# THEORY OF CATEGORIES

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## THEORY OF CATEGORIES

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SAMUEL EILENBERG. Automata, Languages, and Machines: Volumes A and B MORRIS HIRSCH N D STEPHEN A SMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELM MAGNUS. Noneuclidean Tesselations and Their Groups FRANCOIS TREVES. Linear Partial Differential Equations Basic WILLIAM BOOTHBY. Introduction to Differentiable Manifolds and Riemannian M. An Geometry ~ A Y T O N GRAY. Homotopy Theory : An Introduction to Algebraic Topology ROBERT ADAMS. A. Sobolev Spaces JOHN BENEDETTO. J. Spectral Synthesis D. V. WIDDER. Heat Equation The EZRA ~ C A LMathematical Cosmology and Extragalactic Astronomy S . IRVING J. DIEUDONN~....

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1. THEORY OF CATEGORIES
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3. Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Samuel Ellenberg and Hymen E a s s Columbia University, N e w York RECENT TITLES SAMUEL EILENBERG. Automata, Languages, and Machines: Volumes A and B MORRIS HIRSCH N D STEPHEN A SMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELM MAGNUS. Noneuclidean Tesselations and Their Groups FRANCOIS TREVES. Linear Partial Differential Equations Basic M. WILLIAM BOOTHBY. Introduction to Differentiable Manifolds and Riemannian An Geometry ~ A Y T O NGRAY. Homotopy Theory : An Introduction to Algebraic Topology ROBERT ADAMS. A. Sobolev Spaces J. JOHN BENEDETTO. Spectral Synthesis D. V. WIDDER. Heat Equation The IRVING EZRA ~ C A LMathematical Cosmology and Extragalactic Astronomy S . J. DIEUDONN~. Treatise on Analysis : Volume 11, enlarged and corrected printing; Volume IV ; Volume V ; Volume VI, in preparation WERNER Gmua, STEPHEN HALPERIN, RAYVANSTONE. AND Connections, Curvature, and Cohomology : Volume 111, Cohoniology of Principal Bundles and Homogeneous Spaces I. MARTIN ISAACS. Character Theory of Finite Groups JAMES BROWN. R. Ergodic Theory and Topological Dynamics C. TRUESDELL. First Course in Rational Continuum Mechanics: Volume 1, General A Concepts GEORGE GRATZER. General Lattice Theory K. D. STROYAN D W. A. J. LUXEMBURG. AN Introduction to the Theory of Infinitesimals B. M. PUTTASWAMAIAHH N D. DIXON.Modular Representations of Finite AND JO Groups MmwN BERGER. Nonlinearity and Functional Analysis : Lectures on Nonlinear Problems in Mathematical Analysis CHARALAMBOS D. ALIPRANTIS D OWEN AN BURKINSHAW. Locally Solid Riesz Spaces In preparation J A NMIKUSINSKI. Bochner Integral The MICHIEL HAZEWINKEL. Groups and Applications Formal THOMAS Set Theory JECH. SIGURDUR HELGASON.Differential Geometry, Lie Groups, and Symmetric Spaces CARL DEVITO. Functional Analysis L.
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5. THEORY OF CATEGORIES BARRY MITCHELL 1965 ACADEMIC PRESS New York and London A Subsidiary of Harcourt Brace Jovanovich, Publishers
6. COPYRIGHT 0 1965, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED m ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRIlTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMlC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWI LIBRARY CONGRESS OF CARD CATALOG NUMBER: 65-22761
7. Preface A number of sophisticated people tend to disparage category theory as consistently as others disparage certain kinds of classical music. When obliged. to speak of a category they do so in an apologetic tone, similar to the way some sa,y, “ I t was a gift-I’ve never even played it” when a record of Chopin Nocturnes is discovered in their possession. For this reason I add to the usual prerequisitethat the reader have a fair amount of mathematical sophistication, the further prerequisite that he have no other kind. Functors, categories, natural transformations, and duality were introduced in the early 1940’sby Eilenbergand MacLane [ 10,11]. Originally, the purpose of these notions was to provide a technique for clarifying certain coficepts, such as that of natural isomorphism. Category theory as a field in itself lay relatively dormant during the following ten years. Nevertheless some work was done by MacLane [28, 291, who introduced the important idea of defining kernels, cokernels, direct sums, etc.,in terms of universal mapping properties rather than in terms of the elements of the objects involved. MacLane also gave some insight into the nature of the duality principle, illustrating it with the dual nature of the frees and the divisibles in the category of abelian groups (the projectives and injectives, respectively, in that category). Then with the writing of the book “Homological Algebra” by Cartan and Eilenberg [6], it became apparent that most propositions concerning finite diagrams of modules could be proved in a more general type of category and, moreover, that the number of such propositions could be halved through the use of duality. This led to a full-fledged investigation of abelian categories by Buchsbaum [3] (therein called exact categories). Grothendieck’s paper [20] soon followed, and in it were introduced the important notions of A.B.5 category and generators for a category. (The latter idea had been touched on by MacLane [29].) Since then the theory has flourished considerably,not only in the direction of generalizing and simplifying much of the already known theorems in homological algebra, but also in its own right, notably through the imbedding theorems and their metatheoretic consequences. In Chapters 1-111 and V, I have attempted to lay a unified groundwork for the subject. The other chapters deal with matters of more specific interest. Each chapter has an introduction which gives a summary of the material to follow. I shall therefore be brief in giving a description of the contents. In Chapter I, certain notions leading to the definition of abelian category are introduced. Chapter I1 deals with general matters involving diagrams, limits, and functors. In the closing sections there is a discussion of generators, vii
8. viii PREFACE projectives, and small objects. Chapter I11 contains a number of equivalent formulations of the Grothendieck axiom A.B.5 (herein called C,) and some of its consequences. In particular the Eckmann and Schopf results on injective envelopes [8] are obtained. Peter Freyd’s proof of the group valued imbedding theorem is given in Chapter IV. The resulting metatheorem enables one to prove certain statements about finite diagrams in general abelian categories by chasing diagrams of abelian groups. A theory of adjoint functors which includes a criterion for their existence is developed in Chapter V. Also included here is a theory of projective classes which is due to Eilenberg and Moore [ 121. The following chapter is devoted to applications of adjoints. Principal among these are the tensor product, derived and coderived functors for group-valued functors, and the full imbedding theorem. The full imbedding theorem asserts that any small abelian category admits a full, exact imbedding into the category of R-modules for some ring R. The metatheory of Chapter IV can thus be extended to theorems involving the existence of morphisms in diagrams. Following Yoneda [36], in Chapter VII we develop the theory of Ext in terms of long exact sequences. The exactness of the connected sequence is proved without the use of projectives or injectives. The proof is by Steven Schanuel. Chapter VIII contains Buchsbaum’s construction for satellites of a.dditive functors when the domain does not necessarily have projectives [5]. The exactness of the connected sequence for cosatellites of half exact functors is proved in the case where the codomain is a C, category. I n Chapter I X we obtain results for global dimension in certain categories of diagrams. These include the Hilbert syzygy theorem and some new results on global dimension of matrix rings. Here we find the main application of the projective class theory of Chapter V. Finally, in Chapter X we give a theory of sheaves with values in a category. This is a reorganization ofsome work done by Gray [ 191, and gives a further application of the theory of adjoint functors. We shall be using the language of the Godel-Bernays set theory as presented in the appendix to Kelley’s book “General Topology” [25]. Thus we shall be distinguishing between sets and classes, where by definition a set is a class which is a member of some other class. A detailed knowledge of the theory is not essential. The wordsfarnib and collection will be used synonymously with the word set. With regard to terminology, what has previously been called a direct product is herein called a product. In the category of sets, the product of a family is the Cartesian product. Generally speaking, if a notion which com- mutes with products has been called a gadget, then the dual notion has been called a cogadget. In particular what has been known as a direct sum here goes under the name of coproduct. The exceptions to the rule are monomorphism- epimorphism, injective-projective, and pullback-pushout. In these cases euphony has prevailed. In any event the words left and right have been eliminated from the language.
9. PREFACE ix The system of internal references is as follows. Theorem 4.3 of Chapter V is referred to as V, 4 3 if the reference is made outside of Chapter V, and as . 4 3 otherwise. The end of each proof is indicated by 1. . I wish to express gratitude to David Buchsbaum who has given me much assistance over the years, and under whose supervision I have worked out a number of proofs in this book. I have received encouragement from MacLane on a number of occasions, and the material on Ext is roughly as presented in one of his courses during 1959. The value of conversations with Peter Freyd cannot be overestimated, and I have made extensive use of his very elegant work. Conversations with Eilenberg, who has read parts of the manuscript, have helped sharpen up some of the results, and have led to the system of terminology which I have adopted. I n particular he has suggested a proof of IX, 7.2 along the lines given here. This has replaced a clumsier proof of an earlier draft, and has led me to make fairly wide use of the pro- jective class theory. This book has been partially supported by a National Science Foundation Grant at Columbia University. I particularly wish to thank Miss Linda Schmidt, whose patience and accuracy have minimized the difficulties in the typing of the manuscript. B. MITCHELL Columbia University, New York May, 1964
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11. Contents PREFACE . vii I. Preliminaries Introduction . . 1 1. Definition . . 1 2. The Nonobjective Approach . . 2 3. Examples . . 3 4. Duality . . . 4 5. Special Morphisms . . 4 6. Equalizers . . 7 7. Pullbacks, Pushouts . . . 9 8. Intersections . . 10 9. Unions . . I1 10. Images . . 12 11. Inverse Images. . . . . 13 12. Zero Objects . . 14 13. Kernels . . . 14 14. Normality . . 16 15. Exact Categories . . 18 16. The9Lemma . . . . 20 17. Products . . . 24 18. Additive Categories . . 28 19. Exact Additive Categories . , . . 32 20. Abelian Categories . . . 33 21. The Category of Abelian Groups 3 . . 34 Exercises. . . . . 36 1 . Diagrams and Functors 1 Introduction . . . . . 41 1. Diagrams . . 42 2. Limits . . . . 44 3. Functors . . . . 49 4. Preservation Properties of Functors . . 50 5. Morphism Functors . . . 53 6. Limit Preserving Functors . . . 54 7. Faithful Functors . . . . . 56 8. Functors of Several Variables . . 58 9. Natural Transformations . . . . . 59 10. Equivalence of Categories . . . 61 11. Functor Categories . . . . 63 xi
12. xii CONTENTS 12. Diagrams as Functors . . . . . . . 65 13. Categories of Additive Functors; Modules . . . . . 67 14. Projectives, Injectives . . 69 15. Generators . . . . 71 16. Small Objects . . 74 Exercises. . . . . . . . . 76 III. Complete Categories Introduction . . . . . . 81 1. Cf Categories . . . . 81 2. InjectiveEnvelopes . . . 86 3. Existence of Injectives . . . . . . . 88 Exercises. . . . . 90 IV. Group Valued Functors Introduction . . . . . 93 1. Metatheorems . . . . . . . . 93 2. The Group Valued Imbedding Theorem . . 97 3. An Imbedding for Big Categories . . . 101 4. Characterization of Categories of Modules . . . . 104 5. Characterization of Functor Categories . . 106 Exercises. . . . . . 112 V. Adjoint Functors Introduction . . 117 1. Generalities . . . . 117 2. Conjugate Transformations . . . . 122 3. Existence of Adjoints . . . . . 124 4. Functor Categories . . . . 126 5. Reflections . . . . 128 6. Monosubcategories . . . . . . 131 7. Projective Classes . . . . . . 136 Exercises. . . . . . 139 V . Applications of Adjoint Functors I Introduction . . . . . . . 141 I. Application to Limits . . . . 141 2. Module-Valued Adjoints . . . 142 3. The Tensor Product . . . . . 142 4. Functor Categories . . . . . . . . 145 5. Derived Functors .. . . . . . . . 147 6. The Category of Kernel, Preserving Functors. . . 150 7. The Full Imbedding Theorem . . . * . . 151 8. Complexes . . . . . . . . . 152 Exercises. . . . . . . . 156
13. CONTENTS MI. Extensions Introduction . . . . . . . . 161 1. Ext1 . . . . . 161 2. The Exact Sequence (Special Case) . 167 3. Ext" . 4. TheRelation - 5. The Exact Sequence . . . . . . . . . . . . 170 175 178 6. Global Dimension . . 179 7. Appendix: Alternative Description of Ext . 182 Exercises. . . . . . . . . . . 187 VIII. Satellites Introduction . . . . . 191 1. Connected Sequences of Functors . . 191 2. Existence of Satellites . . * . . . 194 3. The Exact Sequence . . . . 199 4. Satellites of Group Valued Functors . . . . 204 5. Projective Sequences. . . . . 207 6. SeveralVariables . . . . . . 208 Exercises. . . . . . . . 21 1 IX. Global Mmension Introduction . . . . . . . . . . 215 1. Free Categories . . . . . . . . 215 2. Polynomial Categories . . . . . . . 219 3. Grassmann Categories . 220 4. Graded Free Categories . . . . . . 22 1 5. Graded Polynomial Categories . . . 224 6. Graded Grassmann Categories . 225 7. Finite Commutative Diagrams . . . . 22 7 8. Homological Tic Tac Toe . . . . . . 229 9. NormalSubsets . . . 233 10. Dimension for Finite Ordered Sets . 234 Exercises. . . . . 24 1 X. Sheaves Introduction . . . . 245 1. Preliminaries . . . . . . . 245 2. St-Categories . . . . . 248 3. Associated Sheaves . . . . . 250 4. Direct Images of Sheaves . 25 1 5. Inverse Images of Sheaves . . . . . 253 6. Sheaves in Abelian Categories 254 7. Injective Sheaves . . 257 8. InducedSheaves . . . . 258 Exercises. . . . . . . . 261 BIBLIOGRAPHY . . . . 267 SUBJECTINDEX . . . . . . . . . . 269
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15. THEORY OF CATEGORIES
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17. CCHAPTER I] Preliminaries Introduction I n this chapter the basic notions and lemmas involving finite diagrams are given. These notions are equalizers, pullbacks, intersections, unions, images, inverse images, kernels, normality, products, and the duals of these. I n general, the material is organized in such a way as to lead up to a very eco- nomical characterization of abelian categories (20.1) although some of the propositions will not be needed until much later. I n the last section we give a discussion of the category of abelian groups, and the technique of diagram chasing is illustrated with the 5 lemma. 1. Definition A category is a class d,together with a class A which is a disjoint union of the form A = u [A,B],. (A,B)€dXd To avoid logical difficulties we postulate that each [ A , B], is a set (possibly void. When there is danger of no confusion we shall write [ A , B] in place of [A, B],.) Furthermore, for each triple ( A , B, C ) of members of d we are to have a function from [B,C ] x [A, B ] to [A, C ] . The image of the pair (8, a ) under this function will be called the composition of / by a , and will be I denoted by p a . The composition functions are subject to two axioms. (i) Associativity : Whenever the compositions make sense we have (rB).= r(B.). (ii) Existence o identities: For each A E&’ we have an element 1 E [A, A] f , such that l A a = a and /31A = /3 whenever the compositions make sense. The members of d are called objects and the members of A are called morphisms. If a E [ A , B] we shall call A the domain of a and B the 1
18. 2 I. PRELIMINARIES codomain, and we shall say “ a is a morphism from A to B.” This last state- a ment is represented symbolically by “ a : A +B,” or sometimes “ A + B.” When there is no need to name the morphism in question, we shall simply write A + B . Observe that 1, can be the only identity for A , for if e is another we must have e =el, = 1,. We sometimes write 1, : A = A in the case of identity morphisms. I f d is a set, then the category will be called small. I n this case A,being a union of sets indexed over a set, is also a set. We shall commit a common notational inconsistency in denoting the above category by the underlying class d. 2. The Nonobjective Approach Reluctant as we are to introduce any abstraction into the theory, we must remark that there is an alternative definition for category which dispenses with the notion of objects. A category can be defined as a class A, together with a binary operation on A, called composition, which is not always defined (that is, a function from a subclass of d x A to A).The image of the pair (8, a ) under this operation is denoted by pa (if defined). An element e E A is called an identity if eu = a and ,Re = #? whenever the compositions are defined. We assume the following axioms : (1) If either ( y p )u or y(/3u)is defined, then the other is defined, and they are equal. (2) If y/? and pa are defined and #? is an identity, then y a is defined. (3) Given a E A, there are identities e, and t?R in A such that e,a and aeRare ? defined (and hence equal a ) . (4) For any pair ofidentities eL and eR, the class {a E ( e L a ) e R is defined} is a set. Clearly our first definition of category gives us a category of the second type. Conversely, given a class A satisfying the postulates (1) to (4), proceed to we show how this can be associated with a category of the first type. We index the class of identities in A by a class d, denoting the identity corresponding to il E d by 1,. Now if a E A, then there can be only one identity such that a l , is defined. For if 1,. is another, then we have al,. = (a1,)lA’, and so by (1) the composition l,l,, is defined. Since both are identities we must then have 1, = 1, The unique A E &’such that al, is defined is called the domain . of a. Similarly, the unique B ~d such that lBa is defined is called the codomain of a. We denote by [ A , B ] the class of members o f d with domain A and codomain B . Then [ A , B] is a set by (4), by (3) A is the disjoint and union u [A, B ] . Next we show that #?u is defined if and only if the codomain of a is the
19. 3. EXAMPLES 3 domain of p. Suppose that pa is defined, and let B be the codomain of a. Then / lBa) is defined, and so 81, is defined by (1). I n other words, B is the I ( domain of 8. Conversely, if the codomain of a is the same as the domain of 8, then pa is defined by (2). Therefore composition can be regarded as a union of functions of the type [B,C] x [ A ,B] +A.We must show, finally, that the image ofsuch a function is in [ A , C]. That is, we must show that if pa is defined, then the domain of pa is the domain of a,and the codomain of pa is the codomain of p. But this follows easily from ( 1). 3. Examples 1. The category Y whose class of objects is the class of all sets, where [ A , BIY is the class of all functions from A to B, is called the category of sets. It is not small. 2. A similar definition applies to the c a te goryy ofall topological spaces, where the morphisms from space A to space B are the continuous functions from A to B. 3. The category Y o sets with base point is the category whose objects of are ordered pairs ( A , u ) where A is a set and a E A . A morphism from ( A , a ) to ( B , 6) is a functionffrom A to B such thatf(a) = b. 4. Replacing sets by topological spaces and functions by continuous func- tions in example 3 we obtain the category Yoof topological spaces with base point. 5 . If A has only one identity (so that composition is always defined), we call the category a semigroup and we replace the word “composition” by “multiplication.” Hence an alternative word for “ category ” would be “ semigroupoid ”-a semigroup where multiplication is not always defined. If pa = ap for every pair of morphisms in a semigroup, then the semigroup is called abelian. I n this case composition is usually called addition, and a + 6 is written in place of ap. Furthermore, the identity is called zero and is denoted by 0. 6 . We shall say that a category d’is a subcategoryof a category &xi‘ under the following conditions : (i) d’ d .c (ii) [ A , BIN c [ A , B ] , for all ( A , B ) ~ d d’, x ’ (iii) The composition of any two morphisms in d’ the same as their is composition in d. (iv) 1, is the same in d’as in d for all A E d‘. If furthermore [ A , BIM = [ A , B], for all ( A , B ) ~ d d’ say that d’ x ’ we is a full subcategory of d. 7. An ordered class is a category d with at most one morphism from an object to any other object. If A and B are objects in an ordered class and

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