THEORY OF CATEGORIES This Page Intentionally Left Blank Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Samuel Ellenberg and Hymen Eass Columbia University, New York RECENT TITLES SAMUELEILENBERG. Automata, Languages, and Machines: Volumes A and B MORRISHIRSCH AND STEPHENSMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELMMAGNUS. Noneuclidean Tesselations and Their Groups FRANCOISTREVES. Basic Linear Partial Differential Equations WILLIAMM. BOOTHBY.An Introduction to Differentiable Manifolds and Riemannian Geometry ~AYTONGRAY. Homotopy Theory : An Introduction to Algebraic Topology ROBERTA. ADAMS.Sobolev Spaces JOHNJ. BENEDETTO.Spectral Synthesis D. V. WIDDER.The Heat Equation IRVINGEZRA S~CAL. Mathematical Cosmology and Extragalactic Astronomy J. DIEUDONN~.Treatise on Analysis : Volume 11, enlarged and corrected printing; Volume IV ; Volume V ; Volume VI, in preparation WERNERGmua, STEPHENHALPERIN, AND RAYVANSTONE. Connections, Curvature, and Cohomology : Volume 111, Cohoniology of Principal Bundles and Homogeneous Spaces I. MARTINISAACS. Character Theory of Finite Groups JAMESR. BROWN.Ergodic Theory and Topological Dynamics C. TRUESDELL.A First Course in Rational Continuum Mechanics: Volume 1, General Concepts GEORGEGRATZER. General Lattice Theory K. D. STROYANAND W. A. J. LUXEMBURG.Introduction to the Theory of Infinitesimals B. M. PUTTASWAMAIAHAND JOHN D. DIXON.Modular Representations of Finite Groups MmwN BERGER.Nonlinearity and Functional Analysis : Lectures on Nonlinear Problems in Mathematical Analysis CHARALAMBOSD. ALIPRANTIS AND OWENBURKINSHAW. Locally Solid Riesz Spaces In preparation JANMIKUSINSKI. The Bochner Integral MICHIELHAZEWINKEL. Formal Groups and Applications THOMASJECH. Set Theory SIGURDURHELGASON. Differential Geometry, Lie Groups, and Symmetric Spaces CARLL. DEVITO. Functional Analysis This Page Intentionally Left Blank THEORY OF CATEGORIES BARRY MITCHELL 1965 ACADEMIC PRESS New York and London A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT 0 1965, BY ACADEMICPRESS, 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 LIBRARYOF CONGRESSCATALOG CARD NUMBER: 65-22761 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, “It 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 prerequisite that 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’s by Eilenberg and 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 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. In Chapter IX 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. 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. In 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 This Page Intentionally Left Blank 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.