓB,A from § 4.1 are obtained by restricting the domains of the functions ΦA and ΓB. We prefer to work now with these single-indexed seesaw functions because by getting rid of an index, which is not absolutely essential, we simplify our exposition (without hiding indices as we did in § 4.1.4). Then 〈A, B, F, G, Φ, Γ〉 will now be called a junction between A and B. A junctor from a junction 〈A, B, F, G, Φ, Γ〉 to a junction 〈A', B ', F', G', Φ', Γ '〉 will now be a pair (NA, NB) such that NA is a functor from the deductive system A to the deductive system A', and NB a functor from the deductive system B to the deductive system B '; moreover, the following naturalness equalities hold: NAF = F'NB, NAΦAg = Φ'NA ANB g,

NBG = G'NA, NBΓB f = Γ 'NB BNA f.

A junction 〈A, B, F, G, Φ, Γ〉 is an adjunction iff (i) 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories; (ii) F and G are functors; (iii) the following equalities hold in A and B: (ΦΓ ') (ΓΦ')

ΦAΓB f = f, for f : FB → A, ΓBΦAg = g, for g : B → GA,

(ΦΦ') (ΦΦ")

ΦA(g2 ° g1) = ΦAg2 ° Fg1, ΦA2(G f ° g) = f ° ΦA1g, for f : A1 → A2,

(ΓΓ ') (ΓΓ ")

ΓB( f2 ° f1 ) = G f2 ° ΓBf1, ΓB1( f ° Fg) = ΓB2f ° g, for g : B1 → B2.

This is for all purposes the same as the definition of rectangular \\ adjunction in § 4.1.4. Either (ΦΦ') and (ΦΦ"), or (ΓΓ ') and (ΓΓ "), could actually be omitted, since these equalities are derivable from the

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remaining equalities. The derived equalities would have the same obvious links between cuts as the primitive equalities (see below). § 4.7.2. Cut elimination in free rectangular \\ adjunctions We define the free junction 〈A, B, F, G, Φ, Γ〉 generated by a pair of graphs (G, H) as in § 4.3, save that instead of clause (4) of the definition of arrow terms we now have if g : B → GA is an arrow term of B, then ΦAg: FB → A is an arrow term of A; if f : FB → A is an arrow term of A, then ΓB f : B → GA is an arrow term of B. Note that since in the free junction F and G are one-one functions on objects, the indices of ΦA and ΓB are actually superfluous. The free adjunction between the categories A* and B* generated by (G, H) is obtained from this free junction by proceeding analogously to what we had in § 4.4. Note that now we have to envisage the congruence rules from f1 = f2 in A*, infer ΓB f1 = ΓB f2 in B*, from g1 = g2 in B*, infer ΦAg1 = ΦAg2 in A*. The definition of linked cuts from § 1.8.1 is easily adapted to equalities among arrow terms of A* and B* by taking that in all the equalities of (iii) of the preceding section the n-th cut on the left-hand side is linked to the n-th cut on the right-hand side; the additional congruence rules we have just mentioned preserve the links of the premises. We can then prove Cut Disintegration for the arrow terms of A* and B*. Proof of Cut Disintegration for A* and B*. We now define the degree of an arrow term as n1+n2, where n1 is the number of occurrences of Φ and Γ in the arrow term, and n2 the number of

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occurrences of 1, ° , F (applied to arrow terms), G (applied to arrow terms), Φ and Γ in the arrow term. So occurrences of Φ and Γ are counted twice. (We could also take as degrees pairs (n1, n2), lexicographically ordered). We then proceed as for the proof of § 4.5.1 (which is based on the proofs of § 1.8.1 and § 2.3.1). So, in an arrow term of A* or B*, take a nonatomic cut whose subterm is f2 ° f1. Then we replace the cases considered in the proof of § 4.5.1 by the following additional cases. (3.4) f1 is Fg and f2 is ΦAg'. Then ΦAg' ° Fg = ΦA(g' ° g) by (ΦΦ'), and the degree of the main cut of g' ° g is strictly smaller than the degree of the main cut on the left-hand side linked to it. (4) f1 is ΦA1g. Then f2 ° ΦA1g = ΦA2(G f2 ° g) by (ΦΦ"), and the degree of the main cut of G f2 ° g is strictly smaller than the degree of the main cut on the left-hand side linked to it. (We counted occurrences of Φ twice to obtain this.) (G.4) f1 is a generative arrow term of A* and f2 is ΦAg. This is excluded because f1 would then have a target of the form FB. We treat analogously a nonatomic cut whose subterm is g2 ° g1, i.e. an arrow term of B*, by using (ΓΓ ') and (ΓΓ ").q.e.d. For this proof we don’t need the equalities (ΦΓ ') and (ΓΦ'), provided all of (ΦΦ'), (ΦΦ"), (ΓΓ ') and (ΓΓ ") are assumed as primitive. We also don’t need (fun 1) for F and G. So, as for rectangular || adjunctions in the (cc), (aa) and (ca) formulations of § 4.5.5, the notion now characterized by Cut Disintegration is a kind of rectangular \\ junction more general than adjunction. (To confirm that, we have to show that (ΦΓ ') and (ΓΦ') cannot be derived from the other equalities of rectangular \\ adjunctions, including all of (ΦΦ'), (ΦΦ"), (ΓΓ ') and (ΓΓ ").) As before, we can establish Total Cut Molecularization for free rectangular \\ adjunctions generated by arbitrary graphs. For that we don’t need (cat 2). And, as in § 4.5.4, we need (cat 2 mol ) only for

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Total Cut Right-Molecularization Molecularization.

and

Total

Cut

Left-

§ 4.7.3. Decidability in free rectangular \\ adjunctions The rectangular \\ notion of adjunction of § 4.7.1 is not practical for solving the theoremhood decision problem for free adjunctions. The rules g : B → GA

f : FB → A

ΦA g : F B → A

ΓB f : B → GA

on which this notion is based, don’t satisfy the subformula property. (We can connect Φ with elimination rules and Γ with introduction rules in natural deduction, provided the functor F is “structural”; cf. § 6.1.) However, the rectangular \\ notion of adjunction can serve well to solve the commuting problem for free adjunctions. The corresponding normal forms have the shape Q1…Qnh, where n ≥ 0, every Qi is either F, or G, or ΦA, or ΓB , and QiQi+1 is neither ΦAΓB nor ΓBΦA; in case we deal with free adjunctions generated by arrowless graphs, h is 1C and Qn is neither F nor G, and in case we deal with free adjunctions generated by arbitrary graphs, h can also be a molecular arrow term. In addition to the reductions corresponding to (cat 1 right), (cat 1 left), (fun 1) and (fun 2) of § 4.6.3 (in both the A* and B* variant), and eventually the (cat 2 mol) reductions, we now have to envisage reductions oriented in the left-to-right directions of (ΦΓ ') and (ΓΦ'), and right-to-left directions of (ΦΦ'), (ΦΦ"), (ΓΓ ') and (ΓΓ "). Strong normalization and the Church-Rosser property are easily established. (For strong normalization we can measure the complexity of an arrow term by pairs (n1, n2), lexicographically ordered, where n1 is the sum of all the degrees of topmost nonmolecular cuts—with Φ and Γ counted twice, as in the proof of Cut Disintegration in the preceding section—and n2 is the degree of the arrow term. In the presence of (cat 2 mol), we need a third parameter n3 in triples 〈n1, n2, n3〉, lexicographically ordered, where n3 is the sum of the numbers associated

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with right parentheses in molecular arrow terms, as in the parenthetical remark after the introduction of (cat 2 mol) reductions in § 4.6.4; cf. also § 6.4.) We can also prove Associativity Elimination and Associativity Molecularization, rather more easily than in § 4.6.3 and § 4.6.4. § 4.8. TRIANGULAR ADJUNCTIONS We shall now consider whether the triangular notion of adjunction behaves with respect to cut elimination as the rectangular || notion. As for the rectangular \\ notion, we shall find that, though Cut Disintegration can be proved, it does not characterize exactly triangular adjunction, but a more general kind of triangular junction. Throughout this part (§ 4.8), “junction” and “adjunction” should be understood in the sense of the definitions we are now going to give. § 4.8.1. Triangular junctions and adjunctions Let 〈A, 1, ° 〉 and 〈B, 1, ° 〉 be deductive systems, let F be a graphmorphism from B to A and let G a function from the objects of A to the objects of B. Next, let ϕa be a function that to an arrow f : A1 → A2 of A assigns the arrow ϕa f : FGA1 → A2 of A. So ϕa is like the homonymous antecedental transformation of § 4.2, save that we cannot say that it is such a transformation because we haven’t assumed that G is a graph-morphism (this as with the objectual transformation ϕ of § 4.1.6; cf. § 1.3). Finally, as in § 4.7, let there be for every object B of B a function ΓB assigning to an arrow f : FB → A of A the arrow ΓB f : B → GA of B, and let Γ be the family of all the functions ΓB. Then we say that 〈A, B, F, ϕa, Γ〉 is a junction between A and B. A junctor from a junction 〈A, B, F, ϕa, Γ〉 to a junction 〈A', B ', F', ϕa', Γ '〉 will now be a pair (NA, NB) such that NA is a functor from the deductive system A to the deductive system A', and NB a functor from

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the deductive system B to the deductive system B '; moreover, the following naturalness equalities hold: NAF = F'NB, NAϕa f = ϕa'NA f,

NBGA = G'NAA, NBΓB f = Γ 'NB BNA f.

A junction 〈A, B, F, ϕa, Γ〉 is an adjunction iff (i) 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories; (ii) F is a functor; (iii) the following equalities hold in A and B: (ϕa 1)

ϕa( f2 ° f1) = f2 ° ϕa f1,

(β")

ϕa f2 ° FΓB f1 = f2 ° f1,

(ΓΓ ")

ΓB1( f ° Fg) = ΓB2 f ° g, for g : B1 → B2,

(Γϕa)

ΓGAϕa1A = 1GA.

The equalities (β") and (Γϕa) correspond, respectively, to the equalities (β') and (Γϕ) of § 4.1.6. The equalities (ΓΓ ") and (Γϕa) are interderivable with the equality ΓB(ϕa1A ° Fg) = g,

the analogue of (η) from § 4.1.6 (but this last equality is not practical for cut elimination). It is straightforward to show that this notion of adjunction is equivalent to the triangular notion of § 4.1.6, and hence to all the other notions of adjunction. § 4.8.2. Cut elimination in free triangular adjunctions We define the free junction 〈A, B, F, ϕa, Γ〉 generated by a pair of graphs (G, H) as in § 4.3, save that in clauses (3) and (4) we omit the parts dealing with G and γc; moreover we have clause (4) from § 4.7.2 with the part dealing with Φ omitted. The free adjunction between the categories A* and B* generated by (G, H) is obtained from this free

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junction by proceeding analogously to what we had in § 4.4 and § 4.7.2. The definition of linked cuts from § 1.8.1 is easily adapted to equalities among arrow terms of A* and B* by taking that in all the equalities of (iii) of the preceding section the n-th cut on the left-hand side is linked to the n-th cut on the right-hand side (the equality (Γϕa) plays no role in that); the additional congruence rules just preserve the links of the premises. We can then prove Cut Disintegration for the arrow terms of A* and B*. Proof of Cut Disintegration for A* and B*. We now define the degree of an arrow term as n1+n2, where n1 is the number of occurrences of Γ in the arrow term, and n2 the number of occurrences of 1, ° , F (applied to arrow terms), ϕa and Γ in the arrow term; so occurrences of Γ are counted twice. (We could also take as degrees pairs (n1, n2), lexicographically ordered). We then proceed as for the proof of § 4.5.1 (which is based on the proofs of § 1.8.1 and § 2.3.1). So, in an arrow term of A* or B*, take a nonatomic cut whose subterm is the arrow term f2 ° f1 of A*. Then, besides cases in the proof of § 4.5.1 that we find in the present context, too, we have the following additional case.

(3.4.Γ) f1 is FΓB f ' and f2 is ϕa f. Then ϕa f ° FΓB f ' = f ° f ' by (β"), and the degree of the main cut of f ° f ' is strictly smaller than the degree of the main cut on the left-hand side linked to it. Take now a nonatomic cut whose subterm is the arrow term g2 ° g1 of B*. Then we have the following cases. (1) - (2) If g2 is 1 or g2" ° g2', then we use (cat 1 left) and (cat 2) as in cases (1) and (2) of the proof in § 1.8.1. (Γ) g2 is ΓB2 f. Then ΓB2 f ° g1 = ΓB1( f ° Fg1) by (ΓΓ "), and the degree of the main cut of f ° Fg1 is strictly smaller than the degree of the

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main cut on the left-hand side linked to it. (We counted occurrences of Γ twice to obtain this.) (G) If g2 is a generative arrow term of B*, then g1 can be only 1 or g1" ° g1', and we proceed as for cases (1) and (2). q.e.d. For this proof we don’t need the equality (Γϕa) (but we used (fun 1) in case (3.4.1) of the proof of § 4.5.1, which is repeated here). So, as for rectangular || adjunctions in the (cc), (aa) and (ca) formulations of § 4.5.5, and as for rectangular \\ adjunctions in § 4.7.2, the notion now characterized by Cut Disintegration is a kind of triangular junction more general than adjunction. (To confirm that, we have to show that (Γϕa) cannot be derived from the other equalities of triangular adjunctions.) As before, we can establish Total Cut Molecularization for free triangular adjunctions generated by arbitrary graphs. For that we don’t need (cat 2). And, as in § 4.5.4, we need (cat 2 mol ) only for Total Cut Right-Molecularization and Total Cut Left-Molecularization. § 4.8.3. Decidability in free triangular adjunctions The triangular notion of adjunction of § 4.8.1 is not very practical for solving the theoremhood decision problem for free adjunctions. Among the rules f : A1 → A2

f : FB → A

ϕaf : F GA 1 → A 2

ΓB f : B → GA

on which this notion is based, the first satisfies the subformula property, but the second doesn’t. This difficulty could perhaps be overcome, since the number of occurrences of F in the premise is bounded by the number of occurrences of G in the conclusion. (We can connect ϕa with rules for introducing a connective on the left in Gentzen’s sequent systems, and Γ with rules for introducing a connective on the right, provided the functor F is “structural”; cf. §§ 6.1-3.) The triangular notion of adjunction can serve, too, for solving the commuting problem for free adjunctions. The corresponding normal

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forms have the shape Q1…Qnh, where n ≥ 0, every Qi is either F, or ϕa, or ΓB ; in case we deal with free adjunctions generated by arrowless graphs, h is 1C, while Qn is not F and Qn-1Qn is not ΓGAϕa, and in case we deal with free adjunctions generated by arbitrary graphs, h can also be a molecular arrow term. In addition to the reductions corresponding to (cat 1 right), (cat 1 left), (fun 1), (fun 2) and (ϕa 1) of § 4.6.3, on the A* side, plus reductions corresponding to (cat 1 right) and (cat 1 left) on the B* side, and eventually the (cat 2 mol) reductions in both A* and B*, we now have to envisage reductions oriented in the left-to-right directions of (β") and (Γϕa), and right-to-left direction of (ΓΓ "). Strong normalization and the Church-Rosser property are easily established. We can also prove Associativity Elimination and Associativity Molecularization, rather more easily than in § 4.6.3 and § 4.6.4. Of course, we could deal in a completely analogous manner with triangular ◄ adjunctions: there, γ would be replaced by γc. We could also envisage triangular ► adjunctions where ϕ is replaced by ϕc, instead of ϕa, as we had it above, and triangular ◄ adjunctions where γ is replaced by γa. Perhaps these notions are worth considering, but with them we would be further removed from the spirit of Gentzen’s systems. § 4.9. CUT ELIMINATION WITH OTHER NOTIONS OF ADJUNCTION Of the notions of adjunction of § 4.1, we still haven’t considered in the context of cut elimination the hexagonal notion, the rectangular // notion and the seesaw notion, i.e. the notion where only seesaw functions are primitive (besides functions on objects). This last notion is the easiest to deal with, since it does not enable cut elimination. We cannot eliminate cuts in arrow terms like ΦA1GA ° ΦFGA1GFGA and ΓGFB1FGFB ° ΓB1FB , which correspond to ϕA ° ϕFGA and γGFB ° γB. We have here the single-indexed seesaw functions ΦA and ΓB of § 4.7.1, with which we shall continue working below. Having these

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functions, rather than the double-indexed ones, doesn’t make any difference for the matters we are discussing. For cut elimination with the hexagonal notion of adjunction we have to replace the natural transformations ϕ and γ by natural antecedental or consequential transformations, and, as for the rectangular || notion, we would have four formulations: (ac), (cc), (aa) and (ca). In the (ac) formulation of hexagonal adjunction, the equalities (fun 2), (ϕa 1), (γc 1), (ΦΦ'), (ΦΦ"), (ΓΓ '), (ΓΓ ") and the following two equalities: ϕa f ° Fg = ΦA(G f ° g), G f ° γcg = ΓB( f ° Fg)

are sufficient for Cut Disintegration. However, these equalities don’t yield (ΦΓ ') and (ΓΦ'), nor (fun 1), and do not amount to adjunction. In the (ca) formulation of hexagonal adjunction, the equalities (fun 2), (ϕc 1), (ϕc 2), (γa 1), (γa 2), (ΦΦ'), (ΦΦ"), (ΓΓ ') and (ΓΓ ") are sufficient for Cut Disintegration. Again, these equalities don’t amount to adjunction. The situation is similar with the (cc) and (aa) formulations of hexagonal adjunction. The rectangular // notion also has four formulations. In the (ac) formulation, the equalities (ϕa 1), (γc 1), (ΦΓ '), (ΓΦ'), the following analogues of the rectangular equalities: ϕa f ° ΦAγcg = f ° ΦAg, ΓBϕa f ° γcg = ΓBf ° g,

and the equalities corresponding to (fun 2) ΦFB2γcg2 ° ΦFB1γcg1 = ΦFB2γc(g2 ° g1),

ΓGA2ϕa f2 ° ΓGA1ϕa f1 = ΓGA1ϕa( f2 ° f1)

are sufficient for Cut Disintegration. These equalities don’t yield the analogue of (fun 1) ΦFBγc1B = 1FB.

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In all that we haven’t considered formulations that would have all of ϕa, ϕc, γa and γc primitive. Such a formulation would transform the second two equalities displayed in the section on rectangular // adjunction (§ 4.1.5) into ΦAg = ϕcAΦFGAγcg,

ΓB f = γaBΓGFBϕa f.

We shouldn’t exclude that another interesting cut-free formulation of adjunction may still be found, but here we will not pursue the matter further. Abandoning this taxonomy of notions of adjunction, we can conclude that of all the standard notions only the rectangular || notion in the (ac) formulation is exactly characterized by Cut Disintegration. § 4.10. MODEL-THEORETICAL NORMALIZATION IN ADJUNCTIONS If there is a functor F from a category C to a category M, we may say that C is modelled by M. This terminology is appropriate when the modelled category C is an abstract category, like a freely generated category of some sort, while the modelling category M is concrete, either in the sense of being a subcategory of the category of sets with functions, or in a looser sense (cf. the end of § 4.10.1 below). Moreover, we may expect from the functor F to preserve the specific structure of C, and not only identity and composition. If the functor F is faithful, the modelling will be called faithful. The case when C is represented by M, i.e. when C is isomorphic to M, is just a special kind of faithful modelling. (Faithfulness gives what logicians call completeness, while soundness follows from functoriality.) Model-theoretical methods of normalization may be traced to an idea of Martin-Löf (see [Coquand & Dybjer 1997] and [Cubric et al. 1998], which contain brief surveys of related work). We normalize an arrow term f of C by having a computable function underlying the faithful functor F from C to M, and another computable function ν assigning to arrows of M arrow terms of C in normal form, such that

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in C we have f = ν(F f ) (see the beginning of § 4.10.2). Techniques of faithful modelling can provide a simple decision procedure for the commuting of diagrams in C, and also other gain, as we will see in the present part of this chapter, in §§ 4.10.1-2, and also later, in § 5.9-10 and § 6.9. There is some practical gain in model-theoretical methods of cut elimination and normalization, but the syntactical methods that were foremost in this chapter, and will remain such in the next one, are simple enough and easier to generalize. (Generalization may go either from free structures of some sort generated by arrowless graphs to free structures of the same sort generated by arbitrary graphs, or from free structures of one sort to free structures of another sort.) Moreover, these syntactical methods are presupposed in our demonstration that the model-theoretical methods work (see § 4.10.1 and § 5.9), though one may perhaps envisage a different demonstration. Finally, for the proof-theoretical characterization of categorial notions we need Cut Disintegration rather than Total Cut Elimination, and only the latter is delivered model-theoretically. This is why we took first the syntactical approach, and dwelt on it in more detail. § 4.10.1. A simple decision procedure for commuting in adjunctions We shall now present a decision procedure of a model-theoretical inspiration for the commuting problem in free adjunctions, which is simpler than the syntactical decision procedures of §§ 4.6.3, 4.6.4, 4.7.3 and 4.8.3. This decision procedure is based on drawing links between symbols in the object terms that are the source and target of an arrow term. However, to show that this decision procedure works we rely on reduction to cut-free normal form such as we had before. More precisely, we rely just on the normal form theorem, but not on strong normalization, the Church-Rosser property and the uniqueness of normal form (see § 4.6.3). This new decision procedure is not tied to a particular formulation of the notion of adjunction: variants of it exist for all notions of adjunction we have envisaged, including those which are not “cut-

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free”, i.e. which don’t permit cut elimination. We shall describe here this procedure for the rectangular || notion of adjunction of § 4.1.3, which is not “cut-free”. To simplify matters, we concentrate on the commuting problem in such free adjunctions generated by arrowless graphs. This technique enriched with further devices could perhaps be extended to the commuting problem in free adjunctions generated by arbitrary graphs, but for this more general case it doesn’t seem to be decisively simpler than the technique described in § 4.6.4. So take the rectangular || notion of adjunction of § 4.1.3, and consider the free adjunction between A* and B* generated by a pair of arrowless graphs. To every arrow term h : C1 → C2 of A* or B* we assign a set of links Λ(h), which is a set of unordered pairs of occurrences of F or G in C1 or C2. (We could also conceive links that are ordered pairs.) We define Λ(h) by induction on the complexity of h. If h is 1C, then C1 and C2 are two copies of C, and in Λ(h) we put the links between the n-th F of C1 and the n-th F of C2, as well as the links between the n-th G of C1 and the n-th G of C2. If h is ϕA: FGA → A, then Λ(h) is obtained by adding to a copy of Λ(1A) the link between the first F and the first G of FGA, counting from the left. If h is γB: B → GFB, then Λ(h) is obtained by adding to a copy of Λ(1B) the link between the first G and the first F of GFB. If h is of the form Fg : FB1 → FB2, then Λ(h) is obtained by adding to a copy of Λ(g) the link between the first F of FB1 and the first F of FB2. If h is of the form G f : GA1 → GA2, then Λ(h) is obtained by adding to a copy of Λ( f ) the link between the first G of GA1 and the first G of GA2. Finally, if h : C1 → C2 is of the form h2 ° h1 for h1: C1 → C3 and h2: C3 → C2, then every link {x1, xn} of Λ(h) is obtained from a finite sequence of links {x1, x2}, {x2, x3}, {x3, x4}, … , {xn-1, xn}, with n ≥ 2, in which links of Λ(h1) alternate with links of Λ(h2) (in this sequence there is at least one link from Λ(h1) or Λ(h2)); the vertices x1 and xn are occurrences of F or G in C1 or C2, and if n ≥ 3, then all of x2, … ,

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xn-1 are occurrences of F or G in C3. (It is easy to see that there can be no sequence of links of the form {x1, x2}, {x2, x3}, … , {x2n-1, x2n}, {x2n, x1}, with n ≥ 1, where all of x1, … , x2n are occurrences of F or G in C3.) Note that for h : C1 → C2 every F or G in C1 or C2 is linked in Λ(h) to exactly one other occurrence of F or G in C1 or C2. The links we have just defined are related to the graphs of [Eilenberg & Kelly 1966], which one also finds in [Kelly & Mac Lane 1971] (see also [D. & Petric 1997, section 2]). The term “link” is used in knot theory for a collection of knots, and this is, of course, a different notion from our notion of link. In knot theory, a set of our links is a special kind of tangle (see [Murasugi 1996, Chapter 9]). Categories of tangles have played recently a prominent role in the theory of quantum groups, in low-dimensional topology and in knot theory (see [Kassel 1995, Chapter 12], [Kauffman & Lins 1994], and references therein). Once links have been defined for arrow terms of A* and B* in one formulation of adjunction, the links of arrow terms of A* and B* in another formulation of adjunction are obtained by taking the arrow terms of the latter formulation as defined in terms of the former. So the links of the “cut-free” (ac) formulation of the rectangular || notion of adjunction of § 4.2 are defined by relying on the definitions ϕa f =def f ° ϕA, γcg =def γB ° g,

which means that for ϕa f : FGA1 → A2 we obtain Λ(ϕa f ) by adding to a copy of Λ( f ) the link between the first F and the first G of FGA1, while for γcg: B1 → GFB2 we obtain Λ(γcg) by adding to a copy of Λ(g) the link between the first G and the first F of GFB2. A cut-free arrow term of A* or B* of the form Q1…Qn1E where n ≥ 0, every Qi is either F, or G, or ϕa, or γc, and E is a generative object term will be said to be in atomic normal form. (This normal form, which we call atomic because E is generative, corresponds to the thin normal form of natural-deduction proofs—with eliminations

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preceding introductions—where the middle part is atomic; cf. §§ 0.3.5-6. In the lambda calculus, an analogous normal form is sometimes called long.) We need this notion of normal form for the proof of the following lemma concerning links of arrow terms of A* and B* in the normal form of § 4.6.3. Lemma. Suppose h1 and h2 are arrow terms of A* or B* in normal form, which are of the same type. Then Λ(h1) = Λ(h2) iff h1 is the same arrow term as h2. Proof. Suppose h1 and h2 are arrow terms of A* or B* of the same type in the normal form of § 4.6.3, and let h1' and h2' be arrow terms in atomic normal form such that h1 = h1' and h2 = h2'; the arrow terms h1' and h2' are obtained from h1 and h2 by perhaps applying (fun 1). It is clear that the links of h1 are equal to the links of h1', and the links of h2 to those of h2'. Since the types of h1 and h2 are equal, if h1' is Q1…Qn1E, then h2' must be P1…Pm1E for the same generative object term E, every Pi being either F, or G, or ϕa, or γc. The sequence Q1…Qn cannot be a proper initial segment of P1…Pm, nor can the latter sequence be a proper initial segment of the former; otherwise, the types of h1 and h2 would differ. If h1 and h2 are different arrow terms, then the sequences Q1…Qn and P1…Pm must be different, and this can only be if for some i we have that Qi and Pi are different. Let j be the smallest i such that Qi and Pi are different. This means either that one of Qj and Pj is F while the other is ϕa, or that one of Qj and Pj is G while the other is γc. In both cases, Λ(h1) will differ from Λ(h2). So, by contraposition, if Λ(h1) = Λ(h2), then h1 is the same arrow term as h2. The converse implication is, of course, trivial. q.e.d.

Returning now to arrow terms of the rectangular || notion of adjunction of § 4.1.3, we can demonstrate the following proposition, which is of a kind called “coherence result” by categorists (cf. [Mac Lane 1971, VII.2] and [Kelly & Mac Lane 1971]), but which logicians would consider rather to be a completeness result.

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Proposition. Suppose f1 and f2 are arrow terms of A* of the same type, while g1 and g2 are arrow terms of B* of the same type. Then we have f1 = f2 in A* iff Λ( f1) = Λ( f2), and we have g1 = g2 in B* iff Λ(g1) = Λ(g2). Proof. To prove these equivalences from left to right we proceed by induction on the length of the derivation of f1 = f2 in A* and g1 = g2 in B*. In this induction, the basis is essential, while the induction step is trivial. For this basis we have to check all the equalities of the definition of adjunction of § 4.1.3 to ascertain that the links of the left-hand side are equal to the links of the right-hand side. Let us take as an example just the equality (ϕγF), for which we have the following links: FγΒ ϕFB

FB

FB

FGFB FB

F1Β FB

To prove the equivalences of the Proposition from right to left, we translate arrow terms of A* and B* into arrow terms of the “cut-free” (ac) formulation of the rectangular || notion of adjunction of § 4.2. Let f1' and f2' be the normal forms in the sense of § 4.6.3 of the translations of f1 and f2. If Λ( f1) = Λ( f2), then Λ( f1') = Λ( f2'), since Λ( f1) = Λ( f1') and Λ( f2) = Λ( f2'), by the left-to-right direction of the Proposition. Then by the left-to-right direction of the Lemma proved in this section we conclude that f1' is the same arrow term as f2', and hence we have f1 = f2 in A*. We proceed analogously with g1 and g2. q.e.d.

So, to answer the question whether for h1 and h2 of the same type h1 = h2 is satisfied in A* or B*, it is enough to draw the links of Λ(h1) and Λ(h2), and check whether they are equal.

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To extend this decision procedure to equalities h1 = h2 in free adjunctions generated by arbitrary graphs, we need to check not only whether Λ(h1) = Λ(h2), but also whether the molecular arrow terms involved in h1 and h2 can be equated. These molecular arrow terms become apparent by reducing h1 and h2 to the normal form of § 4.6.4. The Proposition proved in this section guarantees that there is a faithful functor from A* to the category whose objects are finite, possibly empty, sequences of alternating F’s and G’s, which, if they are nonempty, must begin with F; the arrows of this category are links. (Note that this category is not concrete in the technical sense of § 1.9—it is not a subcategory of the category of sets with functions— but it is concrete in a loose sense.) Analogously, it is guaranteed that there is a faithful functor from B* to the category whose objects are such sequences of F’s and G’s where nonempty sequences begin with G; arrows are again links. If each of the arrowless generating graphs G and H has at least one object, then these functors are onto on objects and on arrows, and if both of G and H are arrowless with a single object, then these functors are isomorphisms. § 4.10.2. Normalizing via links and uniqueness of normal form Consider again the adjunction between A* and B* generated by a pair of arrowless graphs. The links of the preceding section provide a simple model-theoretical method for normalizing an arrow term of A* or B*. This method consists in drawing the links of the arrow term, and then constructing in an obvious way out of these links an arrow term in atomic normal form with the same links (atomic normal form is defined before the Lemma of the preceding section). We pass trivially from the atomic normal form to the normal form of § 4.6.3 by applying (fun 1). The normal form theorem of § 4.6.3 and the Proposition of the preceding section guarantee that links are always such that this construction will succeed. It is easy to see that this method is constructive: namely, the function that produces the links out of the arrow term and the function that assigns to the links an arrow term in normal form are computable functions.

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The following example shows how the latter function works. If A is a generative object term, out of the links FGFG A FGFG A

we construct the arrow term in atomic normal form ϕaFγcG1A, which yields the normal form ϕaFγc1GA. The links of the preceding section can also provide an alternative proof of the uniqueness of normal form of § 4.6.3. There we established with the help of the Church-Rosser property for our reductions and with the help of Associativity Elimination that if h1 = h2 in A* or B*, and h1' and h2' are normal forms of, respectively, h1 and h2, then h1' is the same arrow term as h2'. This implication could also be proved as follows. Suppose we have h1 = h2 in A* or B*, and h1' and h2' are normal forms of the arrow terms h1 and h2. Then by the left-to-right direction of the Proposition of § 4.10.1 we can conclude that Λ(h1) = Λ(h2), as well as Λ(h1) = Λ(h1') and Λ(h2) = Λ(h2'). So Λ(h1') = Λ(h2'). Since h1' and h2' are of the same type and in normal form, we can conclude according to the left-to-right direction of the Lemma of § 4.10.1 that h1' is the same arrow term as h2'. With that we have eschewed using the Church-Rosser property and Associativity Elimination. This new proof is easier than what we had before. However, the full proof should include the inductive proof of the left-to-right direction of the Proposition of § 4.10.1, which we have only sketched. We have preferred to work in § 4.6.3 with the Church-Rosser property in order to show that this standard technique, typical for the lambda calculus, can also be applied here. The Associativity Elimination that we proved directly in § 4.6.3 and needed for uniqueness of normal form, can also be established with the help of links. To derive

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(cat 2)

(h3 ° h2) ° h1 = h3 ° (h2 ° h1)

from the remaining equalities of A* and B* we can reduce both sides to the same normal form h'. The possibility of this reduction is guaranteed above, with the help of links, by the Lemma and the Proposition of § 4.10.1. Then we derive (cat 2) by establishing (h3 ° h2) ° h1 = h', h3 ° (h2 ° h1) = h' as in the reduction procedure, which does not involve (cat 2). However, besides Associativity Elimination, we proved also its generalization, Associativity Molecularization, and we used that together with the Church-Rosser property for uniqueness of normal form in free adjunctions generated by arbitrary graphs. This other uniqueness could perhaps be established by some sort of generalization of the Lemma and of the Proposition of § 4.10.1, which would also imply Associativity Molecularization, but this would not involve simply the links introduced in § 4.10.1. § 4.11. THE MAXIMALITY OF ADJUNCTION As an application of our cut-elimination results for adjunctions we shall demonstrate in this section that the notion of adjunction is maximal in the sense that any equality between arrow terms of junctions not assumed for adjunction yields, in the presence of the equalities of adjunction, the preordering equalities of § 4.6.2, namely ϕaFγcg = FGFg, ϕaFG f = FGϕa f,

γcGϕa f = GFG f, γcGFg = GFγcg

Take the rectangular || notion of adjunction in the (ac) formulation of § 4.2 and consider the free adjunction between A* and B* generated by a pair of arrowless graphs. Then let f1 and f2 be arrow terms of A*, both of the same type A1 → A2, and let g1 and g2 be arrow terms of B*, both of the same type B1 → B2. Suppose f1 = f2 is not satisfied in A* and g1 = g2 is not satisfied in B*.

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Let there be an adjunction between the categories A and B. We say that f1 = f2 holds in A iff for every junctor (NA, NB) from the adjunction between A* and B* to the adjunction between A and B we have that NA f1 = NA f2 is satisfied in A. We say, analogously, that g1 = g2 holds in B iff for every junctor (NA, NB) from the adjunction between A* and B* to the adjunction between A and B we have that NBg1 = NBg2 is satisfied in B. (This notion of holding is universal, which means that an equality of A* or B* is satisfied for every evaluation of the generative object terms, and not only for some evaluation.) We can then prove the following proposition. Maximality of Adjunction. If f1 = f2 holds in A or g1 = g2 holds in B, then we have the preordering equalities in A and B. Proof. Every arrow term h of A* or B* can be brought into a unique cut-free atomic normal form h', which is of the shape Q1…Qn1E where n ≥ 0, every Qi is either F, or G, or ϕa, or γc, and E is a generative object term. (We defined atomic normal form before the Lemma of § 4.10.1.) The unique atomic normal form is easily obtained from the unique cut-free normal form of § 4.6.3 by applying (fun 1). It is clear that if f1 = f2 holds in A, then f1' = f2' holds in A, and if g1 = g2 holds in B, then g1' = g2' holds in B. Note that since f1 = f2 isn’t satisfied in A* and g1 = g2 isn’t satisfied in B*, the arrow term f1' must be different from the arrow term f2' and the arrow term g1' must be different from the arrow term g2'. Let the length of an arrow term Q1…Qn1E in atomic normal form be n. First we show that if f1' = f2' holds in A, then the length of f1' must be equal to the length of f2'. Note first that the reductions going from left to right of the preordering equalities of § 4.6.2 don’t change the length. Of course, they also don’t change the type of the arrow term. With these reductions we can bring f1' into another normal form f1† of the shape Q1…Qn1E where among the Qi’s we don’t have occurrences of both ϕa and γc (i.e., we have either only occurrences of

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ϕa without any occurrence of γc, or only occurrences of γc without any occurrence of ϕa, or neither occurrences of ϕa nor occurrences of γc). We likewise reduce f2' to the normal form f2†. The arrow terms f1† and f2† must have the same number of occurrences of ϕa and the same number of occurrences of γc, because, otherwise, their types would be different. They must also have the same number of occurrences of F and G, for the same reason. We proceed quite analogously to show that if g1' = g2' holds in B, then the length of g1' must be equal to the length of g2'. (A similar balance in equalities can be achieved with normal forms in other formulations of adjunction.) The length of the equality f1' = f2' is the length of f1' and f2'. Suppose now f1' = f2' holds in A and the length of f1' = f2' is greater than 3. (The arrow terms f1' and f2' are, as above, different arrow terms of A* in atomic normal form.) We want to show that there is then an equality h1 = h2 holding in A such that h1 and h2 are different arrow terms of A* in atomic normal form and the length of h1 = h2 is strictly smaller than the length of f1' = f2'. We have the following cases.

(1) If f1' = f2' is of the form Fg1 = Fg2, then we have the following subcases. (1.1) If Fg1 = Fg2 is of the form FG f1" = FG f2", then the arrow term f1" must be different from the arrow term f2", and we have ϕa1A ° FG f1" = ϕa 1A ° FG f2" ϕa f1" = ϕa f2", by (ϕa 2).

This last equality is the desired h1 = h2. (1.2) If Fg1 = Fg2 is of the form Fγcg1' = Fγcg2', then the arrow term g1' must be different from the arrow term g2'. By composing, as above, both sides with ϕa1A on the left, and by applying (ϕaγcF), we obtain Fg1' = Fg2', which is the desired equality h1 = h2. (1.3) If Fg1 = Fg2 is of the form FG f1" = Fγcg2', then again by composing both sides with ϕa1A on the left, and by applying

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(ϕa 2) and (ϕaγcF), we obtain ϕa f1" = Fg2', which is the desired equality h1 = h2. (2) If f1' = f2' is of the form ϕa f1" = ϕa f2", then the arrow term f1" must be different from the arrow term f2", and we have ϕa f1" ° Fγc1B = ϕa f2" ° Fγc1B f1" = f2", by (ϕaγcF).

This last equality is the desired h1 = h2. (3) If f1' = f2' is of the form ϕa f = Fg, then we have the following subcases. (3.1) If Fg is of the form FG f ', then we have the following subcases. (3.1.1) If FG f ' is of the form FGFg", then we have

(γc 2).

ϕa f ° Fγc1B = FGFg" ° Fγc1B f = Fγcg", by (ϕaγcF), (fun 2) and

If f is of the form ϕa f ", then ϕa f " = Fγcg" is the desired equality h1 = h2. If, on the other hand, f is of the form Fg', then we have the following. If the arrow term g' is different from the arrow term γcg", then Fg' = Fγcg" is the desired equality h1 = h2. If, on the other hand, g' is γcg", then f1' = f2' is the equality (ρA *)

ϕaFγcg" = FGFg".

Since the length of (ρA *) is greater than 3, we have the following subcases. (3.1.1.1) If g" is of the form G f ", then we have FGϕa f " = FGϕa 1A ° FGFGf ", by (ϕa 2) and (fun 2) = FGϕa 1A ° ϕaFγcGf ", by (ρA *) = ϕaFGf ", by (ϕa 1), (fun 2) and (ϕaγcG) and FGϕa f " = ϕaFGf " is the desired equality h1 = h2. (3.1.1.2) If g" is of the form γcg'', then we have

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ϕa Fγcg'' = ϕa 1A ° FGFγcg'', by (ϕa 2)

(ϕa 1)

= ϕa 1A ° ϕaFγcγcg'', by (ρA *) = ϕa 1A ° ϕaFγc(Gϕa 1A' ° γcγcg''), by (ϕaγcG) = ϕa 1A ° FγcGϕa 1A' ° ϕaFγcγcg'', by (γc 1), (fun 2) and = ϕa 1A ° FγcGϕa 1A' ° FGFγcg'', by (ρA *) = FGFg'', by (ϕaγcF) and (fun 2)

and ϕa Fγcg'' = FGFg'' is the desired equality h1 = h2. (3.1.2) If FG f ' is of the form FGϕa f ", then we have ϕa f ° Fγc1B = FGϕa f " ° Fγc1B f = FG f ", by (ϕaγcF), (fun 2) and (ϕaγcG).

If f is of the form ϕa f '', then ϕa f '' = FG f " is the desired equality h1 = h2. If, on the other hand, f is of the form Fg', then we have the following. If the arrow term g' is different from the arrow term G f ", then Fg' = FG f " is the desired equality h1 = h2. If, on the other hand, g' is G f ", then f1' = f2' is the equality (πA *)

ϕaFGf " = FGϕa f ".

Since the length of (πA *) is greater than 3, we have the following subcases. (3.1.2.1) If f " is of the form Fg", then we have ϕa Fγcg" = ϕa FGFg" ° FGFγc1B, by (γc 2), (fun 2), (ϕa 1) and (ϕa 2)

= FGϕa Fg" ° FGFγc1B, by (πA *) = FGFg", by (fun 2) and (ϕaγcF) and ϕa Fγcg" = FGFg" is the desired equality h1 = h2. (3.1.2.2) If f " is of the form ϕa f '', then we have ϕa FG f '' = ϕa F(Gϕa f '' ° γc1B), by (ϕaγcG) = ϕa FGϕa f '' ° FGFγc1B, by (fun 2) and (ϕa 2)

= FGϕa ϕa f '' ° FGFγc1B, by (πA *)

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= FGϕa f '', by (fun 2) and (ϕaγcF) and ϕa FG f '' = FGϕa f '' is the desired equality h1 = h2. (3.2) Let (γc)n abbreviate a sequence of n ≥ 0 occurrences of γc. Suppose Fg is of the form F(γc)ng', where n ≥ 1 and the arrow term g' does not begin with γc. Since the source of ϕa f is of the form FGA, the arrow term g' must be of the form G f '. So Fg is of the form F(γc)nG f '. The arrow term f ' cannot be 1E, because the source of ϕa f is of the form FGFB (the length of ϕa f being greater than 3). We have the following subcases. (3.2.1) If f ' is of the form Fg", then we have

(γc 2).

ϕa f ° Fγc1B = F(γc)nGFg" ° Fγc1B f = F(γc)n+1g", by (ϕaγcF), (fun 2), (γc 1) and

If f is of the form ϕa f ", then ϕa f " = F(γc)n+1g" is the desired equality h1 = h2. If, on the other hand, f is of the form Fg'', then we have the following. If the arrow term g'' is different from the arrow term (γc)n+1g, then Fg'' = F(γc)n+1g" is the desired equality h1 = h2. If, on the other hand, g'' is (γc)n+1g, then f1' = f2' is the equality (ρA *n)

ϕaF(γc)n+1g" = F(γc)nGFg".

Then we have (ϕa 1)

ϕaF(γc)ng" = ϕa 1A ° ϕa F(γc)n+1g", by (ϕaγcF) and

= ϕa 1A ° F(γc)nGFg", by (ρA *n) = F(γc)n-1GFg", by (ϕaγcF) and this equality (ρA *n-1) is the desired h1 = h2. (3.2.2) If f ' is of the form ϕa f ", then we have

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ϕa f ° Fγc1B = F(γc)nGϕa f " ° Fγc1B f = F(γc)nG f ", by (ϕaγcF), (fun 2) and (γc 1) and a c (ϕ γ G).

If f is of the form ϕa f ", then ϕa f " = F(γc)nG f " is the desired equality h1 = h2. If, on the other hand, f is of the form Fg", then we have the following. If the arrow term g" is different from the arrow term (γc)nG f ", then Fg" = F(γc)nG f " is the desired equality h1 = h2. If, on the other hand, g" is (γc)nG f ", then f1' = f2' is the equality (πA *n)

ϕaF(γc)nGf " = F(γc)nGϕa f ".

Then we have (ϕa 1)

ϕa F(γc)n-1G f " = ϕa 1A ° ϕa F(γc)nG f ", by (ϕaγcF) and

= ϕa 1A ° F(γc)nGϕa f ", by (πA *n) = F(γc)n-1Gϕa f ", by (ϕaγcF) and this equality (πA *n-1) is the desired h1 = h2. So the existence of h1 = h2 has been demonstrated in every case. Note that (ρ A 1 ) (πA 1)

ϕaFγc1B = FGF1B, ϕaFG1A = FGϕa1A,

for B and A generative object terms, are the only equalities h1 = h2 with h1 and h2 different arrow terms of A* in atomic normal form of length 3. There are no such equalities of length smaller than 3. To show that these two equalities yield the preordering equalities (ρA) (πA)

ϕaFγcg = FGFg, ϕaFGf = FGϕa f

note that with the help of (ϕa 1), (ϕa 2), (γc 1) and (γc 2) we have the following equalities in A*:

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ϕaFγcg = FGFg ° ϕaFγc1B1,

= ϕaFγc1B2 ° FGFg,

ϕaFGf = FG f ° ϕaFG1A, FGϕa f = FG f ° FGϕa1A.

(If (ρA 1) and (πA 1) hold in A, then they are satisfied for every object B of B and every object A of A.) That (ρA) yields (πA) is demonstrated as in case (3.1.1.1) above. The converse deduction is made as in case (3.1.2.1). That (ρA) yields ( ρ B) (πB)

γcGϕaf = GFG f, γcGFg = GFγcg

is shown by the deduction γcGFg = GFGF1B ° γcGFg, by (fun 1)

(γc 1).

= GϕaFγc1B ° γcGFg, by (ρA) = GFγcg, by (ϕaγcG), (fun 2) and

The equivalence of (ρB) and (πB) is demonstrated analogously to the equivalence of (ρA) and (πA) above, and that (ρB) yields (ρA) and (πA) is shown by a deduction analogous to that we have just made. The derivations above that depend only on the equalities of adjunctions can be checked easily with the help of the links of arrow terms of § 4.10.1, but we have preferred to volunteer some additional information by mentioning the equalities involved. We show quite analogously that if g1' = g2' holds in B and the length of g1' = g2' is greater than 3, then there is an equality h1 = h2 holding in B such that h1 and h2 are different arrow terms of B* in atomic normal form and the length of h1 = h2 is strictly smaller than the length of g1' = g2'. q.e.d.

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Maximality of Adjunction is not essentially tied to the (ac) formulation of the rectangular || notion of adjunction. Analogous propositions can be demonstrated with other notions of adjunction. Maximality of Adjunction says that all the equalities of junctions that don’t follow from the equalities of adjunction are of the same strength as the preordering equalities. Any such equality yields the preordering equalities, and the preordering equalities yield this equality, as it follows from the Proposition proved in § 4.6.2. So all the equalities of junctions that are not equalities of adjunction are in the same bag. And any of these equalities when added to the equalities of adjunction makes trivial the resulting special notion of adjunction, in the sense of the Proposition in § 4.6.2. Maximality of Adjunction can also be established by relying on links, as we shall see in § 5.11.

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CHAPTER 5

COMONADS

§ 5.1. DEFINITIONS OF COMONAD Before considering cut elimination in comonads later in this chapter, we survey in this introductory part the definitions of comonad. We shall give the standard definition of this notion and present several alternative definitions, of equivalent notions. The principle guiding this survey will be the adjunction that obtains sometimes between the category of our comonad and a subcategory of it, related to the Kleisli category and to the category of free coalgebras of the comonad, which we will call the delta category. This adjunction defines the comonad, and since adjunction can be formulated in various ways, as we saw in § 4.1, we may envisage various definitions of comonad. After extracting as many interesting definitions as we could find, we compare the delta category of a comonad to the Kleisli and Eilenberg-Moore categories of the comonad. These last categories play an essential role in the adjunctions involving the category of adjunctions and the category of comonads, which we shall consider in the next introductory part (§ 5.2). After these two introductory parts, we enter into the business of cut elimination in comonads. Of course, we could as well deal throughout with monads. Our only reason for preferring comonads is that, from a logical point of view, they seem to bear a certain primacy over monads, as the universal quantifier and the necessity operator bear a primacy over the existential quantifier and the possibility operator. § 5.1.1. Standard definition of comonad Suppose we are given the following:

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a deductive system 〈A, 1, ° 〉, a graph-morphism D from A to A, a transformation ε from D to the identity graph-morphism IA, a transformation δ from D to the composite graph-morphism DD. So in ε we have the arrows εA: DA → A, and in δ the arrows δA: DA → DDA. Then we say that 〈A, D, ε, δ〉 is a comonograph. We may say that this is a comonograph in A, and later we use sometimes the same form of speaking with comonads. To simplify the notation, we don’t mention the identity and composition of 〈A, 1, ° 〉, taking them for granted (as we did in similar situations, with formations and junctions, in the preceding chapters). A monograph would be a comonograph with arrows reversed— sources become targets and targets sources. Note that the function D on objects in a comonograph resembles the necessity operator of the modal logic S4, or a topological interior operation. In a monograph it would resemble the possibility operator of S4, or a closure operation. The appropriate morphisms between comonographs will be called comonofunctors. A comonofunctor from a comonograph 〈A, D, ε, δ〉 to a comonograph 〈A', D', ε', δ'〉 is a functor N from the deductive system A to the deductive system A' such that the following naturalness equalities hold: ND = D'N, NεA = ε'NA, NδA = δ'NA. A comonad is a comonograph 〈A, D, ε, δ〉 such that 〈A, 1, ° 〉 is a category, D is a functor, ε and δ are natural transformations, the following equalities hold:

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(εδ)

εDA ° δA = 1DA,

(εδD)

DεA ° δA = 1DA,

(δδ)

DδA ° δA = δDA ° δA.

A monad (also called a triple) is a comonad with arrows reversed. Note that if in a comonad 〈A, D, ε, δ〉 the object A1 is the same object as A2, then δA1 = δA2, and if δA1 = δA2, then DA1 is the same object as DA2; we also have, of course, that if A1 is the same object as A2, then DA1 is the same object as DA2. But the converse implications need not obtain, as we show below. We will call a comonad δ-injective iff the function δ is one-one, i.e., it establishes a bijection between the objects A of A and the arrows δA. Next, we will call a comonad δ-preordered iff there is a bijection between the arrows δA and the objects DA of A. (The notions we have just defined make sense also for comonographs in general.) It is clear that in a comonad D is one-one on objects iff the comonad is δ-injective and δ-preordered, but we cannot infer that D is one-one on objects from the sole assumption that the comonad is δ-injective, nor from the sole assumption that it is δ-preordered. For that we have the following counterexamples. That δ-preorder does not imply δ-injectivity is shown by the trivial comonad in the category of sets with functions where DA is the empty set for every set A. That δ-injectivity does not imply δ-preorder is shown by the comonad defined as follows. Take the full subcategory of the category of sets with functions whose objects are all nonempty finite ordinals with the additional object {1}. The ordinal Dn is n+1 and D{1} is {0, 1}, i.e. the ordinal 2. Since D{0} is also 2, the function D is not one-one on objects. For the function f : A → B, the function D f : DA → DB is defined by taking that D f (x) is f (x) if x ∈ A, and the unique element of DB - B if x A, i.e. if x ∈ DA - A. This defines the functor D. For the natural transformations ε and δ, the function εA: A → DA is inclusion and δA: DDA → DA is defined by taking that δA(x) is x if x ∈ A, and the unique element of DA - A if x A. To obtain our comonad it remains to take domains as targets and codomains as

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sources of arrows. This comonad is δ-injective, but δ{0}(2) = 1, while δ{1}(2) = 0; so it is not δ-preordered. (This comonad should be compared with the concrete category of finite ordinals considered in connection with links in free comonads, near the end of § 5.9.) We are interested in δ-injectivity and δ-preorder because in the free comonads we shall consider later in this chapter D is one-one on objects, and hence these comonads are δ-injective and δ-preordered. § 5.1.2. The delta category Let 〈A, D, ε, δ〉 be a comonad, and for an arrow f : DA → A' of A let the arrow ∆δA f : DA → DA' be defined by ∆δA f =def D f ° δA.

The operations ∆δA are in one-to-one correspondence with the arrows

δA, since δA is equal to ∆δA1DA. When our comonad is δ-preordered,

the subscript δA of ∆δA is induced by the source of f, and we can

assume we have a single operation ∆, without subscript, defined on all the arrows f : DA → A' for A an arbitrary object of A. We recover ∆δA

by restricting the domain of ∆. When the comonad is not δ-preordered, we cannot obtain ∆δA by restricting the domain of a single function ∆, because ∆δA can differ from ∆δA though DA1 is the 1

2

same object as DA2. In order not to have double, or even triple, indices, we can replace ∆δA by ∆A, but, irrespectively of whether the comonad is δ-preordered

or not, we will often omit the subscript of ∆, taking it for granted; we will write it only when we deem it is necessary. For 〈A, D, ε, δ〉 an arbitrary comonad, consider the subgraph A∆ of A whose objects are the objects of A of the form DA and whose arrows are the arrows of A of the form ∆f. In A∆, there is an identity made of the arrows 1DA of A and the composition of ∆f1: DA1 → DA2

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and ∆f2: DA2 → DA3 is defined as the arrow ∆f2 ° ∆f1 of A. To ensure that 1DA and ∆f2 ° ∆f1 are indeed arrows of A∆ we check that the following equalities hold in A: (∆ε) (∆ ° )

∆AεA = 1DA, ∆( f2 ° ∆f1) = ∆f2 ° ∆f1

(∆A stands for ∆δA in (∆ε), and, similarly, the symbols ∆ in (∆ ° ) should bear appropriate subscripts). It is clear that A∆ is a category with this identity and this composition; namely, it is a subcategory of A. We call A∆ the delta category of our comonad. When the comonad 〈A, D, ε, δ〉 is δ-preordered, there is an adjunction between A and A∆ where the left-adjoint functor F from A∆ to A is inclusion I and the right-adjoint functor G from A to A∆ is D. To show that D f is of the form ∆f ' we check that in every comonad, for every f : A → A', we have D f = ∆A( f ° εA). The counit ϕ of this adjunction is just ε, so that ϕA is εA, and the unit γ is defined via δ, with γDA being δA. (Note that we could not determine γDA uniquely if the comonad were not δ-preordered. Note also that in the comonad with nonempty finite ordinals of the preceding section, which is not δ-preordered, we don’t have the equality Dε{0} ° δ{1} = 1{0,1}.) That this adjunction obtains indeed will be shown in the next three sections. Later, in § 5.1.6 and § 5.1.7, we shall compare the delta category to the Kleisli category and to the category of free coalgebras of a comonad. Before that, in the next three sections, we find the delta category handy to survey various possibilities of defining a comonad. Although we shall look for these alternative definitions in a context where the comonad is δ-preordered, the definitions we shall extract can serve to define comonads in general, without assuming δ-preorder.

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§ 5.1.3. Primitive notions in comonad Let us now consider how for a δ-preordered comonad 〈A, D, ε, δ〉 the adjunction between A and A∆ mentioned in the preceding section could be expressed in various ways according to the definitions of adjunction in § 4.1. First, the primitive notions we might have to express this adjunction are displayed in square brackets in the hexagon of Figure 6 (which we take over from § 4.1.1). Besides the notions we have already encountered, we find in square brackets the inclusion functor I and the seesaw functions E, corresponding to Φ, which will be defined below. The six definitional equalities of § 4.1.1 connecting these notions would now read: (F a

I)

for ∆f : DA' → DA ∆f = EDA(δA ° ∆f ),

(Ga

a

F[I]

for f : A → A' D) D f = ∆A( f ° εA),

γ [δ]

a

Φ [E]

Γ [∆]

ϕ [ε]

Ga [D]

Fig. 6. Definitional dependences in comonads

(ϕε)

εA = EAD1A,

for ∆ f : DA' → DA (ΦE ) EA∆ f = εA ° ∆ f,

(γδ)

δA = ∆A1DA,

(Γ∆)

for f : DA → A' ∆A f = D f ° δA.

When ∆ is understood as Γ, it may be indexed with two subscripts as in § 4.1.1, or just with the first subscript of these two, as in § 4.7.1,

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the second subscript being unessential. However, the first subscript is now unessential, too, since F is the inclusion functor, and we don’t write it. The subscript A of ∆A, which replaces ∆δA, is not the first

subscript of ∆ understood as Γ; this first subscript would be DA. The subscript A of EA is the second subscript of E understood as Φ. So we follow with E the style of § 4.7.1, noting just this second subscript. We must first settle what E stands for. The equality (ΦE ) would permit us to get rid of E in (F aI) and (ϕε) if δA ° ∆ f and D1A were equal to arrows of the form ∆ f '. Now, for D1A this follows immediately from (GaD), while for δA ° ∆ f we have δA ° ∆ f = (δA ° D f ) ° δA', by (Γ∆) and (cat 2) = DD f ° (δDA' ° δA'), by (nat) for δ and (cat 2)

= (DD f ° DδA') ° δA', by (δδ) and (cat 2) = ∆∆ f , by (fun 2) and (Γ∆).

So we may take that E is defined by (ΦE ). The possible choices of primitives for our adjunction would be the following, taking into account that F is now inclusion and doesn’t figure anywhere: hexagonal : 〈D, ε, δ, E, ∆〉 rectangular || : 〈D, ε, δ〉 rectangular \\ : 〈D, E, ∆〉 rectangular // : 〈ε, δ, E, ∆〉 triangular ► : 〈ε, ∆〉 triangular ◄ : 〈D, δ, E〉 The rectangular || choice is the choice of the standard definition. The rectangular \\ choice boils down to 〈ε, ∆〉, since ε can be defined in terms of D and E, while D can be defined in terms of ε and ∆, and E can be defined in terms of ε alone. The rectangular // choice boils down to 〈ε, ∆〉, too, since δ can be defined in terms of ∆ alone, and E can be defined in terms of ε alone. Finally, the triangular ◄ choice

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boils down to 〈D, ε, δ〉, since ε can be defined in terms of D and E, while E can be defined in terms of ε alone. We should mention also the seesaw choice 〈E, ∆〉. This boils down to 〈ε, ∆〉, since εA can be defined as EA1DA, and E is definable in terms of ε alone. The hexagonal choice is of course full of redundances, but we shall nevertheless consider this choice in the next section. Besides that, we are left with only two interesting choices: the standard choice 〈D, ε, δ〉 and 〈ε, ∆〉. § 5.1.4. Hexagonal comonads With the hexagonal choice of primitives, we assume for a comonad 〈A, D, ε, δ, ∆, E〉 that 〈A, 1, ° 〉 is a category, D is a functor from A to A, ε is a natural transformation from D to the identity functor IA, δ is a natural transformation from D to the composite functor DD, ∆ is a function assigning to every object A of A a function ∆A that maps the arrows f : DA → A' of A to the arrows ∆A f : DA → DA' of A, E is a function assigning to every object A of A a function EA that maps the arrows ∆A' f : DA' → DA of A to the arrows EA∆A' f : DA' → A of A, the equalities (F aI), (GaD), (ϕε), (γδ), (ΦE ) and (Γ∆), with subscripts appropriately assigned to ∆ in (F aI) and (ΦE ), all hold, and, moreover, the equality (δδ) holds. The equality (δδ) is assumed not because of the adjunction, but in order to insure that A∆ is closed under composition. It is also used in

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order to guarantee that E can be defined by (ΦE ) in (F aI), as we have shown above. Let us show now that this hexagonal notion of comonad is equivalent to the standard 〈A, D, ε, δ〉 notion. With (ΦE ), the equality (F aI) reads ∆ f = εDA ° (δA ° ∆ f ). This equality clearly follows from (εδ), (cat 1 left) and (cat 2). Conversely, (εδ) follows from this equality as follows. Since from (GaD) with (fun 1) and (cat 1 left) we have ∆AεA = 1DA (i.e. the equality (∆ε) mentioned above), our equality with (cat 1 right) will give (εδ). Therefore, (F aI) amounts to (εδ). With (Γ∆), the equality (GaD) reads D f = D( f ° εA) ° δDA. This equality follows from (εδD), (fun 2), (cat 2) and (cat 1 right). Conversely, (εδD) immediately follows from this equality with (cat 1 left) and (fun 1). Therefore, (GaD) amounts to (εδD). The equalities (F aI) and (GaD) are more important than the remaining four equalities (ϕε), (γδ), (ΦE ) and (Γ∆), which boil down to definitions. So, our hexagonal notion of comonad is equivalent to the standard 〈A, D, ε, δ〉 notion. To prove quite strictly the equivalence of these two notions, we would have to demonstrate an equivalence of categories, which would actually be an isomorphism of categories. Note that in the hexagonal definition a δ-preordered comonad is defined by assuming that A and A∆ are categories and that the functors I and D are adjoints, I being left-adjoint and D right-adjoint. An adjunction between A and B where the left adjoint F is the inclusion functor from B into A is called a coreflection of A in its subcategory B. So a δ-preordered comonad in A is defined by assuming that there is a coreflection of a category A in its subcategory A∆.

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The standard 〈A, D, ε, δ〉 notion of comonad of § 5.1.1 corresponds to the rectangular || notion of adjunction of § 4.1.3. The equality (εδ) corresponds to (ϕγF) and (εδD) to (ϕγG), while (δδ) is related to (nat) for γ. § 5.1.5. Triangular comonads With the 〈ε, ∆〉 choice of primitives, we can imitate the definition of triangular adjunction of § 4.1.6 to define comonads. We define a triangular comonad 〈A, ε, ∆〉 by assuming that

A is a category, D is a function from the objects of A to the objects of A, ε is an objectual transformation from D to the identity function on the objects of A, ∆ is a function assigning to every object A of A a function ∆A that maps the arrows f : DA → A' of A to the arrows ∆A f : DA → DA' of A, the following equalities hold: (ε∆) (∆ ° ) (∆ε)

εA ° ∆f = f, i.e. EA∆ f = f, ∆( f2 ° ∆ f1) = ∆ f2 ° ∆ f1, ∆A εA = 1DA,

with subscripts appropriately assigned to ∆ in (ε∆) and (∆ ° ). These three equalities correspond to the equalities that were mentioned in § 4.1.6 as a possible choice for defining triangular adjunction: (ε∆) corresponds to (β'), while (∆ ° ) corresponds to (ΓΓ") and (∆ε) to (Γϕ). The new notion of comonad is equivalent to the standard 〈A, D, ε, δ〉 notion, via the definitions (GaD), (γδ) and (Γ∆). (A definition of monad analogous to this triangular notion of comonad may be found in [Manes 1976, 1.3, Exercise 12, p. 32]; however,

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there, the analogue of δ-preorder seems to be presupposed, whereas we don’t presuppose δ-preorder.) The triangular notion of comonad becomes more transparent if for f1: DA1 → A2 and f2: DA2 → A3 we introduce the definition given by the equality (

)

∆ O

f2

∆ O

f1 = f2 ° ∆ f1.

The symbol O∆ should inherit the subscript A1 from the ∆ of ∆ f1, but we omit this index, as we omitted it in ∆. We call O∆ delta composition. With delta composition, (∆ ° ) reads (∆

∆ O

)

∆( f2

∆ O

f1) = ∆ f2 ° ∆ f1.

Conversely, we may define ∆ in terms of delta composition by the equality (∆)

∆ f = 1DA

∆ O

f,

where ∆ inherits the index of O∆ . With delta composition primitive, a comonad could be defined as being 〈A, ε, O∆ 〉, where A, D and ε are as for the triangular 〈A, ε, ∆〉 notion above, O∆ is a family of functions indexed by objects A1 of A that assign to a pair ( f1: DA1 → A2, f2: DA2 → A3) of arrows of A the arrow f2 O∆ f1: DA1 → A3 of A (with the index A1 omitted), and the following equalities hold: (cat 1 right O∆ ) (cat 1 left O∆ ) (cat 2 O∆ ) (shift)

f O∆ εA = f, εA O∆ f = f, ( f3 O∆ f2) O∆ f1 = f3 O∆ ( f2 O∆ f1), ( f3 ° f2) O∆ f1 = f3 ° ( f2 O∆ f1).

The first three equalities are clearly analogous to the corresponding categorial equalities, ε behaving as identity. The fourth equality can be replaced by either of the following two equalities: (shift 1) (shift ε)

f3 ° (1DA O∆ f1) = f3 O∆ f1, ( f3 ° εA ) O∆ f1 = f3 O∆ f1.

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(With (shift ε), the equality (cat 1 left O∆ ) becomes superfluous.) The 〈A, ε, O∆ 〉 notion of comonad and the triangular 〈A, ε, ∆〉 notion are equivalent, via the definitions (∆) and ( O∆ ). (A definition of monad analogous to the (shift ε) variant of our 〈A, ε, O∆ 〉 notion may be found in [Manes 1976, 1.3, Definition 3.2, p. 24]; the other variants are based on [D. 1996, section 4.1].) If we don’t economize on primitives, and take both ∆ and delta composition as primitives, then an equivalent notion of comonad is obtained by defining it as 〈A, ε, ∆, O∆ 〉, where A, D, ε, ∆ and O∆ are as before and the equalities (ε∆), (∆ O∆ ) and (∆ε) hold. Now the defining equalities (∆) and ( O∆ ) become derivable (this definition is from [D. 1996, section 4.1]). It seems that the definitions of comonad with delta composition primitive are not practical for cut elimination: with them we should eliminate not only ordinary composition but also delta composition. Note that we are certainly not allowed to suppose that we have now exhausted all possible ways of defining comonads. But the definitions through the adjunction between A and A∆ are well covered, and among these definitions we find the standard definition and other definitions mentioned in the literature. § 5.1.6. The Kleisli category Let 〈A, D, ε, δ〉 be a comonad. Then consider the graph AD whose objects are all the objects of A, while its arrows are obtained by taking that for every object A of A and every arrow f : DA → A' of A, the pair ( f, A), which we abbreviate by f A, is an arrow of AD of type A → A'. (We need a bijection κ that assigns to the pairs ( f, A) the arrows κ( f, A): A → A' of AD. So κ( f, A) may be identified with the pair ( f, A). We cannot identify κ( f, A) just with f instead of ( f, A), because, if D is not one-one on objects, then f could have more than one source in AD. Definitions of Kleisli category in the literature, including Kleisli’s own definition of [1965], usually don’t make this clear.)

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The graph AD has an identity, whose arrows 1A: A → A are defined as εAA, and composition in AD is defined as follows in terms of delta composition: f2A2 ° f1A1 =def ( f2

∆ O

f1)A1,

with the omitted index of O∆ being A1. Let us call the graph AD with this identity and this composition the Kleisli deductive system of the comonad 〈A, D, ε, δ〉. It is clear that due to (cat 1 right O∆ ), (cat 1 left O∆ ) and (cat 2 O∆ ) of the preceding section, this deductive system is a category. This category is called the Kleisli category of the comonad 〈A, D, ε, δ〉. A category isomorphic to AD is A∆', which is related to the delta category A∆ of § 5.1.2, and is defined as follows. Its objects are again the objects of A, while its arrows are obtained by taking that for every pair (A1, A2) of objects of A and every arrow h : DA1 → DA2 of A such that (homo δ)

Dh ° δA1 = δA2 ° h,

the triple 〈h, A1, A2〉, which we abbreviate by hA1,A2, is an arrow of A∆' of type A1 → A2. The identity arrows 1A: A → A of A∆' are defined as 1DAA,A and composition is defined by h2A2,A3 ° h1A1,A2 =def (h2 ° h1)A1,A3.

The equality (homo δ), which is a kind of naturalness condition, could alternatively be written as ∆A1h = ∆A21DA2 ° h. (This is analogous to the equality (solid) of § 1.9.4.) Other conditions equivalent to (homo δ) are ∆A1(εA2 ° h) = h, i.e. ∆ A1EA2h = h, for some f : DA1 → A2 of A, ∆A1 f = h.

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The isomorphism between the categories AD and A∆' is obtained by the functor K from AD to A∆' such that KA = A and for f : DA1 → A2 K f A1 = (∆A1 f )A1,A2. The inverse K-1 of K is defined by K-1A = A and for h : DA1 → DA2 K-1hA1,A2 = (εA2 ° h)A1 = (EA2h)A1. If D is one-one on objects, then it is clear that the category A∆' is isomorphic to the delta category A∆. If our comonad is just δ-preordered (see § 5.1.1), we can ascertain only that the categories A∆' and A∆ are equivalent (see § 1.5), and without δ-preorder we cannot ascertain even that (the isomorphisms needed for the equivalence fail to be natural transformations; see also the remarks about the category ADfree at the end of the next section). The 〈A, ε, O∆ 〉 definition of comonad from the preceding section shows that we could define a comonad by assuming that its Kleisli deductive system is a category and that the (shift) equality holds. This equality expresses the adjunction between A and AD, which we shall examine in § 5.2.3. § 5.1.7. The Eilenberg-Moore category Let 〈A, D, ε, δ〉 be a comonad. Then consider the graph AD whose objects are the arrows d : A → DA of A such that (εd) (δd)

εA ° d = 1 A , δA ° d = Dd ° d.

An arrow of AD with source d1: A1 → DA1 and target d2: A2 → DA2 is made of an arrow h : A1 → A2 of A such that (homo)

Dh ° d1 = d2 ° h.

To prevent the same arrow from having more than one source or more than one target, the arrow h in AD should be indexed by d1 and d2.

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Formally, the arrows of AD will be triples 〈h, d1, d2〉, but we shall take the indices d1 and d2 for granted and omit them (usually, they are not even mentioned). The identity in AD is made of the arrows 1A: A → A of A and composition is defined as composition in A. We can check that the equality (homo) holds when d1 and d2 are equal and for h we put an identity arrow; it holds also for h2 ° h1 if it holds for h1 and h2. So AD is a category, which is called the Eilenberg-Moore category of the comonad 〈A, D, ε, δ〉. For d : A → DA and f : A → A' let ∆d f =def D f ° d. It is clear that the operation ∆δA we had in § 5.1.2 and later is just an

instance of ∆d . To define the Eilenberg-Moore category of a comonad we can assume (εd ') (δd ') (homo')

εA ° ∆d f = f, δA ° ∆d f = ∆d ∆d f, ∆d1h = ∆d21A2 ° h

instead of (εd), (δd) and (homo). The full subcategory ADfree of AD whose objects are all the arrows δA: DA → DDA of A is called the category of free coalgebras of the comonad. This category is isomorphic to the category A∆' of the previous section, and hence also to the Kleisli category AD, when the comonad is δ-injective (see § 5.1.1). In the absence of δ-injectivity, we can ascertain only that these categories are equivalent. We pass from ADfree to a category isomorphic to A∆' by replacing the objects δA of ADfree with pairs (δA, A), and the arrows h : DA1 → DA2 of ADfree with triples 〈h, A1, A2〉. When we omit the superfluous δA from the pairs (δA, A), we obtain exactly A∆'. (In the usual presentation of Eilenberg-Moore categories, objects are said to

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be pairs (d, A), or rather (A, d), where A is the source of d : A → DA and d satisfies (εd) and (δd). These pairs are in one-to-one correspondence with the arrows d. Mentioning the source of d in the pair is not essential: it seems to be there for heuristical reasons. However, replacing (δA, DA) by (δA, A) makes a difference.) It is clear that when the comonad is δ-preordered (see § 5.1.1), the category ADfree is isomorphic to the delta category A∆. In the absence of δ-preorder, we cannot ascertain even that ADfree and A∆ are equivalent categories. (In section 4.2 of [D. 1996] it is stated that it can be shown without the supposition that D is one-one on objects that ADfree and A∆ are isomorphic. What should have been said is that this can be shown sometimes even without making this supposition. Imprecise statements about equivalence with A∆ in the last paragraph of the same section, and in the penultimate paragraph of section 6.2 of the same paper, are also misleading. I can adduce, as an excuse and a solace, that there are in the literature, including textbooks, similar misleading statements about related matters, made by better known authors.) With a δ-preordered comonad, we also have the adjunction between A and A∆ of § 5.1.3 and § 5.1.4. This adjunction should be compared with the adjunction between A and AD, which we shall examine in § 5.2.3. (If in the definition of the latter adjunction in § 5.2.3, we first replace everywhere d by δA, and then replace δA by DA as the argument of FD, the value of GD and the index of γD, we obtain our adjunction between A and A∆.) § 5.2. ADJUNCTION BETWEEN ADJUNCTIONS AND COMONADS We shall now try to clarify the relationship between the notions of comonad and adjunction. It will appear that comonads may be understood as a special kind of adjunction, since the category of comonads (with comonofunctors as arrows) is isomorphic to a full

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subcategory of the category of adjunctions (with junctors as arrows). Moreover, there are two adjunctions involving these two categories. First, we have a functor that associates in a standard manner a comonad to an adjunction. After investigating some aspects of this functor, we show that it has a left adjoint, which associates to a comonad the adjunction with the Kleisli category, and a right adjoint, which associates to a comonad the adjunction with the EilenbergMoore category. At the end (§ 5.2.4), we show how the usual presentation of these matters, via the category of resolutions of a comonad, where the Kleisli category is tied to the initial object and the Eilenberg-Moore category to the terminal object, is a simple corollary of our presentation (which can be derived from [Auderset 1974]). We shall not dwell much on the proofs in this part, since they are pretty straightforward (though sometimes lengthy), and since they deal with matters not essential for later parts of the work. § 5.2.1. The comonad of an adjunction To codify matters for the sake of this part of the work, let us now reformulate the notions of junction, junctor and adjunction in the rectangular || style of § 4.1.3. Suppose we are given the following: two deductive systems, 〈A, 1, ° 〉 and 〈B, 1, ° 〉, a graph-morphism F from B to A and a graph-morphism G from A to B, a transformation ϕ from FG to IA and a transformation γ from IB to GF. Then 〈A, B, F, G, ϕ, γ〉 is a junction. A junctor from a junction 〈A, B, F, G, ϕ, γ〉 to a junction 〈A', B ', F', G', ϕ', γ'〉 is a pair (NA, NB) such that NA is a functor from the deductive system A to the deductive system A', and NB a functor from the deductive system B to the deductive system B '; moreover, the following naturalness equalities hold:

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NAF = F'NB, NAϕA = ϕ'NA A,

NBG = G'NA, NBγB = γ'NB B.

An adjunction is a junction 〈A, B, F, G, ϕ, γ〉 such that 〈A, 1, ° 〉 and 〈B, 1, ° 〉 are categories, F and G are functors, ϕ and γ are natural transformations, the equalities (ϕγF) and (ϕγG) hold (see § 4.1.3). To every adjunction 〈A, B, F, G, ϕ, γ〉 we may associate the comonad 〈A, FG, ϕ, FγG〉, where the composite functor FG is the functor D of the comonad, ϕA is εA and FγGA is δA. (We may analogously associate to the adjunction a monad in B.) It is routine to check that 〈A, FG, ϕ, FγG〉 is indeed a comonad. It is called the comonad of the adjunction 〈A, B, F, G, ϕ, γ〉. § 5.2.2. Reflections and coreflections in comonads An adjunction between A and B where the right adjoint G is the inclusion functor from A into B is called a reflection of B in its subcategory A. We have seen in § 5.1.4 that a δ-preordered comonad in a category A is defined by a coreflection of A in its subcategory A∆, the delta category of the comonad. However, with comonads of adjunctions we may have in some interesting (and in logic rather common) cases also a reflection of a category isomorphic to A∆ in its subcategory A. We shall now consider this matter. Let us first prove the following proposition. Proposition 1. Let 〈A, B, F, G, ϕ, γ〉 be an adjunction where G is oneone on objects. Then the Kleisli category AFG of the comonad 〈A, FG, ϕ, FγG〉 of the adjunction is isomorphic to the full subcategory G(A) of B whose objects are all the objects of B of the form GA.

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Proof. First we show that for f1: FGA1 → A2 and f2: FGA2 → A3 in the comonad 〈A, FG, ϕ, FγG〉 we have

(

∆ O

ΦΓ)

f2

∆ O

f1 = ΦGA1,A3( ΓGA2,A3 f2 ° ΓGA1,A2 f1),

where the omitted index of f2

∆ O

∆ O

is A1. Indeed, we have

f1 = f2 ° (FG f1 ° FγGA1), by definition

= f2 ° FΓGA1,A2 f1, by (fun 2) and (Γ) of § 4.1.1, and we obtain ( O∆ ΦΓ) by applying (ΦΓ') and (ΓΓ") from § 4.1.4. We now define a functor N from AFG to G(A) in the following way. For every object A of AFG, which is by definition an object of A, let NA be GA. For every arrow f A1: A1 → A2 of AFG, for which, by definition, we have an arrow f : FGA1 → A2 of A, let N f A1 be ΓGA1,A2f : GA1 → GA2. To check that N is a functor we have

NϕAA = ΓGA,AϕA = 1GA, by (ϕ), (fun 1) and (ΓΦ') of § 4.1.4, N( f2

∆ O

f1)A1 = ΓGA1,A3( f2

∆ O

f1)

= ΓGA2,A3 f2 ° ΓGA1,A2 f1, by (

∆ O

ΦΓ) and (ΦΓ') of § 4.1.4

= N f A2 ° N f A1 . (Note that, according to the definition in § 5.1.6, the omitted index of ∆ O in the last derivation must be A1.) Relying on the fact that G is one-one on objects, we define the functor N-1 from G(A) to AFG by taking that N-1GA is A and that for g : GA1 → GA2 the arrow N-1g is (ΦGA1,A2g)A1. It remains to use the equalities (ΦΓ') and (ΓΦ') to verify that N-1N f A1 = f A1 and NN-1g = g.

q.e.d.

The following proposition is an immediate corollary of Proposition 1. Proposition 2. Let 〈A, B, F, G, ϕ, γ〉 be an adjunction where G is a bijection on objects. Then the categories AFG and B are isomorphic.

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We know from § 5.1.6 that if in a comonad 〈A, D, ε, δ〉 we have that D is one-one on objects, then the Kleisli category AD of the comonad is isomorphic to the subcategory A∆ of A, the delta category of the comonad. With the comonad 〈A, FG, ϕ, FγG〉 of an adjunction, for f : FGA1 → A2 we have ∆A1 f = FΓGA1,A2 f. So A∆ will be denoted in this case by AFΓ. We can then state the following as a corollary of Proposition 1: Proposition 3. Let 〈A, B, F, G, ϕ, γ〉 be an adjunction where both F and G are one-one on objects. Then the categories AFΓ and G(A) are isomorphic.

The point of this proposition is that AFΓ is a subcategory of A. So in all adjunctions 〈A, B, F, G, ϕ, γ〉 where F is one-one on objects and G is a bijection on objects, B is isomorphic to a subcategory of A. Note that in such an adjunction A may actually be a subcategory of B, so that the adjunction is a reflection of B in its subcategory A. But we can assert that B is also isomorphic to a subcategory of A, namely AFΓ, and that there is a coreflection of A in this subcategory. (The situation we have just described obtains sometimes in the adjunction of deductive completeness, a strengthening of the deduction theorem, originally called functional completeness in [Lambek 1974]; see also [Lambek & Scott 1986, I.6-7] and [D. 1996]. Then B is the polynomial category generated by A and an indeterminate arrow.) It is instructive to see above that the isomorphism from B to AFΓ is the functor F, the left adjoint in the adjunction.

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§ 5.2.3. The adjunctions involving the categories of adjunctions and comonads Let Adj be the category whose objects are adjunctions, with arrows being junctors (this category should not be confused with the category bearing the same name in [Mac Lane 1971, IV.8], where arrows are adjunctions), and let Com be the category whose objects are comonads, with arrows being comonofunctors. Consider now the functor C from Adj to Com that assigns to an adjunction 〈A, B, F, G, ϕ, γ〉 the comonad 〈A, FG, ϕ, FγG〉 of the adjunction, and to a junctor (NA, NB) the comonofunctor NA (we may readily check that NA is indeed a comonofunctor). The functor C has a left adjoint F that assigns to a comonad 〈A, D, ε, δ〉 the adjunction between A and the Kleisli category AD of this comonad, namely the adjunction 〈A, AD, FD, GD, ϕD, γD〉, which is defined as follows: FDA =def DA, FD f A =def ∆A f, ϕD A =def εA,

GDA =def A, GD f =def ( f ° εA)A, γD A =def (1DA)A.

If NA is a comonofunctor from a comonad 〈A, D, ε, δ〉 to a comonad 〈A', D', ε', δ'〉, then FNA is the junctor (NA, NAD) from the adjunction between A and AD to the adjunction between A' and A'D ', where NAD is defined as follows: NADA =def NAA,

NAD f A =def (NA f )NAA.

For an adjunction J = 〈A, B, F, G, ϕ, γ〉 let ϕJ be the junctor (NA, NB) from FCJ to J where NA is the identity functor IA and the functor NB is defined by NBA =def GA,

NB f A =def G f ° γGA = ΓGA,A' f.

The arrows ϕJ of Adj make a natural transformation ϕ from FC to IAdj. It is easy to check that for every comonad S = 〈A, D, ε, δ〉 the

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comonad CFS is identical to S; so the identity comonofunctor IA is an arrow from S to CFS in Com. It is trivial that the arrows IA make a natural transformation I from ICom to CF. That F is left adjoint to C means that 〈Adj, Com, F, C, ϕ, I〉 is an adjunction. In this adjunction, the unit is the identity of the category Com. We can infer that Com is isomorphic by F to a full subcategory of Adj (cf. [Mac Lane 1971, IV.4, pp. 92-93]). The functor C has also a right adjoint G that assigns to a comonad 〈A, D, ε, δ〉 the adjunction between A and the Eilenberg-Moore category AD of this comonad, namely the adjunction 〈A, AD, FD, GD, ϕD, γD〉, which is defined as follows: FDd =def source(d), FDh =def h , ϕDA =def εA,

GDA =def δA, GDf =def D f, γDd =def d.

If NA is a comonofunctor from a comonad 〈A, D, ε, δ〉 to a comonad 〈A', D', ε', δ'〉, then GNA is the junctor (NA, NAD) from the adjunction between A and AD to the adjunction between A' and A'D ', where NAD is defined as follows: NADd =def NAd,

NADh =def NAh.

For an adjunction J = 〈A, B, F, G, ϕ, γ〉 let now γJ be the junctor (NA, NB) from J to GCJ where NA is the identity functor IA and the functor NB is defined by NBB =def FγB,

NBg =def Fg.

The arrows γJ of Adj make a natural transformation γ from IAdj to GC. It is easy to check that for every comonad S = 〈A, D, ε, δ〉 the comonad CGS is identical to S; so the identity comonofunctor IA is an arrow from CGS to S in Com. It is trivial that the arrows IA make a natural transformation I from CG to ICom.

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That G is right adjoint to C means that 〈Com, Adj, C, G, I, γ〉 is an adjunction. In this adjunction, the counit is the identity of the category Com. We can infer that Com is isomorphic by G to a full subcategory of Adj (following the terminology of [Mac Lane 1971, IV.4, pp. 9293], the functor C is a left-adjoint-left-inverse of G; the category Com is isomorphic to a full reflective subcategory of Adj). One could expect that adjunctions similar to those with C, F and G treated in this section may be obtained by taking instead of Adj the category of junctions (with junctors as arrows) and instead of Com the category of comonographs (with comonofunctors as arrows). § 5.2.4. The category of resolutions Take a functor C from a category A to a category B, and for a given object B of B consider the set of objects A of A such that CA = B and the set of arrows f of A such that C f = 1B. These two sets make the graph of a subcategory AB of A. An object A is initial in a graph iff from A to every object in the graph there is exactly one arrow; A is terminal iff from every object to A there is exactly one arrow. If C has a left adjoint F such that the unit of the adjunction is the identity of B, then AB has an initial object FB, and if C has a right adjoint G such that the counit of the adjunction is the identity of B, then AB has a terminal object GB. To show that FB is initial, take an object A of AB; then it can be shown that ϕA: FCA → A is the unique arrow of AB from FB to A. For suppose there is another arrow f : FCA → A in AB; since C f ° γB = C f = 1B, because γB is an identity arrow and f is in AB, and since CϕA ° γB = 1B, by the equality (ϕγG) of § 4.1.3, we obtain

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ϕFB ° F(C f ° γB) = ϕFB ° F(CϕA ° γB), from which with (fun 2), (nat) and the equality (ϕγF) of § 4.1.3, the equality f = ϕA follows. Analogously, in the other adjunction, the one with G, the arrow γA: A → GCA is the unique arrow of AB from A to GB. So by taking the functor C from Adj to Com and by fixing a comonad S in Com we obtain a subcategory AdjS of Adj. We may call the category AdjS the category of resolutions of S, by analogy with the terminology usual when one deals with monads instead of comonads. For a comonad S = 〈A, D, ε, δ〉, the adjunctions in AdjS are all between the category A and a category B, and the junctors (NA, NB) in AdjS all have for NA the identity functor on A. The category AdjS has an initial object FS and a terminal object GS, according to what we have said above. The arrow ϕJ: FCJ → J is the unique arrow of AdjS from FS to an adjunction J of AdjS, and γJ: J → GCJ is the unique arrow of AdjS from J to GS. These arrows correspond to what in the case of monads is called comparison functors. Suppose a functor C from a category A to a category B has both a left adjoint F and a right adjoint G. Then the functors FC and GC from A to A are adjoint, FC being left adjoint and GC right adjoint. (Analogously, CF and CG from B to B are adjoint, CF being left adjoint and CG right adjoint.) This is a consequence of the fact that two successive adjunctions compose to give a single adjunction (see [Mac Lane 1971, IV.8, p. 101]). By taking that A is Adj and B is Com, we obtain that the functors FC and GC from Adj to Adj are adjoint. The functors CF and CG are uninteresting, since they are the identity functor from Com to Com. (Matters of this section and of the preceding one should be compared with [Pumplün 1970].)

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§ 5.3. COMONOGRAPHS, COMONOFUNCTORS AND COMONADS We shall now define a notion of comonad, equivalent to the standard notion, which will be adapted to cut elimination. This notion differs from the standard notion of § 5.1.1 only by replacing the natural transformation ε by a natural antecedental transformation, and the natural transformation δ by a natural consequential transformation. So this notion of comonad corresponds to our main formulation, i.e. the (ac) formulation, of rectangular || adjunction in § 4.2. In the standard notion of comonad, with the natural transformations ε and δ primitive, we cannot eliminate cuts in arrow terms like εA ° εDA and δDA ° δA. Before introducing our new notion of comonad, we introduce appropriate notions of comonograph and comonofunctor. Let 〈A, 1, ° 〉 be a deductive system and let D be a graph-morphism from A to A. Next, let εa be an antecedental transformation from D to the identity graph-morphism IA and δc a consequential transformation from D to the composite graph-morphism DD. Note that here we can take that εa is a single operation on the arrows of A, assigning to an arrow f : A1 → A2 of A the arrow εa f : DA1 → A2 of A. The functions εaA are obtained by restricting the domain of εa. When D is one-one on objects, as it will be in the free comonographs and free comonads of § 5.4 and § 5.6, we can likewise take that δc is a single function on the arrows of A of the form f : A1 → DA2, which maps them to the arrows δc f : A1 → DDA2 of A. In general, however, although DA2 may be the same object as DA2', we have to distinguish δcA2 from δcA2' when A2 is different from A2'. We shall, nevertheless, usually omit the subscript of δc, taking it for granted. This will simplify considerably our notation, and cannot produce anything unwholesome in free comonads. Then we say that 〈A, D, εa, δc〉 is a comonograph. As before, to simplify the notation, we don’t mention the identity and composition of 〈A, 1, ° 〉. We say that this is a comonograph in A, and later we use sometimes the same form of speaking with comonads.

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A comonofunctor from a comonograph 〈A, D, εa, δc〉 to a comonograph 〈A', D', εa', δc'〉 is a functor N from the deductive system A to the deductive system A' such that the following naturalness equalities hold: ND = D'N, Nεa f = εa'N f, NδcA f = δc'NAN f. A comonofunctor is a comonograph isomorphism iff it is an isomorphism of deductive systems. A comonograph 〈A, D, εa, δc〉 is a comonad iff (i) 〈A, 1, ° 〉 is a category; (ii) D is a functor; (iii) εa is a natural antecedental transformation and δc a natural consequential transformation, which means that the following equalities hold in A: (εa 1) εa( f2 ° f1) = f2 ° εa f1, (εa 2) εa( f2 ° f1) = εa f2 ° D f1, DD f2 ° δc f1;

(δc 1) δc( f2 ° f1) = δc f2 ° f1, (δc 2) δc(D f2 ° f1) =

(iv) finally, the following equalities hold in A: (εaδc) (εaδcD) (δcδc)

εa f2 ° δc f1 = f2 ° f1, Dεa f2 ° δc f1 = D f2 ° f1, Dδc1DA ° δc f = δcδc f,

with the subscripts of δc omitted. It is straightforward to show that the notion of comonad we have just introduced is equivalent to the standard notion of § 5.1.1, and hence to all other notions of comonad we have considered in § 5.1. To achieve this precisely we should establish an equivalence of categories, which would in fact be an isomorphism of categories.

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Let (δc)n be an abbreviation for a sequence of n occurrences of δc, where n ≥ 0 (if n = 0, this is, of course, the empty sequence). If n > 1, the omitted subscripts of various occurrences of δc in (δc)n will be different. We can then show that for every comonad we have the following equalities: (δnD)

D(δc)nD f2 ° δc f1 = (δc)n+1(D f2 ° f1),

(δnεa)

D(δc)nεa f2 ° δc f1 = D(δc)n f2 ° f1,

(δnδc)

D(δc)n1DA ° δc f = (δc)n+1 f.

We derive these equalities by induction on n. For (δnD) we use (δc 2) in case n = 0, and in case n > 0 we have D(δc)nD f2 ° δc f1 = D(δc1DA ° (δc)n-1D f2) ° δc f1, by (cat 1 left) and (δc 1) = Dδc1DA ° (D(δc)n-1D f2 ° δc f1), by (fun 2) and (cat 2) = Dδc1DA ° (δc)n(D f2 ° f1), by the induction hypothesis = (δc)n+1(D f2 ° f1), by (δcδc). For (δnεa) we use (εaδcD) in case n = 0, and in case n > 0 we have D(δc)nεa f2 ° δc f1 = Dδc1DA ° (D(δc)n-1εa f2 ° δc f1), as above = Dδc1DA ° (D(δc)n-1 f2 ° f1), by the induction hypothesis = D(δc)n f2 ° f1, as above. For (δnδc) we use (fun 1) and (cat 1 left) in case n = 0, and in case n > 0 we have D(δc)n1DA ° δc f = Dδc1D nA ° (D(δc)n-11DA ° δc f ), as above = Dδc1D nA ° (δc)n f, by the induction hypothesis = (δc)n+1 f, by (δcδc). We could also derive (δnδc) indirectly from (δnD) by replacing f2 in (δnD) by 1A and then using (fun 1) and (cat 1 left).

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The equality (δ0D) is just (δc 2), while (δ0εa) is (εaδcD) and (δ1δc) is (δcδc). So we could have assumed (δnD), (δnεa) and (δnδc), or just (δnD) and (δnεa), instead of (δc 2), (εaδcD) and (δcδc) in the definition of comonad. § 5.4. FREE COMONOGRAPHS The notion of free comonograph S generated by a graph G, which we shall now define, will be linguistic, as were the notions of free deductive system from § 1.6, free graph-morphism between deductive systems from § 2.1, free formation from § 3.3 and free junction from § 4.3. We first introduce atomic symbols for the objects of G, called the generative object terms of A (these symbols are in one-to-one correspondence with the objects of G). As before, we don’t specify what exactly these symbols are: it is enough to know that they exist. Then we define inductively the object terms of A by the following clauses: (G) every generative object term of A is an object term of A; (1) if A is an object term of A, then DA is an object term of A. Next we define the arrow terms of A. Every arrow term of A will have a unique type, which is a pair (A1, A2) of object terms of A. To indicate that an arrow term f of A has the type (A1, A2) we write f : A1 → A2, as before. We introduce atomic symbols for the arrows of G, called the generative arrow terms of A (these symbols are in oneto-one correspondence with the arrows of G). As with object terms, we don’t specify the exact form of these symbols. If a generative arrow term stands for an arrow of G of type A1 → A2, its type is made of the pair of generative object terms that stand for A1 and A2. Then we define inductively the arrow terms of A by the following clauses: (G) every generative arrow term of A is an arrow term of A;

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(1) if A is an object term of A, then 1A : A → A is an arrow term of A; (2) if f1: A1 → A2 and f2: A2 → A3 are arrow terms of A, then f2 ° f1: A1 → A3 is an arrow term of A; (3) if f : A1 → A2 is an arrow term of A, then D f : DA1 → DA2 is an arrow term of A; (4) if f : A1 → A2 is an arrow term of A, then εa f : DA1 → A2 is an arrow term of A; (5) if A2 is an object term of A and f : A1 → DA2 is an arrow term of A, then δcA2 f : A1 → DDA2 is an arrow term of A. The object terms of A make the objects and the arrow terms of A the arrows of a graph A, with the obvious source and target functions. Moreover, 〈A, 1, ° 〉 is a deductive system, and 〈A, D, εa, δc〉 is a comonograph. This is the free comonograph S generated by G. This comonograph is free in the following sense. Consider the generative graph-morphism H from G to A that assigns to every object of G the corresponding generative object term of A and to every arrow of G the corresponding generative arrow term of A (so H amounts to inclusion, both on objects and on arrows). The free comonograph S and the generative graph-morphism H have the following universal property: For every comonograph S' in A' and every graph-morphism M from G to A' there is a unique comonofunctor N from S to S' such that M = NH. This universal property characterizes S up to comonograph isomorphism. The comonograph S is the image of G under a left adjoint to the forgetful functor from the category of comonographs (with comonofunctors as arrows) to the category of graphs (with

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graph-morphisms as arrows). This forgetful functor assigns to a comonograph in D the graph D. § 5.5. CONNECTIVES With the function D on objects of the free comonograph S we encounter something we may take to be a unary logical connective. (This connective in particular resembles a modal necessity operator.) As usual, an operation on a set X is a function from Xn to X, where n ≥ 0. Let us say that an operation is one-one iff it is a one-one function. (A nullary operation, i.e. an operation from X0 = {Ø} to X, may be identified with an element of X; such operations are trivially one-one.) A connective, on its own, is just a one-one operation on a set, called the set of propositions, or set of formulae. However, connectives are usually members of families of connectives, and a more basic notion is the notion of a family of connectives. Such a family is a family of one-one operations on the set of propositions such that the images of these operations are mutually disjoint. The operation D on the objects of A in the free comonograph S is just the unique member of the family of connectives {D}. We have seen in § 2.1 that free functions are one-one functions. A connective can analogously be characterized as a free operation and a family of connectives as a free family of operations. A set X with a family of operations freely generated by a set of generators E (i.e. set of generative elements, in the style of the terminology we used in this work) is characterized up to isomorphism by the following universal property involving the generative inclusion function h from E to X: For every set X' with a family of operations on X' and every function µ from E to X' there is a unique homomorphism ν from X to X' such that for every e in E we have µ(e) = ν(h(e)) = ν(e). The set X' and its family of operations, which need not be one-one, can be taken as a model; µ is then a basic valuation of propositional letters and ν is the valuation extending µ defined inductively on the set of all propositions.

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We have made these comments on connectives to stress the importance of freely generated structures for logic. But, of course, the notion of connective is not the only logical notion tied to an inductive definition, and such definitions engender free structures. Indeed, an essential ingredient of the spirit of logic is to investigate such notions. (The older name of mathematical logic, “symbolic logic”, should be understood as referring to these linguistic, free structure, preoccupations of logic.) § 5.6. FREE COMONADS We shall first define the notion of free comonad S* generated by a comonograph S. Let S be 〈A, D, εa, δc〉 and consider the categorial equivalence relations ≡ on the arrows of A (in the sense of § 1.7) which in addition to the congruence law of categorial equivalence relations: if f1 ≡ f1' and f2 ≡ f2', then f2 ° f1 ≡ f2' ° f1', satisfy also the following congruence laws: if f1 ≡ f2, then D f1 ≡ D f2, if f1 ≡ f2, then εa f1 ≡ εa f2, if f1 ≡ f2, then δcA f1 ≡ δcA f2, provided δcA f1 and δcA f2 are defined; these equivalence relations satisfy also the conditions derived from the equalities of (ii)-(iv) of the definition of comonad in § 5.3 by replacing = with ≡ (these are the equalities (fun 1) and (fun 2) for D, (εa 1), (εa 2), (δc 1), (δc 2), (εaδc), (εaδcD) and (δcδc); conditions derived from the categorial equalities of (i) are satisfied because we have categorial equivalence relations). Let us call such equivalence relations comonadic. It is straightforward to check that the intersection ≡∩ of all comonadic equivalence relations is again a comonadic equivalence relation. Then for every arrow f of A let [ f ] be the equivalence class { f '| f ≡∩ f '}, and on these equivalence classes define the following:

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1A =def [1A ],

[ f2] ° [ f1] =def [ f2 ° f1], D[ f ] =def [ D f ], εa[ f ] =def [ εa f ], δcA[ f ] =def [ δcA f ].

The objects of A together with the equivalence classes we have just introduced make a category 〈A*, 1, ° 〉. It would be more precise if we distinguished notationally the D, εa and δc of A* from those of A, but it is more practical if we don’t do so. Then 〈A*, D, εa, δc〉, where D, εa and δc are not those of A, but those of A* we have just defined, is a comonad. This is the free comonad S* generated by S. Consider now the generative comonofunctor H from S to S*, which is the identity function on objects, and which assigns to every arrow f of A the arrow [ f ] of A*. The free comonad S* and the generative comonofunctor H have the following universal property: For every comonad S' and every comonofunctor M from S to S' there is a unique comonofunctor N from S* to S' such that M = NH. This property characterizes S* up to comonograph isomorphism. The comonad S* is the image of S under a left adjoint to the forgetful functor from the category of comonads (with comonofunctors as arrows) to the category of comonographs (with comonofunctors as arrows). The free comonad S* that is generated by the free comonograph S generated by a graph G will be called the free comonad S* generated by G. For the free comonad S* generated by G, the generative arrows of A* are the arrows [ f ] such that f is a generative arrow term of the graph A of the free comonograph S generated by G. We designate these generative arrows by the generative arrow terms f of A; then we can use the object terms and arrow terms of A to designate the objects and arrows of A*.

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Consider now the generative graph-morphism H* from G to A*, which assigns to every object of G the corresponding object of A* and to every arrow of G the corresponding generative arrow of A*. The graph-morphism H* amounts to inclusion, both on objects and on arrows, and it is faithful. The free comonad S* generated by G has the following universal property involving H*, quite analogous to the universal property of the free comonograph S generated by G, which we have considered in § 5.4: For every comonad S' in A' and every graph-morphism M from G to A' there is a unique comonofunctor N from S* to S' such that M = NH*. This universal property characterizes S* up to comonograph isomorphism. The comonad S* is the image of G under a left adjoint to the forgetful functor from the category of comonads (with comonofunctors as arrows) to the category of graphs (with graphmorphisms as arrows). This forgetful functor assigns to a comonad in D the graph D. If HG is the generative graph-morphism from G to A, where A comes from the free comonograph S generated by G, and HA is the generative comonofunctor from S to S*, then it is clear that for the generative graph-morphism H* from G to A* we have H* = HAHG, and we could derive the universal property for H* from the universal properties for HG and HA. § 5.7. CUT ELIMINATION IN FREE COMONADS Let 〈A*, D, εa, δc〉 be the free comonad generated by G. We shall now show that Cut Disintegration holds for the category A*. The assumptions made for comonads will prove necessary and sufficient for having this Cut Disintegration. This holds for the definition of comonad of § 5.3. We also consider briefly how alternative, equivalent, notions of comonad behave with respect to cut elimination

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(see § 5.7.3). In particular we reformulate the triangular notion from § 5.1.5 so as to obtain Cut Disintegration. However, the assumptions for this notion of comonad, though sufficient for Cut Disintegration, will not all be necessary. Until § 5.7.3, the symbol A* will refer to the category of the free comonad above. § 5.7.1. Cut Disintegration in free comonads Every equality f1 = f2 that holds for the arrow terms f1 and f2 of A* may be derived as follows. It may be axiomatic, which means that it is an instance of h = h, (cat 1 right), (cat 1 left), (cat 2), (fun 1), (fun 2), (εa 1), (εa 2), (δc 1), (δc 2), (εaδc), (εaδcD) or (δcδc); else it is derived from axiomatic or previously derived equalities by the rules of symmetry or transitivity of equality, or the rule (congr ° ) of § 1.8.1, or the additional congruence rules from f1 = f2, infer D f1 = D f2, from f1 = f2, infer εa f1 = εa f2, from f1 = f2, infer δcA f1 = δcA f2, provided δcA f1 and δcA f2 are defined. The first of these congruence rules is just a variant of (congr F) of § 2.3.1, and the second two are variants of (congr τc) of § 3.5.1. We must now extend the definition of linked cuts from § 1.8.1 to equalities among arrow terms of A*. For that we have to take into account the following additional cases. If f1 = f2 is an instance of (fun 1), then no cut occurs on either side, and if it is an instance of (fun 2), (εa 1), (εa 2), (δc 1), (δc 2), (εaδc) or (εaδcD), then the n-th cut of f1 is linked to the n-th cut of f2. If f1 = f2 is an instance of (δcδc), then the n+1-th cut of f1 is linked to the n-th cut of f2; the first cut of f1, i.e. the main ° of Dδc1DA ° δc f, is not linked to any cut of f2. The additional congruence rules just preserve the links of the premises. This definition of linked cuts induces the following links for the derived equalities (δnD), (δnεa) and (δnδc) of § 5.3: if f1 = f2 is an

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instance of the first two equalities, then the m-th cut of f1 is linked to the m-th cut of f2; if f1 = f2 is an instance of (δnδc), then the m+1-th cut of f1 is linked to the m-th cut of f2, the first cut of f1 not being linked to any cut of f2. If, alternatively, the equalities (δc 2), (εaδcD) and (δcδc) are derived, while (δnD), (δnεa) and (δnδc) are primitive, and the links for the latter equalities are defined as we just said, this will induce for the former equalities the links specified in the preceding paragraph. (Nothing changes if only (δnD) and (δnεa) are primitive.) We just take over the other definitions of § 1.8.1, save that in the definition of the degree of an arrow term we also count, besides occurrences of 1 and ° , also occurrences of D (applied to arrow terms), εa and δc. We can then prove Cut Disintegration for the arrow terms of A*. Proof of Cut Disintegration for A*. We proceed as in the proof of § 1.8.1. So, in an arrow term of A*, take a nonatomic cut whose subterm is f2 ° f1. Then we have the following cases in addition to cases analogous to those considered in § 1.8.1 and § 2.3.1. The subscripts of δc are now unimportant, and we shall systematically omit them. In case (3) we suppose that f1 is D f, and in subcases (3.1) - (3.3) we proceed as in the corresponding cases of § 2.3.1, replacing F by D; we have the following additional subcases.

(3.4) f2 is εa f '. Then εa f ' ° D f = εa( f ' ° f ) by (εa 2), and the degree of the main cut of f ' ° f is strictly smaller than the degree of the main cut on the left-hand side linked to it. (3.5) f2 is δc f '. Then δc f ' ° D f = δc( f ' ° D f ) by (δc 1), and the degree of the main cut of f ' ° D f is strictly smaller than the degree of the main cut on the left-hand side linked to it. (3.G) f2 is a generative arrow term. This is excluded because f2 must have a source of the form DA. We also have the following cases.

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(4) f1 is εa f. Then f2 ° εa f = εa( f2 ° f ) by (εa 1), and the degree of the main cut of f2 ° f is strictly smaller than the degree of the main cut on the left-hand side linked to it. (This quite analogous to case (4) of the proof in § 4.5.1.) (5) f1 is δc f. Then we have the following subcases. (5.1) - (5.2) f2 is 1DDA or f2" ° f2'. Then we proceed as in cases (1) and (2) of the proof of § 1.8.1. (5.3) f2 is D f '. Then we have the following subcases. (5.3.1) f ' is (δc)n1DA. Then D(δc)n1DA ° δc f = (δc)n+1 f by (δnδc), and the main cut on the left-hand side is not linked to anything on the right-hand side. (5.3.2) f ' is (δc)n( f '' ° f "). Then D(δc)n( f '' ° f ") ° δc f = D(δc)n f '' ° (D f " ° δc f ) by (δc 1), (fun 2), (congr ° ) and (cat 2), and the degree of the main cut of D f " ° δc f is strictly smaller than the degree of the main cut on the left-hand side linked to it. (5.3.3) f ' is (δc)nD f ". Then D(δc)nD f " ° δc f = (δc)n+1(D f " ° f ) by (δnD), and the degree of the main cut of D f " ° f is strictly smaller than the degree of the main cut on the left-hand side linked to it. (5.3.4) f ' is (δc)nεa f ". Then D(δc)nεa f " ° δc f = D(δc)n f " ° f by (δnεa), and the degree of the main cut on the right-hand side is strictly smaller than the degree of the main cut on the left-hand side linked to it. (5.3.G) f ' is (δc)n f " where f " is a generative arrow term. This is excluded because f " must have a source of the form DA. (5.4) f2 is εa f '. Then εa f ' ° δc f = f ' ° f by (εaδc), and the degree of the main cut of f ' ° f is strictly smaller than the degree of the main cut on the left-hand side linked to it. (5.5) f2 is δc f '. This case is treated like (3.5) above. (5.G) f2 is f ' where f ' is a generative arrow term. This is excluded because f ' must have a source of the form DDA.

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In case (G), where f1 is a generative arrow term, we have subcases analogous to those we had in §§ 1.8.1, 2.3.1 and 4.5.1, except when f2 is δc f , which we treat analogously to (3.5) above. q.e.d. § 5.7.2. Necessary conditions for Cut Disintegration in free comonads Note that in the complete proof of Cut Disintegration in the preceding section we have used all the equalities of (i)-(iv) of the definition of comonad in § 5.3 (including (fun 1), which is involved in the derivation of (δnδc)). We want now to show that all these equalities are also necessary. The necessity of (cat 1 right), (cat 1 left) and (cat 2) is demonstrated as in § 1.8.2. (see also § 4.5.2). All the counterexamples to Cut Disintegration in the remainder of this section are for free comonad-like structures, lacking one of the remaining equalities, that are generated by an arrowless graph with at least one object. In the absence of (fun 1) we have the counterexample D1DA ° δc1DA. In the absence of (fun 2) we have the counterexample Dεa1A ° DεaD1A, and others built analogously. In the absence of (εa 1) we have the counterexample εa1A ° εaD1A, and in the absence of (εa 2) the counterexample εa1A ° Dεa1A. In the absence of (δc 1) we have the counterexample c δ D1DA ° δc1DA, and in the absence of (δc 2) the counterexample DDδc1DA ° δcD1DA. In the absence of (εaδc) we have the counterexample εaD1A ° δc1DA. In the absence of (εaδcD) we have the counterexample Dεa1A ° δc1DA. Finally, in the absence of (δcδc) we have the counterexample Dδc1DA ° δc1DA. § 5.7.3. Cut Disintegration with alternative notions of comonad As we did for the definition of adjunction, we could replace the natural antecedental transformation εa and the natural consequential

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transformation δc by εc and δc, or εa and δa, or εc and δa. This would lead us into considerations very much like those of § 4.5.5. The obvious alternative formulations of comonad enable us to demonstrate Cut Disintegration, but their equalities would not all be necessary for this result, as were the equalities of the formulation of § 5.3 with εa and δc. Rather than rehash these matters, we shall consider a reformulation of the triangular notion of comonad of § 5.1.5, which will also yield Cut Disintegration. This definition of comonad will be analogous to one used in [Lambek 1969] for proving a cut-elimination result in connection with monads freely generated by categories. A comonad will now be a structure 〈A, εa, ∆〉 for which we assume that

A is a category, D is a function from the objects of A to the objects of A, εa is a function mapping the arrows f : A1 → A2 of A to the arrow εa f : DA1 → A2 of A, ∆ is a family of functions ∆A, indexed by the objects A of A, which map the arrows f : DA → A' of A to the arrows ∆A f : DA → DA' of A, the following equalities hold: (εa 1)

εa( f2 ° f1) = f2 ° εa f1,

(εa∆)

εa f2 ° ∆ f1 = f2 ° f1,

(∆ ° )

∆( f2 ° ∆ f1) = ∆ f2 ° ∆ f1,

(∆εa)

∆Aεa1A = 1DA,

with subscripts appropriately assigned to ∆ in (εa∆) and (∆ ° ). In general, we shall omit the subscripts of ∆, except where they are essential. These four equalities correspond respectively to the equalities (ϕa 1), (β"), (ΓΓ") and (Γϕa) of the definition of triangular adjunction in § 4.8.1. It is easy to verify that the notion of comonad we have just

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introduced is equivalent to the triangular 〈A, ε, ∆〉 notion of § 5.1.5, and hence to all other notions of comonad. This new triangular notion of comonad is more economical than the “cut-free” notion of § 5.3, and is also closer to the usual sequent formulation of the modal logic S4. It also enables us to prove in a straightforward manner Cut Disintegration (cf. § 5.8.3 and § 5.8.4). However, not all equalities are now necessary for the proof: the equality (∆εa), which is independent from the remaining equalities, is not needed. We shall find the notion of comonad of the present section useful for deciding the commuting problem for free comonads generated by arbitrary graphs, a matter to which we devote the next part of this chapter. § 5.8. DECIDABILITY IN FREE COMONADS As in § 4.6 we shall now apply our cut-elimination results for comonads to decidability problems. In this part of the work, A* is meant to be the graph of the category of the free comonad generated by a graph, either in the 〈A, D, εa, δc〉 formulation of the notion of comonad of § 5.3 or in the 〈A, εa, ∆〉 formulation of § 5.7.3. In § 5.8.3 and § 5.8.4 we concentrate on the latter formulation. § 5.8.1. Decision problems in free comonads As in § 4.6.1, we can use Total Cut Elimination, which follows from Cut Disintegration for a free comonad generated by an arrowless graph, combined with the subformula property of our formulations of comonad of § 5.3 and § 5.7.3, to devise a procedure for deciding theoremhood—namely, whether for a pair (A1, A2) of objects of A* there is an arrow of the type A1 → A2 in A*. This is the analogue of the decision procedure that cut elimination usually yields in logic, and in this particular case this is the analogue of a decision procedure for a modal logic included in S4. It is easy to see that we have arrows in A* only for types of the form A → A, D…DA → A and DA → D…DA.

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With Total Cut Elimination, we can also address the less trivial commuting problem for A* (see § 4.6.1). This problem is significant because A* is not a preorder, provided it is generated by a graph with at least one object; for example, the arrow δcεa1DA : DDA → DDA of A* is different from 1DDA (as will become clear at the end of § 5.8.3). Actually, if these arrows were equated and the generating graph is arrowless, then A* would be a preorder. We shall show this in the next section, and after that we shall return to the commuting problem. § 5.8.2. Free comonads in preorders The special comonads where the equality (idem)

εaD f = Dεa f

is satisfied for every f are called idempotent comonads (cf. [Lambek 1969, section 1]). Instead of the idempotency equality (idem), we could as well assume the equivalent equality obtained by replacing f by 1A . (As a matter of fact, any equality of comonographs not satisfied in the free comonad generated by an arrowless graph would do for defining idempotent comonads; see § 5.10.) Consider the free idempotent comonad in A** generated by an arrowless graph. (This free idempotent comonad is constructed quite analogously to the free comonad: we only have to take into account an additional condition, derived from the equality (idem), for the comonadic equivalence relations of the free comonograph.) Then we can show the following. Proposition. The category A** is a preorder. Proof. For every object A of A**, the objects DA and DDA are isomorphic in A**, with the isomorphisms

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δc1DA : DA → DDA, εa1DA : DDA → DA. To show that, we use (εaδc) and δc1DA ° εa1DA = δc(Dεa1A ° 1DDA ), by (fun 1), (idem) and (δc 1) = DDεa1A ° δc1DDA , by (δc 2) = DεaD 1A ° δc1DDA , by (idem) = 1DDA , by (εaδcD) and (fun 1). (By (δc 1) and (cat 1 left), the left-hand side of the equality we have just derived is equal to δcεa1DA of the preceding section.) Then we can infer that any two objects DA and D…DA are isomorphic in A**. (It is clear that we don’t have this isomorphism in A*: just take the comonad of an adjunction such that D is FG.) For the rest of the proof we proceed analogously to what we had in the proof of the Proposition in § 4.6.2. q.e.d. This proposition can be generalized to free idempotent comonads in A** generated by a treelike graph (cf. § 4.6.2). So the equality (idem) is sufficient for turning into a preorder the category of a free special comonad satisfying this equality. Of course, conversely, if we have a comonad in a preorder, then the equality (idem) must hold. § 5.8.3. The commuting problem in free comonads generated by arrowless graphs We shall not deal with the commuting problem for free comonads in the 〈A, D, εa, δc〉 formulation of § 5.3. In this formulation we cannot have a unique arrow term in normal form for every arrow simply by eliminating all cuts and all subterms of the form D1A, because we also have the equality εaδc f = δcεa f. Introducing a reduction corresponding to this equality, where the left-hand or right-hand side would be the redex and the other side the contractum, brings in complications. Moreover, reductions based on the equalities (δnD), (δnεa) and (δnδc),

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which we used in the proof of Cut Disintegration in § 5.7.1, are not particularly handy. It is simpler if now we take comonads in the triangular 〈A, εa, ∆〉 formulation of § 5.7.3, where ∆, as it is clear from its definition in terms of D and δc (cf. § 5.1.2), takes care exactly of these problematic reductions of the 〈A, D, εa, δc〉 formulation. So let us consider the free comonad in the 〈A, εa, ∆〉 formulation generated by an arrowless graph. This is a comonad in the category called again A*. The commuting problem is significant in this context, as it was before: the equality εa∆εa1A = ∆εaεa1A, which corresponds to an instance of the equality (idem) of the preceding section, is does not hold in A* (as we shall show below, at the end of this section). In this equality, and also below, we omit the subscripts of ∆, which are not essential, and can be recovered from the context. Free comonads generated by arrowless graphs are not trivial as were free categories generated by arrowless graphs. Categories of these comonads are not discrete and they are not preorders, provided the generating graph is not empty, i.e., it has at least one object. However, the graphs of these categories can be obtained as disjoint unions of graphs of categories of free comonads generated by arrowless graphs with a single object. An arrow term of A* will be said to be in normal form iff there is no cut in it and no subterm of the form ∆εa1A. So an arrow term in normal form has the shape Q1…Qn1A, where n ≥ 0, every Qi is either εa or ∆, and Qn-1Qn is not ∆εa. With the help of Cut Disintegration, and the ensuing Total Cut Elimination, together with (∆εa), we easily obtain an ordinary normal form theorem: namely, the assertion that for every arrow term f of A* there is an arrow term f ' of A* in normal form such that f = f ' in A*. As we did in § 4.6.3, we shall introduce a collection of reductions with respect to which we have that irreducible arrow terms are in

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normal form. For these reductions we can also prove a strong normalization theorem: namely, the assertion that every sequence of reduction steps is finite. Moreover, we can establish the Church-Rosser property for this collection of reductions, which yields a unique normal form for every arrow term of A*. Strong normalization and the Church-Rosser property, together with Associativity Elimination, which guarantees that every arrow of A* is designated by a unique term in normal form, yield a decision procedure for the commuting problem for A*. Let us now introduce our collection of reductions. Here are the redexes and contracta, where the redexes are either subterms of topmost cuts, i.e., they are of the form f2 ° f1 with no cuts in f1 and f2, or they are of the form ∆εa1A: redexes f ° 1A 1A ° f f 2 ° εa f 1 εa f 2 ° ∆ f 1 ∆ f2 ° ∆ f1 ∆εa1A

contracta f f a ε ( f2 ° f1) f2 ° f1 ∆( f2 ° ∆ f1) 1DA

(cat 1 right) (cat 1 left) (εa 1) (εa∆) (∆ ° ) (∆εa)

In the rightmost column we mention the corresponding equality. Note that with these reductions we have covered all the equalities of the definition of comonad of § 5.7.3 except the equality (cat 2). It is clear that the redexes cover all the possible forms of topmost cuts. This entails that an irreducible arrow term, i.e. one that has no redex as a subterm, will have no topmost cuts, and hence no cuts at all. Since it also cannot have subterms of the form ∆εa1A, it will be in normal form. The strong normalization theorem for these reductions is established quite analogously to what we had in § 4.6.3. The ChurchRosser property for these reductions is established by straightforward considerations of all possible cases. It remains to establish that every arrow of A* is designated by a unique arrow term in normal form.

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This will follow, as in § 4.6.3, from the following form of Associativity Elimination. Associativity Elimination for A*. The equality (cat 2) is derivable from the remaining equalities assumed for the arrow terms of A* without (∆εa). Proof. We proceed as in the proof of Associativity Elimination of § 4.6.3. The following cases remain to be considered in the induction step.

(1) f1 is εa f1'. Then we proceed as in case (2) of the proof of § 4.6.3, using (εa 1). (2) f1 is ∆ f1'. Then we have the following subcases. (2.1) f2 is εa f2'. Then we have ( f3 ° εa f2') ° ∆ f1' = ( f3 ° f2') ° f1', by (εa 1) and (εa∆) = f3 ° ( f2' ° f1'), by the induction hypothesis = f3 ° (εa f2' ° ∆ f1'), by (εa∆). (2.2) f2 is ∆ f2'. Then we have the following subcases. (2.2.1) f3 is εa f3'. Then we have (εa f3' ° ∆ f2') ° ∆ f1' = ( f3' ° f2') ° ∆ f1', by (εa∆) = f3' ° ( f2' ° ∆ f1'), by the induction hypothesis = εa f3' ° (∆ f2' ° ∆ f1'), by (εa∆) and (∆ ° ). (2.2.2) f3 is ∆ f3'. Then we have (∆ f3' ° ∆ f2') ° ∆ f1' = ∆(( f3' ° ∆ f2') ° ∆ f1'), by (∆ ° ) = ∆( f3' ° (∆ f2' ° ∆ f1')), by the hypothesis = ∆( f3' ° ∆( f2' ° ∆ f1')), by (∆ ° ) = ∆ f3' ° ∆( f2' ° ∆ f1'), by (∆ ° ) = ∆ f3' ° (∆ f2' ° ∆ f1'), by (∆ ° ).q.e.d.

induction

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(Besides this syntactical proof of Associativity Elimination for A*, there is a shorter, model-theoretical, proof, which we mention in § 5.9.) As before, once we have eliminated (cat 2), the Church-Rosser property of our reductions implies that if two arrow terms of A* are equal, then they have the same normal form. (An alternative proof of that, eschewing the Church-Rosser property and Associativity Elimination, may be given along the lines of § 4.10.2 with the help of the links of § 5.9.) Since the converse holds, too, this yields a procedure for deciding equality of arrow terms of A* by reducing them to normal form. For example, εa1DA and ∆εaεa1A, which are both in normal form, must be different in A*. Solving the theoremhood or commuting problem in one formulation of a notion solves it for all equivalent formulations, i.e. for all equivalent notions. For example, to decide whether two arrow terms in the 〈A, D, εa, δc〉 formulation of comonad are equal, we can translate them into the 〈A, εa, ∆〉 formulation and decide the question there. § 5.8.4. Decidability in free comonads generated by arbitrary graphs By proceeding as in § 4.5.4, we can prove Total Cut Molecularization, Total Cut Right-Molecularization and Total Cut Left-Molecularization for free comonads generated by arbitrary graphs in both the 〈A, D, εa, δc〉 formulation of § 5.3 and the 〈A, εa, ∆〉 formulation of § 5.7.3. With the latter formulation we don’t need the equality (∆εa) to prove either of these results. We can combine Total Cut Molecularization and the subformula property of these two formulations of comonad, together with a hypothetical decision procedure for a variant of the theoremhood problem concerning the generating graph of the free comonad—a problem about paths, analogous to the problem displayed at the beginning of § 4.6.4—so as to obtain a decision procedure for the theoremhood problem in free comonads generated by arbitrary graphs.

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We also have a decision procedure for the commuting problem in these free comonads, independently of any presuppositions concerning the generating graph. We shall now describe this procedure. As in the preceding section, let us concentrate on the 〈A, εa, ∆〉 notion of comonad, and with this notion let us consider the free comonad in A* generated by an arbitrary graph. Arrow terms of A* will now be said to be in normal form iff every cut in them is molecular and they have no subterms of the form ∆εa1A. If instead of “molecular” we put “right-molecular”, we obtain right-normal forms. (Alternatively, we could as well put “left” instead of “right”, and work with left-normal forms.) We reduce an arrow term to right-normal form with the collection of reductions of the preceding section applied to subterms of topmost nonmolecular cuts or subterms of the form ∆εa1A; moreover, we have the (cat 2 mol) reduction of § 4.6.4. We obtain strong normalization and the Church-Rosser property for this collection of reductions as before. It remains to establish that every arrow term of A* is designated by a unique arrow term in right-normal form. This will follow, as in § 4.6.4, from the following version of Associativity Molecularization. Associativity Molecularization for A*. The equality (cat 2) is derivable from (cat 2 mol) and the remaining equalities assumed for the arrow terms of A* without (∆εa).

To prove this lemma, we proceed as for the proof of Associativity Elimination and Associativity Molecularization in § 4.6.3 and § 4.6.4, and as for the proof of Associativity Elimination in the preceding section. The cases to be considered are those of the preceding section: no additional case can arise with molecular arrow terms because of incompatibilities in the targets and sources in composition. As before, once we have replaced (cat 2) by (cat 2 mol), we can establish that two arrow terms of A* are equal iff they have the same right-normal form. This yields a procedure for deciding equality of these arrow terms.

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§ 5.9. MODEL-THEORETICAL NORMALIZATION IN COMONADS In this section, which is parallel to § 4.10, we use a faithful modelling of free comonads generated by arrowless graphs to obtain a simple decision procedure for commuting and a simple method of normalizing arrow terms. With the help of this modelling, the uniqueness of normal form can be demonstrated, as in § 4.10.2, without invoking the Church-Rosser property of reductions and Associativity Elimination. In the next section we use this modelling to demonstrate the maximality of the notion of comonad, which is analogous to the maximality we have demonstrated syntactically for the notion of adjunction in § 4.11. So, let us first present the model-theoretical decision procedure for the commuting problem in free comonads, which is simpler than the syntactical decision procedure of § 5.8.3 and § 5.8.4. This decision procedure, analogous to the decision procedure for commuting in free adjunction of § 4.10.1, is based on drawing links between symbols in the object terms that are the source and target of an arrow term. However, to show that this decision procedure works we rely on reduction to cut-free normal form such as we had before. This new decision procedure is not tied to a particular formulation of the notion of comonad: variants of it exist for all notions of comonad we have envisaged, including those which are not “cut-free”, i.e. which don’t permit cut elimination. We shall describe here this procedure for the standard notion of comonad of § 5.1.1, which is not “cut-free”. To simplify matters, we concentrate on the commuting problem in such free comonads generated by arrowless graphs. Extending this technique to free comonads generated by arbitrary graphs doesn’t seem to yield a clear advantage over the technique of § 5.8.4. So take the standard notion of comonad of § 5.1.1, and consider the free comonad in A* in this sense generated by an arrowless graph. To every arrow term f : A1 → A2 of A* we assign a set of links Λ( f ), which is a set of ordered pairs (x2, x1) where x2 is an occurrence of D

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in A2 and x1 an occurrence of D in A1. We define Λ( f ) by induction on the complexity of f. If f is 1A, then A1 and A2 are two copies of A, and in Λ( f ) we put the links between the n-th D of A2 and the n-th D of A1. If f is εA: DA → A, then in Λ( f ) we put just a copy of the links in Λ(1A). So nothing is linked to the first D of DA, counting from the left. If f is δA: DA → DDA, then Λ( f ) is obtained by adding to a copy of Λ(1A) the link between the first D of DDA and the first D of DA and also the link between the second D of DDA and the first D of DA. So two occurrences of D in DDA are linked to the first D of DA. If f is of the form D f ': DA1' → DA2', then Λ( f ) is obtained by adding to a copy of Λ( f ') the link between the first D of DA2' and the first D of DA1'. Finally, if f is of the form f2 ° f1 for f1: A1 → A3 and f2: A3 → A2, then every link (x2, x1) of Λ( f ) is obtained from a link (x3, x1) of Λ( f1) and a link (x2, x3) of Λ( f2). Let Dn, where n ≥ 0, stand for a sequence of n occurrences of D. Every arrow term f of A* is of some type DnE → DmE for E a generative object term. Then the links of Λ( f ) determine an orderpreserving function from {1,…, m} to {1,…, n} (this function goes in the direction opposite to that of the arrow f ). If n = 0, then {1,…, n} is the empty set. The links of Λ(1DnE) determine the identity function from {1,…, n} to {1,…, n}, and those of Λ(εDnE) the inclusion function from {1,…, n} to {1,…, n+1}, i.e. the function that maps i to i. If n = 0, then both of these functions are the empty function. The links of Λ(δDnE) determine the function from {1,…, n+2} to {1,…, n+1} that for i ≤ n+1 maps i to i, while n+2 is mapped to n+1. If the function determined by Λ( f ) is from {1,…, m} to {1,…, n}, then the function determined by Λ(D f ) is from {1,…, m+1} to {1,…, n+1}; it behaves exactly as the function of Λ( f ) for i ≤ m and it maps m+1 to n+1. Finally, the function determined by Λ( f2 ° f1) is obtained by composing the function determined by Λ( f2) with the function determined by Λ( f1).

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Then we can demonstrate the following coherence proposition, analogous to the Proposition of § 4.10.1. Proposition. Suppose f1 and f2 are arrow terms of A* of the same type. Then we have f1 = f2 in A* iff Λ( f1) = Λ( f2). Proof. To prove this equivalence from left to right we proceed by induction on the length of the derivation of f1 = f2 in A*. In this induction, the basis is essential: in it we have to check all the equalities of the definition of comonad of § 5.1.1 to ascertain that the links of the left-hand side are equal to the links of the right-hand side. For example, for (εδ) we have DA δA εDA

DA

DDA

1 DA

DA

DA

To prove the equivalence of the Proposition from right to left, we translate arrow terms of A* into arrow terms of the “cut-free” triangular notion of comonad of § 5.7.3. Links are now defined by relying on the definitions εa f =def f ° εA1, ∆A f =def D f ° δA. This means that for f : A1 → A2 we find in Λ(εa f ) just a copy of the links in Λ( f ), while for f : DA → A' and ∆A f : DA → DA' we obtain Λ(∆A f ) by adding to a copy of Λ( f ) the link between the first D of DA' and the first D of DA. In this “cut-free” formulation of comonad, if two arrow terms of the same type, which are in the normal form of § 5.8.3, are different, then they have different links. (The proof of that is analogous to the proof of the left-to-right direction of the Lemma of § 4.10.1.) So, by contraposition, if two terms of the same type in normal form have the same links, then they are the same term. We pursue further this proof

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by following the pattern of the proof of the Proposition in § 4.10.1. q.e.d.

So, to answer the question whether for f1 and f2 of the same type f1 = f2 is satisfied in A*, it is enough to draw the links of Λ( f1) and Λ( f2), and check whether they are equal. To extend this decision procedure to equalities f1 = f2 in free comonads generated by arbitrary graphs, we need to check not only whether Λ( f1) = Λ( f2), but also whether the molecular arrow terms involved in f1 and f2 can be equated. These molecular arrow terms become apparent by reducing f1 and f2 to the normal form of § 5.8.4. The Proposition proved in this section guarantees that there is a faithful functor from A* to the concrete category whose objects are finite ordinals and whose arrows are order-preserving functions from finite ordinals to finite ordinals, with domains being targets and codomains sources. If the arrowless generating graph of the free comonad in A* has at least one object, then this functor is onto on objects and on arrows, and if the generating graph is arrowless with a single object, then this functor is an isomorphism (according to [Lambek 1969, p. 95], this insight is due to Lawvere; cf. [Lawvere 1969a, pp. 148ff.]). The links of this section provide a simple model-theoretical method for normalizing arrow terms in free comonads generated by arrowless graphs. As in § 4.10.2, this method consists in drawing the links of an arrow term, and then constructing in an obvious way out of these links an arrow term in normal form with the same links. For example, if A is a generative object term, out of the links DDDA DDDDA we construct the arrow term εa∆DA εa∆A ∆A ∆A εa1A, whose normal form is εa∆DA εa∆A ∆A 1DA.

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Our links can also provide an alternative proof of the uniqueness of normal form of § 5.8.3 along the lines of § 4.10.2. In that proof we eschew using the Church-Rosser property and Associativity Elimination. § 5.10. THE MAXIMALITY OF COMONAD As an application of our cut-elimination results for comonads we shall demonstrate in this section that the notion of comonad is maximal in the sense that any equality between arrow terms of comonographs not assumed for comonads yields, in the presence of the equalities of comonads, the idempotency equality (idem) of § 5.8.2. This is analogous to the Maximality of Adjunction we established in § 4.11. However, our proof of the maximality of the notion of comonad will be somewhat different in relying on the links we have introduced in the preceding section (though a syntactical proof, like the proof of § 4.11, which would be somewhat longer, could also be given). Take the standard notion of comonad of § 5.1.1 and consider the free comonad in A* generated by an arrowless graph. Then let f1 and f2 be arrow terms of A*, both of the same type DnA → DmA for A a generative object term. Suppose f1 = f2 is not satisfied in A*. For a comonad in A, we say that f1 = f2 holds in A iff for every comonofunctor N from A* to A we have that N f1 = N f2 is satisfied in A. We can then prove the following proposition, analogous to the Maximality of Adjunction of § 4.11. Maximality of Comonad. If for a comonad in A the equality f1 = f2 holds in A, then this comonad is idempotent. Proof. Since f1 = f2 is not satisfied in A*, by the Proposition of § 5.9 the function φ1 from {1,…, m} to {1,…, n} determined by Λ( f1) is different from the function φ2 from {1,…, m} to {1,…, n} determined by Λ( f2). So for some i in {1,…, m} we must have φ1(i) ≠ φ2(i). Suppose φ1(i) < φ2(i). (In case φ1(i) > φ2(i), we reason analogously.)

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Let g : DmA → DA be an arrow term of A* composed with various instances of εB with the help of the functor D such that the i-th D of DmA counting from the right is linked to the D of DA. Next let h : DDA → DnA be an arrow term of A* composed with various instances of δB with the help of the functor D such that the right D of DDA is linked to the φ1(i)-th D of DnA counting from the right, and the left D of DDA is linked to the φ2(i)-th D of DnA counting from the right. It is straightforward to show that arrow terms of A* such as g and h do exist. Then Λ(g ° f1 ° h) = Λ(εDA) and Λ(g ° f2 ° h) = Λ(DεA). So, by the Proposition of § 5.9, we have g ° f1 ° h = εDA and g ° f2 ° h = DεA in A*. So if f1 = f2 holds in A, then (idem')

εDA = DεA

holds in A. From this equality we deduce (idem) as follows: εaD f = D f ° εDA, by the definition of εa = D f ° DεA, by (idem') = Dεa f, by (fun 2) and the definition of εa. (If (idem') holds in A, then it is satisfied for every object A of A.) q.e.d. (I am indebted to Sava Krstic for helping me to reach this proof.) This proposition says that all the equalities of comonographs that don’t follow from the equalities of comonads are of the same strength as the equality (idem). Any such equality yields (idem), and (idem) yields this equality, as it follows from the Proposition proved in § 5.8.2. So all the equalities of comonographs that are not equalities of comonads are in the same bag. And any of these equalities when added to the equalities of comonads produces idempotent comonads.

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§ 5.11. THE LINKS OF ADJUNCTIONS AND THE LINKS OF COMONADS We have remarked in § 4.5.1 that the graphs of categories in free adjunctions generated by arrowless graphs are disjoint unions of graphs of categories involved in free adjunctions generated by pairs of graphs (G, H) where one of G and H is arrowless with a single object and the other is the empty graph. Take now the free adjunction 〈A, B, F, G, ϕ, γ〉 generated by (G, H) where G is arrowless with a single object C and H is empty. The objects of A are then of the form (FG)nC, where (FG)n, with n ≥ 0, stands for a possibly empty sequence of n blocks of FG. The objects of B are the objects of A with G prefixed. It follows from Propositions 2 and 3 of § 5.2.2 that B is isomorphic to AFΓ, the delta category of the comonad of the adjunction. The objects of the subcategory AFΓ of A are (FG)nC with n ≥ 1, i.e. all the objects of A except C. It follows easily that the adjunction 〈A, B, F, G, ϕ, γ〉 is isomorphic to the adjunction 〈A, AFΓ, I, FG, ϕ, FγG〉, where I is inclusion. This isomorphism is based on the functor F, which maps B isomorphically onto AFΓ. Take now the free comonad 〈A, D, ε, δ〉 generated by the arrowless graph G with a single object C. We can show that this comonad is isomorphic to the comonad of the adjunction above by the comonofunctor that maps D to FG, ε to ϕ and δ to FγG. This isomorphism (which was considered from a 2-categorial point of view by [Auderset 1974] and [Schanuel & Street 1986]) is demonstrated most easily with the help of the links of § 5.9 and § 4.10.1. It suffices to correlate the links on the left-hand side with those on the right-hand side: DA

FGA

εδA

ϕA A

A

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DA

FGA

δA

FγGA

DDA

FGFGA

So the links of comonads of § 5.4 can replace the links of A of our free adjunction. For the links of B it suffices to note that they have an isomorphic copy in AFΓ, and these links reduce again to the links of comonads. When our free adjunction is generated by an empty G and an arrowless H with a single object, we rely analogously on the links of monads, which are obtained by reversing the links of comonads. So the links of adjunctions of § 4.10.1 could be replaced by the links of comonads of § 5.9. We have nevertheless preferred to introduce the former links because the approach through them is more direct, and because of their connection with the graphs of [Eilenberg & Kelly 1966] and with the theory of tangles mentioned in § 4.10.1. By the same token, we could use the links of adjunctions to work with the free comonad. The maximality of adjunction of § 4.11 can now be deduced from the maximality of comonad of § 5.10. Adding any further equality to the free adjunction makes idempotent the comonad of the adjunction, and this yields the preordering equalities. Our direct proof of the maximality of adjunction in § 4.11 shows however that we can obtain the same result by relying on syntactical methods and cut elimination. (Matters of this section are treated in [D. 2008].)

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CHAPTER 6

CARTESIAN CATEGORIES

§ 6.1. RECTANGULAR || CATEGORIES WITH BINARY PRODUCT The fact that we have Cut Disintegration and its corollaries for free adjunctions generated by graphs doesn’t mean that we shall have these results in every adjunction. Cut elimination is not automatically transferred from the free case to particular adjunctions. In general, cut-elimination results for various systems cannot be easily transferred from one case to another. One small change is often enough to block this inference and to require that we start over again our cut-elimination proofs. (Even when we have equivalent formulations of the same system, cut elimination cannot be transferred from one formulation to another, because, as we said at the beginning of § 0.3.1, cut elimination is a property of a particular formulation.) We shall now consider a particular kind of adjunction were Total Cut Elimination fails for free adjunctions generated by arrowless graphs in the (ac) formulation of rectangular || adjunction of § 4.2, though it can be recovered in other formulations. This kind of adjunction is interesting for logic because it gives rise to categories with binary product, the free specimens of which are logical systems with the conjunction connective. This conjunction is both classical and intuitionistic, because classical and intuitionistic conjunction don’t differ. (Sequent systems where conjunction is the only logical constant coincide in classical and intuitionistic logic. These systems can be based either on singular sequents, i.e. such that Y in X ≤ Y is a singleton, or plural sequents, i.e. such that Y can have more than one member.)

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Conjunction may be taken as being the most basic logical constant. It serves to join a finite number of premises into a single premise, which is the first thing we must do when we approach sequents through categories. So Gentzen’s structural rules involving premises can be formulated as rules about conjunction. This connective is the closest to the structural foundations, the substructure, of logic. Implication also matches structural features of logic, but matters are clearer if they are formulated in terms of conjunction, rather than implication. For example, the deduction theorem based on conjunction is easier to explain and generalize than the usual deduction theorem based on implication (see [D. 1996]). We now proceed with the definition of category with binary product. The product of two graphs A and B is the graph A × B whose objects are pairs (A , B) where A is an object of A and B an object of B, and whose arrows are pairs ( f , g): (A 1, B1) → (A 2, B2) where f : A1 → A2 is an arrow of A and g: B1 → B2 an arrow of B. The product of two deductive systems 〈A, 1, ° 〉 and 〈B, 1, ° 〉 is the deductive system 〈A × B, 1, ° 〉 where 1 and ° are defined in terms of the 1 and ° of A and B by 1(A, B) =def (1A, 1B),

( f 2, g2) ° ( f 1, g1) =def ( f 2 ° f 1, g2 ° g1). It is easy to see that the product of two categories is a category. Let F be the diagonal functor from a deductive system 〈B, 1, ° 〉 to the product deductive system 〈B × B, 1, ° 〉, which is defined as follows: FB =def (B , B),

Fg =def (g , g).

Let us suppose that we have a binary operation × on the objects of B and a binary operation, again written ×, on the arrows of B that assigns to the arrows g': B1' → B2' and g": B1" → B2" the arrow g'× g": B1'× B1" → B2'× B2". Then let G be a graph-morphism from B × B to B defined as follows:

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G(B', B") =def B'× B",

G(g', g") =def g'× g".

(The internal product × of the graph-morphism G shouldn’t be confused with the external product × of A × B.) Let K1aB" be operations on the arrows of B that to every g: B' → B assign the arrow K1aB"g: B'× B" → B, and K2aB' be operations on the arrows of B that to every g: B" → B assign the arrow K2aB'g: B'× B" → B. These operations define as follows an antecedental transformation ϕa from FG to IB×B. For g': B1' → B2' and g": B1" → B2" we have ϕa(g', g") =def (K1aB1"g', K2aB1'g"). The arrows K1aB"1B': B'× B" → B' are first projections and the arrows K2aB'1B": B'× B" → B" second projections. This explains the superscripts 1 and 2 of K1a and K2a; the superscript “a” comes from the antecedental transformation ϕa, and K comes from the name of the related constant combinator of combinatory logic. Finally, let γc be a consequential transformation from IB to GF. Then 〈B, 1, ° 〉 is said to be a category with binary product, or ×-category, for short, iff 〈B × B, B, F, G, ϕa, γc〉 is an adjunction. Of course, this should be an adjunction in our main, i.e. (ac) formulation, of rectangular || adjunction of § 4.2. This means that 〈B, 1, ° 〉 is a category (this implies that 〈B × B, 1, ° 〉 is a category, too); (ii) G is a functor (F is a functor anyway); (iii) ϕa is a natural antecedental transformation and γc a natural consequential transformation; (iv) the rectangular equalities (ϕaγcF) and (ϕaγcG) hold. (i)

All these conditions can be expressed by equalities between arrow terms of B, as it is clear for (i), and as will become clear below for (ii), (iii) and (iv).

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For 〈B, 1, ° 〉 a deductive system with the operations × on objects and on arrows, let 〈B2, 1, ° 〉 be the deductive system whose objects are the objects B'× B" of B, and whose arrows are the arrows g'× g": B1'× B1" → B2'× B2" of B, while 1 and ° are defined as follows: 1B'×B" =def 1B' × 1B",

(g2'× g2") ° (g1'× g1") =def (g2' ° g1')×(g2" ° g1"). Let the graph-morphism G from B × B to B be defined as above. Then G is a functor from 〈B × B, 1, ° 〉 to 〈B, 1, ° 〉 iff 〈B2, 1, ° 〉 is a subsystem of 〈B, 1, ° 〉. We also have that G is one-one both on objects and on arrows iff the deductive systems 〈B × B, 1, ° 〉 and 〈B2, 1, ° 〉 are isomorphic by a restriction of G. So, if G is a functor and one-one on objects and on arrows, then 〈B × B, 1, ° 〉 is isomorphic to a subsystem of 〈B, 1, ° 〉. The adjunction 〈B × B, B, F, G, ϕa, γc〉 amounts then to a reflection of 〈B, 1, ° 〉 in its subcategory 〈B2, 1, ° 〉. We shall not respect any more the notational conventions we have previously introduced for adjunctions, and which we still followed for the adjunction of product above. (If we continued applying these conventions, we would overburden the letters for objects and arrows with indices.) Let us use A, B, C, … for objects and f, g, h, … for arrows of a single graph, as we did at the beginning of this work. Let us also write Wc for γc, because γc1A : A → A × A is related to the contraction combinator W of combinatory logic. Then 〈A, 1, ° 〉 is a ×-category iff (i)

〈A, 1, ° 〉 is a category;

(ii)

× is a functor from 〈A × A, 1, ° 〉 to 〈A, 1, ° 〉,

which means that the following equalities hold in A: (fun 1 ×) (fun 2 ×)

1A × 1B = 1A×B,

( f2 ° f1)×(g2 ° g1) = ( f2 × g2) ° ( f1 × g1);

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(iii) we have the following operations on the arrows of A: f:B → C

f:B → C

K 1a A f : B × A → C K 2a A f : A × B → C f:B → C Wc f : B → C × C

which for f1: A1 → B1 and f2: A2 → B2 satisfy the following equalities tied to naturalness: (K1a 1) (K1a 2)

K1aA(g ° f ) = g ° K1aA f, K1aA2(g ° f1) = K1aB2g ° ( f1 × f2),

(K2a 1) (K2a 2)

K2aA(g ° f ) = g ° K2aA f, K2aA1(g ° f2) = K2aB1g ° ( f1 × f2),

(Wc 1) (Wc 2)

Wc(g ° f ) = Wcg ° f, Wc(g ° f ) = (g × g) ° Wc f;

(iv) as well as the following equalities tied to the rectangular equalities: (K1aWcF) (K2aWcF)

K1aBg ° Wc f = g ° f, K2aBg ° Wc f = g ° f,

(KaWcG)

(K1aA2 f1 ×K2aA1 f2) ° Wcg = ( f1 × f2) ° g.

The diagonal functor F cannot be seen in the three rules for K1a, K2a and Wc. This justifies our calling this functor “structural” and “invisible” (cf. the corresponding (ac) rules in § 4.5.5). It should be clear by now how to construct the free ×-category generated by a graph. In such categories the operation × on objects is one-one and is hence a connective in the sense of § 5.5. (To show that × on arrows is then also one-one is a more involved matter. That this is the case for free ×-categories generated by arrowless graphs will follow from the results on normalization of § 6.4.)

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We could envisage formulating Cut Disintegration for these free ×-categories, but the definitions of § 1.8.1 would then have to be reworked. The equalities (fun 2 ×) and (Wc 2) would introduce cuts linked to two other cuts—something we didn’t have before. However, we shall not go into these complications, because if this Cut Disintegration were to imply Total Cut Elimination for free ×-categories generated by arrowless graphs, it would not be provable. (If, on the other hand, it doesn’t imply this Total Cut Elimination, then its interest is limited.) The Total Cut Elimination just mentioned (which we can formulate without reworking the definitions of § 1.8), and hence also the Total Cut Elimination for the corresponding adjunction, simply fail. A counterexample is provided by the arrow term (K2aA1B ×K1aB1A) ° Wc1A×B: A × B → B × A, which is not equal to a cut-free arrow term. There are other such counterexamples, based on terms of the form ( f × g) ° Wc1C that can be instances neither of the right-hand side of (Wc 2) nor of the lefthand side of (KaWcG). § 6.2. TRIANGULAR CATEGORIES WITH BINARY PRODUCT To obtain Total Cut Elimination we have to find another formulation of ×-categories than that of the preceding section. Let us first briefly survey possible choices of primitives in terms of our hexagonal figure of primitive notions in adjunction of § 4.1.1. We had the rectangular || choice in the preceding section in its (ac) formulation; there remains three more formulations of this choice, which we shall consider in § 6.5 and § 6.6 (obtaining Total Cut Elimination in the (aa) and (ca) formulation). For the remaining choices note that the diagonal functor F, being “invisible”, does not count (such was also the inclusion functor F in the adjunction defining comonads in § 5.1.2 and § 5.1.3). Because of that, Φ is definable in terms of ϕ alone: both ϕ and Φ involve just the first and second projection, which in the preceding section were given via the operations K1aA and K2aA.

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On the other hand, γ, which in the preceding section was given via is definable in terms of Γ alone. This Γ is given by pairing, i.e. the binary partial operation on arrows Wc,

f:C→ A

g:C→ B

〈 f, g〉 : C → A × B

which is involved in the following definitional equalities: (Ga×) (γW) (Γ〈 , 〉)

f1 × f2 = 〈K1aA2 f1, K2aA1 f2〉,

Wc f = 〈1B, 1B〉 ° f,

〈 f, g〉 = ( f × g) ° Wc1C.

It is easy to conclude that the rectangular \\, rectangular //, triangular ► and seesaw choices of primitives all boil down to having the two projections and the pairing operation primitive, while the triangular ◄ choice has the same primitives as the rectangular || choice. So there remains the possibility of defining ×-categories with the operations K1aA, K2aA and pairing as primitive. We shall do that now in the style of a triangular 5 adjunction of § 4.8. This is the simplest and, in the present context, best adapted approach to cut elimination. With this definition we obtain Total Cut Elimination, which in the next section we use to consider decidability problems. A category 〈A, 1, ° 〉 will now be called a ×-category iff we have the operations K1aA, K2aA and pairing on the arrows of A that satisfy the equalities (K1a 1) (K1a)

K1aA(g ° f ) = g ° K1aA f, K1aBh ° 〈 f, g〉 = h ° f,

(K2a 1) (K2a)

K2aA(g ° f ) = g ° K2aA f, K2aBh ° 〈 f, g〉 = h ° g,

(distr) (K1aK2a)

〈 f, g〉 ° h = 〈 f ° h, g ° h〉, 〈K1aB1A, K2aA1B〉 = 1A×B.

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These equalities should be compared with the equalities of the definition of triangular adjunction in § 4.8.1. The equalities (K1a 1) and (K2a 1) correspond to (ϕa 1), the equalities (K1a) and (K2a) to (β"), the equality (distr) to (ΓΓ"), and, finally, the equality (K1aK2a) to (Γϕa). It is easy to show that this notion of ×-category is equivalent to the notion of the preceding section. (Note that (fun 1 ×) stands behind (K1aK2a).) Our new notion of ×-category is quite in the spirit of Gentzen’s sequent systems with rules for introducing a connective on the left and on the right of the turnstile. And Total Cut Elimination is now easily obtainable for free ×-categories generated by arrowless graphs. § 6.3. CUT ELIMINATION IN FREE TRIANGULAR CATEGORIES WITH BINARY PRODUCT Let now A* be the graph of a free triangular ×-category, formulated as in the preceding section, generated by an arrowless graph. We shall demonstrate Total Cut Elimination for A*. Proof of Total Cut Elimination for A*. We redefine the notion of degree of an arrow term for this proof. This degree is now the number of occurrences of the symbols 1, ° , K1a, K2a and 〈 , 〉 in this arrow term (here, 〈 , 〉 is counted as a single symbol). If the subterm of a cut is g ° f, then the degree of the cut is, as before, the degree of g ° f. We shall now follow Gentzen’s procedure of eliminating topmost cuts, i.e. cuts in whose subterms g ° f the arrow terms f and g are cutfree. (Because of that we could have omitted counting the occurrences of ° in the definition of degree, above.) Take in an arrow term of A* a topmost cut whose subterm is g ° f. We show that g ° f is either equal to a cut-free arrow term, or it is equal to an arrow term all of whose cuts are topmost and of strictly smaller degree than the degree of g ° f. The possibility of eliminating every topmost cut, and hence every cut, follows then by induction on degree. We have the following cases.

(1) f is 1A. Then g ° 1A = g by (cat 1 right), and g is cut-free.

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(2) f is K1aA f '. Then g ° K1aA f ' = K1aA(g ° f ') by (K1a 1), and the degree of g ° f ' is strictly smaller than the degree of g ° f. The cut of g ° f ' is topmost. (3) f is K2aA f '. This case is treated analogously to (2), using (K2a 1). (4) f is 〈 f ', f "〉. Then we have the following subcases. (4.1) g is 1A. Then 1A ° f = f by (cat 1 left), and f is cut-free. (4.2) g is K1aAg'. Then K1aAg' ° 〈 f ', f "〉 = g' ° f ' by (K1a), and the degree of g' ° f ' is strictly smaller than the degree of g ° f. The cut of g' ° f ' is topmost. (4.3) g is K2aAg'. This case is treated analogously to (4.2), using (K2a). (4.4) g is 〈g', g"〉. Then 〈g', g"〉 ° f = 〈g' ° f , g" ° f 〉 by (distr), and the degrees of g' ° f and g" ° f are strictly smaller than the degree of g ° f. The cuts of g' ° f and g" ° f are topmost. q.e.d. Note that the equalities (cat 2) and (K1aK2a) are not used in the proof we have just given. The fact that this approach to ×-categories works for Total Cut Elimination is explained by noting that the pairing operation on arrows takes care exactly of the problematic cuts of ×-categories in the rectangular || formulation of § 6.1. So one could say that the problematic cuts have not really been eliminated, but rather hidden under pairing. We have not tried to formulate Cut Disintegration and its corollary Particular Cut Elimination for the free ×-categories of this section, because the equality (distr) would require that a single cut be linked to more than one cut (this concerns not only the cuts displayed in (distr), but also those in an h of (distr)). Before, Particular Cut Elimination entailed Total Cut Elimination, because in Particular Cut Elimination every cut save the eliminated one was linked to exactly one cut. So we could eliminate all cuts one by one. Now, Particular Cut Elimination, by itself, would not entail Total Cut Elimination, because cuts that are not eliminated may be linked to more than one cut. So it is not clear

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that eliminating cuts that are not topmost, one by one, leads to the elimination of all cuts. Besides that, Cut Disintegration could probably not be applied here to show the necessity of all our equalities: the equality (K1aK2a) may well prove superfluous. Because of all that we eschew the complications of reformulating Cut Disintegration in the present context. However, we can easily extend the proof of Total Cut Elimination above to a proof of Total Cut Molecularization, and its right and left forms, for free ×-categories generated by arbitrary graphs. For that we eliminate topmost nonmolecular cuts (and we could omit counting occurrences of ° of molecular cuts in the definition of degree). § 6.4. DECIDABILITY IN FREE TRIANGULAR CATEGORIES WITH BINARY PRODUCT Let A* be, as in the preceding section, the graph of the free triangular ×-category generated by an arrowless graph. Combining the Total Cut Elimination of the preceding section with the subformula property of the operations K1a, K2a and pairing, we can obtain a decision procedure for the theoremhood problem for A* (see § 4.6.1). We can also obtain a decision procedure for the commuting problem for A* as we shall show below. This problem is significant because in A* we have K1aA1A: A × A → A, which is different from K2aA1A: A × A → A (as we shall establish below). Actually, if these two arrows are equal in a ×-category, then this category is a preorder. Because for the arrows f : C → A and g: C → A, from K1aA1A = K2aA1A we obtain K1aA1A ° 〈 f, g〉 = K2aA1A ° 〈 f, g〉, which yields f = g with (K1a), (K2a) and (cat 1 left).

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In fact, it can be shown that for all arrow terms f : A → B and g: A → B of A* such that we don’t have f = g in A*, if f = g holds in a ×-category, then this category is a preorder. This shows that the set of equalities of ×-categories is maximal: any further equality makes these categories collapse into preorders. The same maximality is demonstrable for cartesian categories, i.e. ×-categories with a terminal object. This maximality of the notions of ×-category and cartesian category is stronger than the one we proved for the notions of adjunction and comonad in § 4.11 and § 5.10. There we had only that some free special adjunctions and some free special (i.e. idempotent) comonads, with an extra equality, are preorders. Here we have that every special ×-category and every special cartesian category, with an extra equality, is a preorder. We shall, however, not go into the proof of that here, since this proof need not depend upon cut elimination, as the results of § 4.11 and § 5.10 do. (A proof based on normalization in natural deduction and on a coherence theorem for cartesian categories, which also yield uniqueness of normal form and an easy decision procedure for commuting via links, is presented in [D. & Petric 1997a]; see § 6.9.) An arrow term of A* will be said to be in normal form iff it is cutfree and has no subterms of the special forms 〈K1aB1A, K2aA1B〉, 〈K1aA f, K1aAg〉 and 〈K2aA f, K2aAg〉. We shall now introduce a collection of reductions that eliminate cuts from arrow terms, or replace them by cuts of smaller degree, or eliminate subterms of the special forms listed above. With respect to this collection of reductions we shall be able to prove strong normalization and the Church-Rosser property. Here are the redexes and contracta of our collection of reductions, where the redexes are either subterms of topmost cuts, or they are of the special forms mentioned in the preceding paragraph: redexes f ° 1A 1A ° f g ° K1aA f K1aBh ° 〈 f, g〉

contracta f f K1aA(g ° f ) h° f

(cat 1 right) (cat 1 left) (K1a 1) (K1a)

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g ° K2aA f K2aBh ° 〈 f, g〉 〈 f, g〉 ° h 1a 〈K B1A, K2aA1B〉 〈K1aA f, K1aAg〉 〈K2aA f, K2aAg〉

K2aA(g ° f ) h° g 〈 f ° h, g ° h〉 1A×B

K1aA〈 f, g〉 K2aA〈 f, g〉

(K2a 1) (K2a) (distr) (K1aK2a) (K1a distr) (K2a distr)

In the rightmost column we mention the corresponding equalities, which, except for the last two, are all from the definition of ×-category of § 6.2. With our reductions we have covered all the equalities of the definition of ×-category except the equality (cat 2). Conversely, all reductions are covered by equalities that hold in A*. We only need to derive (K1a distr) and (K2a distr). For (K1a distr) we have 〈K1aA f, K1aAg〉 = 〈 f ° K1aA1B, g ° K1aA1B〉, by (cat 1 right) and (K1a 1) = K1aA〈 f, g〉, by (distr), (K1a 1) and (cat 1 right), and we proceed analogously for (K2a distr). Strong normalization for these reductions is a consequence of the fact that cuts in contracta either disappear or are of strictly smaller degree than cuts in the redexes, and of the fact that the degree of contracta of the last three kinds is strictly smaller than the degree of the corresponding redexes. More precisely, we can take as the complexity measure of an arrow term f a triple 〈n1, n2, n3〉 where n1 is the number of cuts in f that are not topmost; next, if d1,…, dk are the degrees of all the topmost cuts in f, then n2 = 3d1 +…+3dk (if there are no topmost cuts in f, and hence no cuts, then n2 = 0), and n3 is the degree of f. The triples 〈n1, n2, n3〉 are lexicographically ordered by an order of type ω3 with the definition 〈n1, n2, n3〉 < 〈m1, m2, m3〉 iff n1 < m1 or (n1 = m1 and n2 < m2) or (n1 = m1 and n2 = m2 and n3 < m3). In the first two reductions, n1 either decreases, or it is kept constant while n2 decreases. In all other reductions n1 is kept constant. It is clear that in the reductions corresponding to (K1a 1), (K1a), (K2a 1) and (K2a) the number n2 decreases (while n3 decreases or is kept

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constant, which is without importance). The number n2 decreases in a (distr) reduction too, because if d1 < d and d2 < d, then 3d1 +3d2 < 3d (although d1 + d2 might be strictly greater than d). This is inspired by a trick of Gentzen from [1938]; the general fact is that if for every i such that 1≤ i ≤ k we have mi < m, then k

Σ

(k+1)m i < (k+1)m .

i=1

The fact that n3 increases in a (distr) reduction is of no consequence. In the (K1aK2a), (K1a distr) and (K2a distr) reductions the number n2 decreases or is kept constant while n3 decreases. (We could have used a complexity measure of this kind for proving strong normalization in § 4.6.3 and § 5.8.3, but there we could do with a simpler kind; the complications we have here are due to the (distr) reductions.) The Church-Rosser property for our reductions is established by rather straightforward considerations of all possible cases. The only somewhat problematic cases arise when 〈K1aB1A, K2aA1B〉 ° h is reduced with the (distr) and (K1aK2a) reductions, and when 〈KiaA f, KiaAg〉 ° h is reduced with the (distr) and (Kia distr) reductions. In these cases we have to make an induction on the complexity of h to show the diamond property. (Since we have strong normalization, we can infer the Church-Rosser property from the weak Church-Rosser property, as in § 4.6.3.) It remains to establish that every arrow of A* is designated by a unique arrow term in normal form. This will follow, as in § 4.6.3 and § 5.8.3, from the following form of Associativity Elimination. Associativity Elimination for A*. The equality (cat 2) is derivable from the remaining equalities assumed for the arrow terms of A* without (K1aK2a). Proof. We proceed as in the proof of Associativity Elimination of § 4.6.3. The following cases remain to be considered in the induction step.

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(1) - (2) f1 is K1aA f1' or K2aA f1'. Then we proceed as in case (2) of the proof of § 4.6.3, using (K1a 1) or (K2a 1). (3) f1 is 〈 f1', f1"〉. Then we have the following subcases. (3.1) f2 is K1aA f2'. Then we have ( f3 ° K1aA f2') ° 〈 f1', f1"〉 = ( f3 ° f2') ° f1', by (K1a 1) and (K1a) = f3 ° ( f2' ° f1'), by the induction hypothesis = f3 ° (K1aA f2' ° 〈 f1', f1"〉), by (K1a). (3.2) f2 is K2aA f2'. Then we proceed as for (3.1), using (K2a 1) and (K2a). (3.3) f2 is 〈 f2', f2"〉. Then we have the following subcases. (3.3.1) f3 is K1aA f3'. Then we have (K1aA f3' ° 〈 f2', f2"〉) ° f1 = ( f3' ° f2') ° f1, by (K1a) = f3' ° ( f2' ° f1), by the induction hypothesis = K1aA f3' ° 〈 f2' ° f1, f2" ° f1〉, by (K1a) = K1aA f3' ° (〈 f2', f2"〉 ° f1), by (distr). (3.3.2) f3 is K2aA f3'. Then we proceed as for (3.3.1), using (K1a) and (distr). (3.3.3) f3 is 〈 f3', f3"〉. Then we have (〈 f3', f3"〉 ° f2) ° f1 = 〈( f3' ° f2) ° f1, ( f3" ° f2) ° f1〉, by (distr) = 〈 f3' ° ( f2 ° f1), f3" ° ( f2 ° f1)〉, by the induction hypothesis = 〈 f3', f3"〉 ° ( f2 ° f1), by (distr). q.e.d. As before, once we have eliminated (cat 2), the Church-Rosser property of our reductions implies that if two arrow terms of A* are equal, then they have the same normal form. Since the converse holds too, this yields a procedure for deciding equality of arrow terms in A* by reducing them to normal form. For example, K1aA1A and K2aA1A, which are both in normal form, must be different in A*.

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We can also extend the foregoing to obtain a decision procedure for the commuting problem in the graph A* of a free ×-category generated by an arbitrary graph. We have first to redefine normal form as having all cuts molecular, instead of being cut-free, besides lacking subterms of the special forms we had before. Right-normal form has all cuts right-molecular. Our collection of reductions now applies to topmost nonmolecular cuts and is enlarged by (cat 2 mol) reductions. To demonstrate strong normalization, the complexity measure of an arrow term f will be 〈n1, n2, n3, n4〉 where n1, n2 and n3 are as above and n4 is obtained as in the parenthetical remark after the introduction of (cat 2 mol) reductions in § 4.6.4. With every right parenthesis in a molecular subterm h of f we associate the number of cuts in h on the right of this parenthesis. Then n4 is the sum of all these numbers; n4 is zero for an arrow term in right-normal form. The quartets 〈n1, n2, n3, n4〉 are ordered lexicographically. The ChurchRosser property is established as before. As in § 4.6.4, it remains to establish, by proving Associativity Molecularization, that every arrow of A* is designated by a unique arrow term in right-normal form. This is done by enlarging the proof of Associativity Elimination in this section. (We only have to check that several additional cases involving molecular arrow terms are excluded because of incompatibilities in the targets and sources in composition.) § 6.5. ALTERNATIVE FORMULATIONS OF RECTANGULAR || CATEGORIES WITH BINARY PRODUCT In § 6.1 we presented the rectangular || notion of ×-category in the (ac) formulation. We shall now consider the remaining formulations of this rectangular || notion. We could have the following as primitives instead of the operations on arrows K1aA, K2aA and Wc: (cc) the operation on arrows Wc, as before, and the partial operations on arrows

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f:C→ B ×A

f:C→ A ×B

K 1 cB f : C → B

K 2 cB f : C → B

(aa) the operations on arrows K1aA and K2aA, as before, and the partial operations on arrows f:C×C → B WaC f : C → B

(ca) the operations on arrows K1cB, K2cB and WaC. Besides these operations we always have the functor × from the category 〈A × A, 1, ° 〉 to the category 〈A, 1, ° 〉. Note first that in the (cc) formulation, with the arrow term (K2cB1A×B ×K1cA1A×B) ° Wc1A×B: A × B → B × A we encounter the same problem with Total Cut Elimination as in our (ac) formulation. To eschew this problem, we can replace Wc by pairing, as in the triangular formulation of § 6.2, which together with K1cB and K2cB yields something analogous to natural-deduction rules for conjunction. In the (aa) formulation pairing can be defined by the equality 〈 f, g〉 = WaC( f × g), but now there is no cut hidden behind pairing, as we had it with the definitional equality (Γ〈 , 〉) of § 6.2. In the (aa) formulation we have the equalities of (i) and (ii) of § 6.1, together with (K1a 1), (K1a 2), (K2a 1), (K2a 2) and (Wa 1) (Wa 2)

WaC(g ° f ) = g ° WaC f, WaC(g ° ( f × f )) = WaBg ° f

(these two equalities say that Wa is a natural antecedental transformation from IA to GF),

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(K1aWaF)

WaAK1aAg = g,

(K2aWaF) (KaWaG)

WaAK2aAg = g, WaA1×A2(K1aA2 f1 ×K2aA1 f2) = f1 × f2

(these three equalities are tied to the rectangular equalities). It is quite easy to show that the (ac) and (aa) notions of ×-category are equivalent (see the definitions of Wa in terms of Wc and vice versa in the next section). In the (aa) formulation, the operations K1aA and K2aA are analogous to Gentzen’s structural rule of thinning, and WaC is analogous to the structural rule of contraction, provided we take × as replacing a comma on the left-hand side of sequents. (This explains our connecting the (aa) formulation of the notion of adjunction in § 4.5.5 with Gentzen’s structural rules.) The (aa) formulation of ×-categories is involved in the definition of cartesian category of [D. 1996, section 2.3]. In this formulation we can prove Total Cut Elimination for free ×-categories generated by arrowless graphs. (The proof of cut elimination in [D. 1996, section 2.3] can serve for that, provided we correct the mistake noted in the errata of this paper; the first two sentences of the proof should read: “First we define a complexity measure, the degree, of an arrow-term gf, which we write d(gf ). This is (n, m) where n is the number of occurrences of W in g and m the number of occurrences of ×, K1, K2, W and K in gf.”) We shall not go into the details of this proof because we give an analogous proof in the next section for Total Cut Elimination in the (ca) formulation. Let us note, however, that the equalities (K1aWaF), (K2aWaF) and (KaWaG) are not necessary for this proof. Let us also note that using this Total Cut Elimination for solving the theoremhood problem runs into difficulties with Wa. These difficulties are like those we have with contraction when, after having proved cut elimination in logic, we want to solve the theoremhood decision problem. In classical and intuitionistic logic, we can overcome these difficulties in the propositional case by proving a lemma that limits the number of applications of contraction (cf. [D. 1987]), but in the predicate calculus we cannot do that, and encounter undecidability.

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§ 6.6. CUT ELIMINATION IN FREE RECTANGULAR || CATEGORIES WITH BINARY PRODUCT We shall now introduce the rectangular || notion of ×-category in the (ca) formulation and prove Total Cut Elimination for it. In the (ca) formulation of ×-categories we have (i)

a category 〈A, 1, ° 〉,

(ii)

a functor × from 〈A × A, 1, ° 〉 to 〈A, 1, ° 〉,

(iii) the partial operations K1cB, K2cB and WaC on arrows of A, which satisfy the following equalities, tied to naturalness: (K1c 1) (K1c 2)

K1cB( f ° g ) = K1cB f ° g, K1cB1(( f1 × f2) ° g) = f1 ° K1cC1g ,

(K2c 1) (K2c 2)

K2cB( f ° g ) = K2cB f ° g, K2cB2(( f1 × f2) ° g) = f2 ° K2cC2g ,

(Wa 1) (Wa 2)

WaC(g ° f ) = g ° WaC f, WaC(g ° ( f × f )) = WaBg ° f,

(iv) as well as the following equalities tied to the rectangular equalities, where f : C → A, g : C → B and h : C → A × B, (K1cWaF) (K2cWaF)

WaCK1cA( f × g ) = f, WaCK2cB( f × g ) = g,

(KcWaG)

WaC(K1cAh ×K2cBh) = h.

Since we have WaCK1cAk = WaC(K1cA(1A×B ° k) ° 1C×C), by (cat 1 right) and (cat 1 left) = (K1cA1A×B ° k) ° WaC1C×C, by (Wa 1) and (K1c 1) = K1cAWaCk, by (cat 2), (Wa 1), (K1c 1), (cat 1 right) and (cat 1 left), we could formulate (K1cWaF) with WaCK1cA replaced by K1cAWaC, and analogously for (K2cWaF).

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The (ca) formulation of ×-categories is equivalent to the (ac) formulation of § 6.1 via the definitions K1aB f =def f ° K1cA1A×B, K1cA f =def K1aB1A ° f, K2aA f =def f ° K2cB1A×B, K2cB f =def K2aA1B ° f, Wc f =def WaC1C×C ° f, WaC f =def f ° Wc 1C. Let now A* be the graph of a free ×-category in the (ca) formulation generated by an arrowless graph. We shall demonstrate Total Cut Elimination for A*. Proof of Total Cut Elimination for A*. We need to redefine the notion of degree of a cut for this proof. Before, we defined the degree for every arrow term, but in fact we were interested only in the degree of arrow terms that are subterms of cuts, i.e. of arrow terms of the form g ° f. In proofs of Cut Disintegration like those of §§ 1.8.1, 4.5.1, and 5.7.1, and in the proof of Total Cut Elimination in § 6.3, we made an induction only on the degrees of these special arrow terms. However, the definition of degree could be given uniformly for arbitrary arrow terms, and we didn’t refrain from giving it in full generality. Even modified definitions of degree, like the one in § 4.5.5, where degrees are pairs, were given for arbitrary arrow terms. (A definition of degree tied to cuts only, like the one we are going to give now, is referred to in the last paragraph of § 6.5.) Now we shall have a definition of degree only for arrow terms of the form g ° f. Moreover, since we shall follow Gentzen’s procedure of eliminating topmost cuts, we define the degree of g ° f only for f and g cut-free. The degree of g ° f is 〈n1, n2, n3〉 where n1 is the number of occurrences of Wa in g, while n2 is the number of occurrences of K1c and K2c in f, and n3 is the number of occurrences of ×, K1c, K2c and Wa in g ° f. These degrees are lexicographically ordered (see § 6.4).

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Once we have this definition of degree, we proceed as follows. Take in an arrow term of A* a topmost cut whose subterm is g ° f. We show that g ° f is either equal to a cut-free arrow term, or it is equal to an arrow term all of whose cuts are topmost and of strictly smaller degree than the degree of g ° f. The possibility of eliminating every topmost cut, and hence every cut, follows then by induction on degree. We have the following cases. (1) f is 1A. Then g ° 1A = g by (cat 1 right), and g is cut-free. (2) f is f '× f ". Then we have the following subcases. (2.1) g is 1A. Then 1A ° f = f by (cat 1 left), and f is cut-free. (2.2) g is g '× g ". Then (g '× g ") ° ( f '× f ") = (g' ° f ')×(g" ° f ") by (fun 2 ×), and the degrees of g' ° f ' and g" ° f " are strictly smaller than the degree of g ° f . The cuts of g' ° f ' and g" ° f " are topmost. (2.3) g is K1cBg'. Then K1cBg' ° f = K1cB(g' ° f ) by (K1c 1), and the degree of g' ° f is strictly smaller than the degree of g ° f. The cut of g' ° f is topmost. (2.4) g is K2cBg'. This case is treated analogously to (2.3), using (K2c 1). (2.5) g is WaBg'. Then WaBg' ° f = WaC(g' ° ( f × f )) by (Wa 2), and the degree of g' ° ( f × f ) is strictly smaller than the degree of g ° f. The cut of g' ° ( f × f ) is topmost. (Here n1 has decreased, while n2 and n3 might have increased.) (3) f is K1cC f '. Then g ° K1cC f ' = K1cB ((g × 1D) ° f ') by (K1c 2), and the degree of (g × 1D) ° f ' is strictly smaller than the degree of g ° f. The cut of (g × 1D) ° f ' is topmost. (Here n2 has decreased, while n1 is unchanged.) (4) f is K2cC f '. This case is treated analogously to (3), using (K2c 2).

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(5) f is WaC f '. Then g ° WaC f ' = WaC (g ° f ') by (Wa 1), and the degree of g ° f ' is strictly smaller than the degree of g ° f. The cut of g ° f ' is topmost. q.e.d. Note that the equalities (cat 2), (fun 1 ×) and the equalities of (iv), tied to the rectangular equalities, are not used in the proof we have just given. We have not tried to formulate Cut Disintegration and its corollary Particular Cut Elimination for A* because the equalities (fun 2 ×) and (Wa 2) would require that a single cut be linked to more than one cut. This leads to the same sort of considerations as those made at the end of § 6.3. The complications we have with the complex degree 〈n1, n2, n3〉 in the proof above are parallel to those Gentzen had with contraction in his proof of cut elimination. This is why he replaced his cut rule by a multiple-cut rule he called “mix” (Mischung in German), and eliminated mix rather than cut. Since we lack the subformula property with the operations K1cB, K2cB and WaC, the Total Cut Elimination we have for the (ca) formulation of ×-categories is not useful for solving the theoremhood decision problem. We could perhaps envisage using it for solving the commuting problem, but we have already solved this problem in § 6.4 for the triangular formulation of ×-categories. And this solution seems to be simpler than what we could expect in the (ca) formulation. Let us see where problems with this last formulation arise. For the normal form in the (ca) formulation we would have to envisage redexes drawn out of (fun 1 ×) and of the equalities of (iv), namely, (K1cWaF), (K2cWaF) and (KcWaG), as well as from WaCK1cAk = K1cAWaCk and WaCK2cBk = K2cBWaCk. If in (fun 1 ×), the arrow term 1A × 1B is a redex and 1A×B the corresponding contractum, and in (K1cWaF), the arrow term WaCK1cA( f × g ) is a redex and f the corresponding contractum, then WaAK1cA(1A × 1A) would reduce to WaAK1cA1A×A and 1A. To get confluence, it then seems better to take 1A×B as a redex and 1A × 1B as the corresponding contractum. But then the complexity could not be measured by degree, since the degree of

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the contractum is greater than the degree of the redex. This problem and others, like those involving the permuting of WaC with K1cA and K2cB, would complicate the demonstration of the Church-Rosser property in the (ca) formulation of ×-categories. These problems need not be unsurmountable, but since in § 6.4 we already solved the commuting problem for ×-categories in another formulation, we need not tackle this matter any more. (Of course, solving the commuting problem in one formulation solves it in all equivalent formulations.) § 6.7. THE TERMINAL OBJECT An object is terminal in a graph iff from every object to this object there is exactly one arrow. To obtain cartesian categories out of categories with binary product we need to add a terminal object to them. Let us first show how having a terminal object is involved in a particular kind of adjunction. Let 〈T, 1, ° 〉 be the trivial category with a single object T and a single arrow 1T: T → T. Consider now a category 〈B, 1, ° 〉 such that 〈T, 1, ° 〉 is a subcategory of it and there is an adjunction 〈T, B, F, G, ϕa, γc〉 where F is the constant functor from 〈B, 1, ° 〉 to 〈T, 1, ° 〉, i.e., we have FB = T,

Fg = 1T,

G is the inclusion functor from 〈T, 1, ° 〉 to 〈B, 1, ° 〉, i.e., we have GT = T,

G1T = 1T,

ϕa is the identity antecedental transformation from FG to IT, i.e., we have ϕa 1 T = 1 T, and γc is a consequential transformation from IB to GF, i.e., we have g: B → A γcA g : B → T

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It is trivial that F and G are functors and that ϕa is a natural antecedental transformation. That γc is a natural consequential transformation means that the following equalities hold in B: (γc 1)

γcA(g2 ° g1) = γcAg2 ° g1,

(γc 2)

γcA(g2 ° g1) = γcBg1,

for g2: B → A. The rectangular equality (ϕaγcF) is trivially satisfied, while the other rectangular equality reduces to (ϕaγcG)

γcTg = g,

for g: B → T. The natural transformation γ from IB to GF, equivalent to γc, is defined by γA =def γcA1A , while γc is defined in terms of γ by γcAg =def γA ° g (see § 3.1). If γ is primitive instead of γc, then the equalities (γc 1), (γc 2) and (ϕaγcG) are taken care of by the single equality for g : A → T,

g = γA,

which says that T is a terminal object in B. This adjunction between T and B is a reflection of the category B in its subcategory T. The category T may be conceived as the “empty product” category: the adjunction between T and B is parallel to the adjunction between B × B and B from § 6.1. We shall not dwell on cut-elimination problems in free categories with terminal object generated by graphs, since these problems are rather trivial. We shall immediately pass to cartesian categories, and consider cut elimination there.

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§ 6.8. CUT ELIMINATION IN FREE CARTESIAN CATEGORIES Cartesian categories are ×-categories, i.e. categories with binary product, that have a terminal object. To prove Total Cut Elimination for free categories of this kind we formulate the assumptions involving binary product in the triangular manner of § 6.2. So we have as primitives K1aA, K2aA and pairing. Besides that, in a cartesian category A we have a special object T and a family of arrows KA: A → T, indexed by the objects A of A, which satisfy (K)

for f : A → T,

f = KA.

(We write K for the γ of the previous section because of the connection between this natural transformation and the combinator K. Together with arrows of types B ×T→ B and T× B→ B, the arrows KA can define first and second projections. Note, however, that the arrows KA bear also some analogy to W. Both are tied to units of adjunctions, and A → T is a kind of null case of A → A × … × A.) By analogy to what we had before, it is straightforward to define free cartesian categories generated by a graph. Note that in the generating graph there is no object T. This object is a nullary operation on objects and, as an operation, it is trivially one-one (see § 5.5). So T is a nullary connective. For the graph A* of a free cartesian category generated by an arrowless graph we can demonstrate Total Cut Elimination by extending the proof of § 6.3. We take the definition of degree exactly as in that proof: so we do not count occurrences of KA in an arrow term. Proof of Total Cut Elimination for A*. We have the following additional cases for g ° f.

(4.5) f is 〈 f ', f "〉 and g is KB. Then g ° f = KA by (K), and KA is cutfree. (5) f is KA. Then we have the following subcases.

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(5.1) g is 1T. Then 1T ° KA = KA by (cat 1 left) or (K), and KA is cutfree. (5.2) - (5.3) g is K1aAg' or K2aAg'. This is excluded because the source of g is T. (5.4) g is 〈g', g"〉. Then we proceed as in case (4.4) of the proof in § 6.3. (5.5) g is KT. Then KT ° KA = KA by (K), and KA is cut-free. q.e.d. Combining the Total Cut Elimination just proved with the subformula property, we can obtain as before a decision procedure for the theoremhood problem in A*. However, we shall encounter difficulties with the commuting problem. If we define normal form by requiring, in addition to what we had in § 6.4, that all subterms of type A → T must be KA, it is not enough to introduce reductions from redexes f : A → T to contracta KA. To obtain the Church-Rosser property we would have to envisage also reductions to cover the equalities between, for example, K1aB〈 f ' ° KA, f " ° KA〉 and 〈 f ' ° KA×B, f " ° KA×B〉, K1aB〈KA, f 〉 and 〈KA×B, K1aB f 〉, 〈K1aT1A, KA×T〉 and 1A×T, 〈KT×T, KT×T〉 and 1T×T, K1aBKA and KA×B. We shall not go much into these complications here, because the commuting problem is not our main concern: we are not particularly concerned by it if its solution does not flow mainly out of cut elimination. And the complications we have above are extraneous to cut elimination: all the candidates for reduction displayed above, except the first, are cut-free. Of course, this does not mean that a solution to the commuting problem for free cartesian categories cannot be found in the formulation we have chosen above, or in another formulation. In the next section, we shall indeed find such a solution, as well as an appropriate notion of cut-free normal form.

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The proof of Total Cut Elimination of this section can easily be extended to yield a proof of Total Cut Molecularization, and its right and left forms, for free cartesian categories generated by arbitrary graphs. § 6.9. MODEL-THEORETICAL NORMALIZATION IN CARTESIAN CATEGORIES A unique cut-free normal form for arrows of a free cartesian category A* generated by an arrowless graph, formulated as in the preceding section, may be easily obtained from the normal form of [D. & Petric 1997a], which is given for another, more standard, not “cut-free”, formulation of cartesian categories. This normal form for A*, analogous to the atomic normal form of § 4.10.1, corresponds to a natural-deduction proof in the thin normal form—with eliminations preceding introductions—where the middle part is atomic (cf. §§ 0.3.5-6). Let a Kia arrow term be an arrow term Q1…Qn1E of A*, where n ≥ 0, every Qi is either of the form K1aA, or of the form K2aA, and E is a generative object term. Arrow terms of A* in normal form are then defined inductively as follows: every Kia arrow term and every arrow term KA is in normal form; if f : C → A and g: C → B are in normal form, then 〈 f, g〉 is in normal form. Note that this normal form requires that we interchange the redexes and contracta of the last three reductions in the list of § 6.4. The uniqueness of this normal form may be established as in § 4.10.2, without invoking the Church-Rosser property, with the help of links of arrow terms of A*, which we shall now briefly consider. The links of [D. & Petric 1997a] are related to a faithful functor Λ from A* to the concrete category Finordop, whose objects are finite ordinals and whose arrows are arbitrary functions from finite ordinals to finite ordinals, with domains being targets and codomains sources. (If the free cartesian category A* is generated by an arrowless graph with a single object, then Finordop is equivalent, but not isomorphic, to A*: it is the skeleton of A*, which means that isomorphic objects of

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A* have become equal in Finordop.) The functor Λ assigns to an object A of A* the ordinal | A|, which is the number of occurrences of

generative object terms in A. Next, Λ assigns to 1A the identity function and to KA the empty function; ° corresponds to composition of functions. If f is of the form K1aAg, then Λ f (i) = Λg(i), and if f is of the form K2aAg : A × B → C, then Λ f (i) = Λg(i) + | A| - 1. Finally, if f is

of the form 〈g, h〉: C → A × B, then for i < | A| we have Λ f (i) = Λg(i) and for i ≥ | A| we have Λ f (i) = Λh(i). The product of Finordop that corresponds by Λ to the product of A* is simply addition. The terminal object of Finordop is zero, i.e. the empty set. It can be proved that for two arrow terms f and g of A* of the same type, f = g in A* iff Λ f = Λg in Finordop. (This amounts to a coherence result envisaged by [Kelly 1972, section 4.1], and proved in [Mints 1980, Theorem 2.2], [Troelstra & Schwichtenberg 1996, Theorem 8.2.3, p. 207] and [Petric 1997].) The proof of this equivalence in [D. & Petric 1997a] proceeds via a lemma analogous to the Lemma of § 4.10.1. This coherence equivalence yields a simple decision procedure for commuting in A*. With the help of this coherence, a maximality result for cartesian categories, which we mentioned in § 6.4, is established in [D. & Petric 1997a]. The proof of this maximality result is pretty analogous to the proof of the Maximality of Comonad in § 5.10, the role of the idempotency equality (idem') being taken by the preordering equality K1aA1A = K2aA1A. Although it is not clear how to use the Maximality of Adjunction of § 4.11 in order to deduce the maximality of cartesian categories, it is plausible that the former maximality explains the latter. Namely, if we know about Maximality of Adjunction, and know that everything in cartesian categories is defined through adjunctions, the maximality of these categories will not appear surprising. The normal form, links, coherence and maximality of cartesian categories considered in this section can easily be adapted to ×-categories. We just have to omit whatever involves the terminal

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object and the arrows KA. Instead of Finordop we then use the analogous category of nonempty finite ordinals. The maximality of cartesian categories and ×-categories is interesting for logic. It tells us that the choice of equalities between deductions in conjunctive logic is optimal. These equalities are wanted, because they are induced by normalization of deductions, and no equality is missing, because any further equality would equate all deductions that share premises and conclusions.

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CONCLUSION

Cut elimination in categories is better adapted to a purely categorial context than to a context that approaches logic. With free adjunctions in the (ac) formulation of the rectangular || notion, and with free comonads formulated analogously, we obtained an exact match between the defining equalities of these notions and equalities necessary and sufficient for cut elimination in the form of Cut Disintegration. In the case of special adjunctions motivated by logic, of which we have considered what is presumably the most basic one—the adjunction of categories with binary product—matters are not so straightforward. In the (ac) formulation of the rectangular || notion for this adjunction, cut elimination fails, while in other formulations it obtains, but at the cost of having equalities unnecessary for cut elimination. Moreover, if we add the terminal object, to get cartesian categories, normalization through cut elimination is not as efficient as it was with free adjunctions and free comonads. We could blame for this situation the fact that we have in that case not a single adjunction, but two adjunctions mixed together. Indeed, problems would get even worse with cartesian closed categories, which correspond to the implicationconjunction fragment of intuitionistic logic, or with bicartesian closed categories, which correspond to the whole of intuitionistic propositional logic (see [Lambek & Scott 1986, Part I]). In these categories we would have more additional adjunctions mixed together. For normalization in these categories, one usually relies on their equivalence with various categories of typed lambda calculuses— normalization is then achieved in these calculuses. If, however, we wished to have a direct normalization in cartesian closed and bicartesian closed categories through cut elimination, we should

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presumably look for inspiration again in Gentzen, who dealt with sequents X ≤ A where X is a collection of formulae, and not only a singleton. This standard kind of sequent system has the virtue of disentangling connectives mixed together. The corresponding categorial framework should then be found in Lambek’s multicategories (see [Lambek 1969, 1989, 1993]). There, we should be able to obtain cut-elimination results not only for multicategories corresponding to intuitionistic logic, but also for multicategories corresponding to other nonclassical logics—in particular, substructural logics. One could expect in that context, too, a tie, or even an exact match, between cut elimination and an appropriate notion of adjointness. Speaking of logics weaker than intuitionistic, we should point out that cut elimination is tied not only to connectives characterized by adjunctions, but also to connectives characterized by weaker conditions. Such a connective is, for example, the noncommutative and nonassociative product of the weakest substructural logic (called nonassociative Lambek’s calculus), which is tied simply to a functor, i.e. bifunctor. As we have shown in this work, cut elimination is also guaranteed by functoriality, and other conditions not yet amounting to adjointness. However, these matters pertaining to logic are better tackled after the more elementary matters treated in the present work have been settled. Our purpose was not to reformulate the usual, or less usual, logical cut-elimination results in terms of categories. Rather, we wished to show that purely categorial notions may be understood in terms of cut elimination.

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___ [1969] Adjointness in foundations, Dialectica vol. 23, pp. 281-296. ___ [1969a] Ordinal sums and equational doctrines, in: B. Eckmann ed., Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics no. 80, Springer, Berlin, pp. 141-155. Mac Lane, S. [1971] Categories for the Working Mathematician, Springer, Berlin. Manes, E.G. [1976] Algebraic Theories, Springer, Berlin. Mints, G.E. [1980] Category theory and proof theory (in Russian), Aktual'nye voprosy logiki i metodologii nauki, Naukova Dumka, Kiev, pp. 252-278 (English translation, with permuted title, in: G.E. Mints, Selected Papers in Proof Theory, Bibliopolis, Naples, 1992). Murasugi, K. [1996] Knot Theory and its Applications, Birkhäuser, Boston. Petrić, Z. [1997] Coherence in substructural categories, Studia Logica vol. 70 (2002), pp. 271-296. Prawitz, D. [1965] Natural Deduction: A Proof-Theoretical Study, Almqvist & Wiksell, Stockholm. ___ [1971] Ideas and results in proof theory, in: J.E. Fenstad ed., Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, pp. 235-307. Pumplün, D. [1970] Eine Bemerkung über Monaden und adjungierte Funktoren, Mathematische Annalen vol. 185, pp. 329-337. Riguet, J. [1948] Relations binaires, fermetures, correspondances de Galois, Bulletin de la Société Mathématique de France vol. 76, pp. 114-155. Schanuel, S., and Street, R. [1986] The free adjunction, Cahiers de Topologie et Géométrie Différentielle Catégoriques vol. 27, pp. 81-83. Smullyan, R.M. [1968] Analytic cut, The Journal of Symbolic Logic vol. 33, pp. 560-564. Szabo, M.E. [1978] Algebra of Proofs, North-Holland, Amsterdam. Troelstra, A.S., and Schwichtenberg, H. [1996] Basic Proof Theory, Cambridge University Press, Cambridge.

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INDEX Entries are numerated by section numbers. (aa) 0.3.5, 4.5.5, 6.5 (ac) 0.3.5, 4.5.5, 6.1 (ac) formulation 4.2 acyclic path 4.6.2 Adj 5.2.3 adjoint functor 4.1.2 adjoint semifunctor 4.1.4 adjunction 0.3.2, 4.1, 4.2, 4.7.1, 4.8.1, 5.2.1 admissible rule 0.3.1 analytic cut 0.3.4 antecedental transformation 3.1 arrow 1.3 arrow term 1.6, 2.1, 3.3, 4.3, 5.4 arrowless graph 1.3 Associativity Elimination 4.6.3, 4.10.2, 5.8.3, 5.9, 6.4 Associativity Molecularization 4.6.4, 5.8.4 atomic cut 1.8.1 atomic normal form 4.10.1, 4.11, 6.9 atomized cut 1.8.1 Auderset, C. 5.2, 5.11 Barendregt, H. 0.1, 4.6.3 bijection 2.4.2 binary product 6.1 (ca) 0.3.5, 4.5.5, 6.5, 6.6 cartesian category 6.8 (cat 1 left) 1.4 (cat 1 left O∆ ) 5.1.5 (cat 1 right) 1.4 (cat 1 right O∆ ) 5.1.5 (cat 2) 1.4 (cat 2 mol) 4.5.4 (cat 2 mol) reduction 4.6.4

269

(cat 2 O∆ ) 5.1.5 categorial equivalence relation 1.7 categorial proof theory 0.1 categorical (see categorial) category 1.4 category of adjunctions 5.2.3 category of comonads 5.2.3 category of resolutions 5.2.4 category with binary product 6.1, 6.5, 6.6 Cayley Representation of Monoids 1.9.7 (cc) 0.3.5, 4.5.5, 6.5 Church-Rosser property 4.6.3, 5.8.3 coherence 4.10.1, 5.9, 6.9 Com 5.2.3 commuting problem 0.2, 4.6.1 comonad 5.1.1, 5.1.4, 5.1.5, 5.3, 5.7.3 comonad of an adjunction 5.2.1 comonadic equivalence relation 5.6 comonofunctor 5.1.1, 5.3 comonograph 5.1.1, 5.3 comonograph isomorphism 5.3 comparison functor 5.2.4 composite graph-morphism 1.3 composition 1.4 concrete category 1.9, 4.10, 4.10.1, 5.9, 6.9 (congr F) 2.3.1 (congr τc) 3.5 (congr ° ) 1.8.1 conjunction 6.1 connective 5.5 consequential transformation 3.1 constant arrow term 4.5.3 constant cut 4.5.3 Constant-Cut Elimination 4.5.3 contraction 6.5 Coquand, T. 4.10 coreflection 5.1.4 counit of an adjunction 0.3.2, 4.1.2

270

Cubric, D. 4.10 cut 0.1, 1.8.1 Cut Disintegration 0.2, 1.8.1, 2.3.1, 3.5.1, 4.5.1, 4.7.2, 4.8.2, 5.7.1 deductive completeness 0.3.3, 5.2.2 deductive embedding 1.9.1 deductive isomorphism 1.9.1 deductive system 1.4 definitional equalities of adjunc-tion 4.1.1, 5.1.3, 6.2 degree 1.8.1, 2.3.1, 3.5.1, 4.5.1, 4.7.2, 4.8.2, 5.7.1, 6.3, 6.6 degree of a cut 1.8.1, 6.6 degree of an arrow term 1.8.1 delta category 5.1.2 delta composition 5.1.5 derivable rule 0.3.1 diagonal functor 6.1 δ-injective comonad 5.1.1 discrete category 1.8.3 double morphism 1.5 δ-preordered comonad 5.1.1 Dybjer, P. 4.10 Eilenberg, S. 4.10.1, 5.11 Eilenberg-Moore category 5.1.7 eliminated cut 1.8.1 embedding 1.3 embedding of deductive systems 1.4 empty graph 1.3, 4.5.1 equivalent categories 1.5 equivalent notions 1.3, 1.5 faithful 1.3 faithful modelling of categories 4.10 family of connectives 5.5 finite ordinals 5.9, 6.9 Finordop 6.9 forgetful functor 0.3.2

271

formation 3.2 formation isomorphism 3.2 formators 3.2 formulation 1.5 free adjunction 4.4 free adjunction generated by arrowless graphs 4.5.1, 4.6.3, 4.10.1, 4.10.2 free category 1.7 free coalgebras 5.1.7 free comonad 5.6 free comonad generated by an arrowless graph 5.8.3, 5.9 free comonograph 5.4 free deductive system 1.6 free formation 3.3 free function 2.1 free functor between categories 2.2 free graph-morphism between graphs 2.1 free graph-morphism between deductive systems 2.1 free junction 4.3 free natural formation 3.4 free natural transformation 3.2, 3.4 Freyd, P.J. 1.9 full 1.3 full subgraph 1.3 full subsytem 1.4 (fun 1) 1.4 (fun 1 ∞) 6.1 (fun 2) 1.4 (fun 2 ∞) 6.1 function 2.4.1 function pair 1.2 functional completeness 0.3.3, 5.2.2 functor 1.4 functorial equivalence relation 2.2 Galois connection 2.4.4 generative arrow 1.7, 2.2, 3.4, 4.4, 5.6 generative arrow term 1.6, 2.1, 3.3, 4.3, 5.4

272

generative comonofunctor 5.6 generative formator 3.3 generative function pair 2.1 generative functor 1.7 generative graph-morphism 1.6, 1.7, 5.4, 5.6 generative graph-morphism pair 2.1, 2.2, 3.3, 3.4, 4.3, 4.4 generative junctor 4.4 generative morpho-functor 2.2 generative object term 2.1, 3.3, 4.3, 5.4 Gentzen, G. 0.1, 0.2, 0.3.4, 0.3.5, 1.8, 1.8.3, 4.5, 4.5.1, 4.5.4, 4.5.5, 4.6.1, 4.8.3, 6.1, 6.2, 6.4, 6.5, 6.6, Conclusion Gentzen’s degree 4.5.1 graph 1.3 graph-morphism 1.3 graph-morphism pair 2.1 grounding graph-morphism 1.9.2 Hayashi, S. 4.1.4 hexagonal 4.1.1 hexagonal adjunction 4.1.2 hexagonal comonad 5.1.4 hexagonal figure of primitives 4.1.1, 5.1.3 hom-set 1.3 (homo) 5.1.7 (homo δ) 5.1.6 Hoofman, R. 2.3.2, 4.1.4 (idem) 5.8.2 (idem') 5.10 idempotency equality 5.8.2 idempotent comonad 5.8.2 identity 1.4 identity arrows 1.4 identity functor 1.4 identity graph-morphism 1.3 IG 1.3 inclusion graph-morphism 1.3 initial object 5.2.4

273

inverse of an arrow 1.5 isomorphic objects 1.5 isomorphism 1.3, 1.5 isomorphism of deductive sys-tems 1.4 junction 4.2, 4.7.1, 4.8.1, 5.2.1 junction isomorphism 4.2 junctor 4.2, 4.7.1, 4.8.1, 5.2.1 Kassel, C. 4.10.1 Kauffman, L.H. 4.10.1 Kelly, G.M. 0.1, 4.10.1, 5.11, 6.9 Kia arrow term 6.9 Kleene, S.C. 0.1 Kleisli, H. 5.1.6 Kleisli category 5.1.6 Kleisli deductive system 5.1.6 Krstic, S. 5.10 Lagado 1.6 Lambek, J. 0.1, 0.3.2, 0.3.3, 4.6.2, 5.2.2, 5.7.3., 5.8.2, 5.9, Conclusion Lawvere, F.W. 0.3.3, 1.1, 4.1.4, 5.9 left adjoint 4.1.2 left compositional lifting 31 left cone 1.9.1 left-cone graph 1.9.1 left-invariable function 1.9.1 left-molecular arrow term 4.5.4 left-normal form 4.6.4, 5.8.4 (left-total) 2.4.1, 2.4.2 (left-unique) 2.4.1, 2.4.2 length of an arrow term 4.11 length of an equality 4.11 lexicographical order 4.5.5, 6.4 lifting graph morphism 1.9.2 link of cuts 1.8.1 linked cuts 1.8.1, 2.3.1, 3.5.1, 4.5.1, 5.7.1 links of arrow terms 4.10.1, 5.9, 6.9

274

Lins, S.L. 4.10.1 Lipkovski, A. 2.4.4 Mac Lane, S. 0.1, 0.3.2, 1.1, 1.6, 2.4.4, 3.1, 4.1, 4.1.4, 4.1.6, 4.2, 4.10.1, 5.2.3, 5.2.4 Manes, E.G. 5.1.5 Martin-Löf, P. 4.10 Mints, G.E. 6.9 mix 6.6 modelling of categories 4.10 model-theoretical methods of normalization 1.9, 4.10, 4.10.1, 4.10.2, 5.9, 6.9 molecular arrow term 4.5.4 molecular cut 4.5.4 molecularized cut 4.5.4 monad 5.1.1 monograph 5.1.1 monoid 1.9.7 morphism 1.2 morphism between functions 1.2 morpho-functor 2.1 morpho-morphism 2.1 multicategory Conclusion Murasugi, K. 4.10.1 (nat) 1.4 natural antecedental transformation 3.1 natural consequential transformation 3.1 natural deduction 0.3.4, 4.5.5 natural equivalence relation 3.4 natural formation 3.2 natural isomorphism 1.5 natural transformation 1.4, 3.1 naturalness 1.2 nonassociative Lambek’s calculus Conclusion normal form 4.6.3, 4.6.4, 4.7.3, 4.8.3, 4.10.1, 4.11, 5.8.3, 5.8.4, 6.4, 6.9 normal form theorem 4.6.3, 5.8.3

275

normalization 0.3.5 object 1.3 object term 1.6, 2.1, 3.3, 4.3, 5.4 objectual transformation 1.3 one-one function 2.4.1 one-one operation 5.5 one-to-one correspondence 2.4.2 onto function 2.4.1 op 1.9.1, 1.9.5 pair of adjunctional equivalence relations 4.4 pairing 6.2 Particular Cut Elimination 1.8.3 Particular Cut Molecularization 4.5.4 path 4.6.2 Petric, Z. 4.10.1, 6.4, 6.9 Prawitz, D. 0.1 preorder 1.9.7, 4.6.1 preordering equality 4.6.2 product 6.1 product of two graphs 6.1 projections 6.1 Pumplün, D. 5.2.4 rank 4.5.1 rectangular || 4.1.1, 4.4, 5.1.4 rectangular || 4.1.1 rectangular || adjunction 4.1.3, 4.4, 4.5.5 rectangular \\ 4.1.1 rectangular \\ adjunction 4.1.4, 4.7.1 rectangular // 4.1.1 rectangular // adjunction 4.1.5 rectangular equalities 4.2, 4.5.5 reduce 4.6.3 reduction relation 4.6.3 reductions 4.6.3, 5.8.3, 6.4 reflection 5.2.2

276

relation 2.4.1 resolution method 4.6.4 resolutions 5.2.4 right adjoint 4.1.2 right cone 1.9.1 right-cone graph 1.9.1 right-molecular arrow term 4.5.4 right-normal form 4.6.4, 5.8.4 (right-total) 2.4.1, 2.4.2 (right-unique) 2.4.1, 2.4.2 Riguet, J. 2.4 Scedrov, A. 1.9 Schanuel, S. 5.11 Schwichtenberg, H. 0.1, 6.9 Scott, P.J. 0.1, 0.3.2, 0.3.3, 4.6.2, 5.2.2, Conclusion seesaw adjunction 4.1.7 seesaw functions 4.1.1, 4.7.1 semifunctor 2.3.2, 4.1.4 (shift) 5.1.5 (shift ε) 5.1.5 (shift 1) 5.1.5 single morphism 1.5 skeletal deductive system 1.9.7 skeleton 6.9 small category 1.1 Smullyan, R.M. 0.3.4 (solid) 1.9.4 solidifiable 1.9.4 source 1.3 Stone Representation of Preorders 1.9.7 Street, R. 5.11 strong normalization 4.6.3, 5.8.3 structural rule 4.5.5, 6.5 structural sequent 0.3.3 subcategory 1.4 subformula property 4.5.5, 4.6.1 subgraph 1.3

277

subsystem 1.4 subterm of the cut 1.8.1 Swift, J. 1.6 Szabo, M.E. 0.1 T 6.7 tangle 4.10.1 target 1.3 terminal object 5.2.4, 6.7 theoremhood problem 0.2, 4.6.1 thick normal form 0.3.5, 4.5.5 thin normal form 0.3.5, 4.5.5 thinning 6.5 topmost cut 1.8.3 topmost nonmolecular cut 4.5.4 Total Cut Elimination 1.8.3, 6.3, 6.6, 6.8 Total Cut Left-Molecularization 4.5.4 Total Cut Molecularization 4.5.4 Total Cut Right-Molecularization 4.5.4 transformation 1.3 treelike graph 4.6.2 triangular 4.1.1 triangular ► 4.1.1 triangular ◄ 4.1.1 triangular adjunction 4.1.6, 4.8.1 triangular comonad 5.1.5, 5.7.3 triple 5.1.1 trivial category 6.7 Troelstra, A.S. 0.1, 6.9 type of an arrow 1.3 type of an arrow term 1.6, 2.1, 3.3, 4.3, 5.4 uniqueness of normal form 4.6.3, 4.6.4, 4.10.2, 5.8.3, 5.8.4, 5.9, 6.4, 6.9 unit of an adjunction 0.3.2, 4.1.2 universal arrow problem 4.1.6

278

Yoneda, N. 1.9 Yoneda functor 1.9.8 Yoneda Lemma 1.9.8