Categories for the working mathematician, saunders mac lane 1

Graduate Texts in Mathematics Editorial Board S Axler F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 TAKEUn/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTUZARING Axiomatic Set Theory HUMPHREYs Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable! 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARlsKIiSAMUEL Commutative Algebra Vol.! ZARlsKIiSAMUEL Commutative Algebra VoU! JACOBSON Lectures in Abstract Algebra ! Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed 35 ALEXANDERIWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRlTZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELUKNAPP Denumerable Markov Chains 2nd ed 41 ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory ! 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELUFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol ! 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed (continued after index) Saunders Mac Lane Categories for the Working Mathematician Second Edition Springer Saunders Mac Lane Professor Emeritus Department of Mathematics University of Chicago Chicago, IL 60637-1514 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.w Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 18-01 Library of Congress Cataloging-in-Publication Data Mac Lane, Saunders, 1909Categories for the working mathematician/Saunders Mac Lane 2nd ed p cm - (Graduate texts in mathematics; 5) Includes bibliographical references and index ISBN 978-1-4419-3123-8 ISBN 978-1-4757-4721-8 (eBook) DOI 10.1007/978-1-4757-4721-8 Categories (Mathematics) QA169.M33 1998 512'.55-dc21 I Title II Series 97-45229 Printed on acid-free paper © 1978, 1971 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Tnc in 1971 Softcover reprint ofthe hardcover 2nd edition 1971 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar Of dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Francine McNeill; manufacturing supervised by Thomas King Typeset by Asco Trade Typesetting Ltd., Hong Kong 987 654 SPIN 10796433 Preface to the Second Edition This second edition of "Categories Work" adds two new chapters on topics of active interest One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them-items of interest in their own right and also in view of their use in string theory in quantum field theory The second new chapter describes 2-categories and the higher-dimensional categories that have recently come into prominence In addition, the bibliography has been expanded to cover some of the many other recent advances concerning categories The earlier 10 chapters have been lightly revised, clarifying a number of points, in many cases due to helpful suggestions from George lanelidze In Chapter III, I have added a description of the colimits of representable functors, while Chapter IV now includes a brief description of characteristic functions of subsets and of the elementary topoi Saunders Mac Lane Dune Acres, March 27, 1997 v Preface to the First Edition Category theory has developed rapidly This book aims to present those ideas and methods that can now be effectively used by mathematicians working in a variety of other fields of mathematical research This occurs at several levels On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category These notions are presented, with appropriate examples, in Chapters I and II Next comes the fundamental idea of an adjoint pair of functors This appears in many substantially equivalent forms: that of universal construction, that of direct and inverse limit, and that of pairs of functors with a natural isomorphism between corresponding sets of arrows All of these forms, with their interrelations, are examined in Chapters III to V The slogan is "Adjoint functors arise everywhere." Alternatively, the fundamental notion of category theory is that of a monoid-a set with a binary operation of multiplication that is associative and that has a unit; a category itself can be regarded as a sort of generalized monoid Chapters VI and VII explore this notion and its generalizations Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces Since a category consists of arrows, our subject could also be described as learning how to live without elements, using arrows instead This line of thought, present from the start, comes to a focus in Chapter VIII, which covers the elementary theory of abelian categories and the means to prove all of the diagram lemmas without ever chasing an element around a diagram Finally, the basic notions of category theory are assembled in the last two chapters: more exigent properties of limits, especially of filtered limits; a calculus of "ends"; and the notion of Kan extensions This is the deeper form of the basic constructions of adjoints We end with the observations that all concepts of category theory are Kan extensions (§ of Chapter X) vii viii Preface to the First Edition I have had many opportunities to lecture on the materials of these chapters: at Chicago; at Boulder, in a series of colloquium lectures to the American Mathematical Society; at St Andrews, thanks to the Edinburgh Mathematical Society; at Zurich, thanks to Beno Eckmann and the Forschungsinstitut fUr Mathematik; at London, thanks to A Frohlich and Kings and Queens Colleges; at Heidelberg, thanks to H Seifert and Albrecht Dold; at Canberra, thanks to Neumann, Neumann, and a Fulbright grant; at Bowdoin, thanks to Dan Christie and the National Science Foundation; at Tulane, thanks to Paul Mostert and the Ford Foundation; and again at Chicago, thanks ultimately to Robert Maynard Hutchins and Marshall Harvey Stone Many colleagues have helped my studies I have profited much from a succession of visitors to Chicago (made possible by effective support from the Air Force Office of Scientific Research, the Office of Naval Research, and the National Science Foundation): M Andre, J Benabou, E Dubuc, F.W Lawvere, and F.E.J Linton I have had good counsel from Michael Barr, John Gray, Myles Tierney, and Fritz Ulmer, and sage advice from Brian Abrahamson, Ronald Brown, W.H Cockcroft, and Paul Halmos Daniel Feigin and Geoffrey Phillips both managed to bring some of my lectures into effective written form Myoid friend, A.H Clifford, and others at Tulane were of great assistance John MacDonald and Ross Street gave pertinent advice on several chapters; Spencer Dickson, S.A Huq, and Miguel La Plaza gave a critical reading of other material Peter May's trenchant advice vitally improved the emphasis and arrangement, and Max Kelly's eagle eye caught many soft spots in the final manuscript I am grateful to Dorothy Mac Lane and Tere Shuman for typing, to Dorothy Mac Lane for preparing the index, and to M.K Kwong for careful proofreading-but the errors that remain, and the choice of emphasis and arrangement, are mine Dune Acres, March 27, 1971 Saunders Mac Lane Contents Preface to the Second Edition Preface to the First Edition v vii Introduction I Categories, Functors, and Natural Transformations Axioms for Categories Categories Functors Natural Transformations Monics, Epis, and Zeros Foundations Large Categories Hom-Sets 10 13 16 19 21 24 27 II Constructions on Categories 31 Duality 31 33 Contravariance and Opposites Products of Categories Functor Categories The Category of All Categories Comma Categories Graphs and Free Categories Quotient Categories III Universals and Limits 36 40 42 45 48 51 55 55 59 Universal Arrows The Yoneda Lemma Coproducts and Colimits Products and Limits 62 68 ix Contents x Categories with Finite Products Groups in Categories Colimits of Representable Functors IV Adjoints 10 Adjunctions Examples of Adjoints Reflective Subcategories Equivalence of Categories Adjoints for Preorders Cartesian Closed Categories Transformations of Adjoints Composition of Adjoints Subsets and Characteristic Functions Categories Like Sets V Limits Creation of Limits Limits by Products and Equalizers Limits with Parameters Preservation of Limits Adjoints on Limits Freyd's Adjoint Functor Theorem Subobjects and Generators The Special Adjoint Functor Theorem Adjoints in Topology VI Monads and Algebras Monads in a Category Algebras for a Monad The Comparison with Algebras Words and Free Semigroups Free Algebras for a Monad Split Coequalizers Beck's Theorem Algebras Are T-Algebras Compact Hausdorff Spaces 72 75 76 79 79 86 90 92 95 97 99 103 105 106 109 109 112 115 116 118 120 126 128 132 137 137 139 142 144 147 149 151 156 157 VII Monoids 161 Monoidal Categories Coherence 161 165 Contents xi Monoids Actions The Simplicial Category Monads and Homology Closed Categories Compactly Generated Spaces Loops and Suspensions VIII Abelian Categories Kernels and Cokernels Additive Categories Abelian Categories Diagram Lemmas IX Special Limits Filtered Limits Interchange of Limits Final Functors Diagonal Naturality Ends Coends Ends with Parameters Iterated Ends and Limits X Kan Extensions Adjoints and Limits Weak Universality The Kan Extension Kan Extensions as Coends Pointwise Kan Extensions Density All Concepts Are Kan Extensions XI Symmetry and Braiding in Monoidal Categories Symmetric Monoidal Categories Monoidal Functors Strict Monoidal Categories The Braid Groups Bn and the Braid Category Braided Coherence Perspectives 170 174 175 180 184 185 188 191 191 194 198 202 211 211 214 217 218 222 226 228 230 233 233 235 236 240 243 245 248 251 251 255 257 260 263 266 Index Ab-category, 17,24,28,29 Abelian - categories, 28 - groups, 24 Absolute - coequalizer, 149 - Kan extension, 249 -limit, 149 Action group, 141 left - of a monoid, - of operators, 124 Addition ordinal -, 175 - of arrows, 195 Ex Additive - category, 196 - functor, 29, 197 - Kan extension, 242 Adjoint arrows, 276 Freyd's - functor theorem, 128, 129 left -, 38, 81 left - left-inverse, 94 right -,81 - equivalence, 93 - functor, 38 - pairs, 95 - square, 103 Adjointness, Adjunct, 79, 81 Adjunction, 80, 83, 276 category of -, 101 counit of -, 83 front and back -,83 mapof-,99 monad defined by -, 139 unit of -,83 - with a parameter, 102 Algebraic system, 75, 124 Algebras, 156 morphisms of T -, 140 structure map of T -, 140 T-,140 variety of -, 124 Amalgamated product, 66 Ambient category, 267 Arity, 124 Arrows addition of -, 195 Ex canonical -, 73 category of -,40 composable pair of -,9, 10, 13,49 connecting -,207 diagonal -, 84 epi -,19 factorization of -, 194, 199 idempotent -, 20, 21 identity -, 7, 8, 10 invertible -, 19 kernel of -, 191 monic -,19 parallel -, 11 regular -,21 Ex universal-, 55, 58,235 Ex weak universal-, 235 zero -, 20, 74, 190, 194 - function, 13 - oniy-metacategory, 303 304 Associative law, general-, 166, 171 - for monad, 138 - for monoid, 162 - for monoidal categories, 162 - for T-algebra, 140 Associativity, Atomic statement, 31 Augmentation, 179 Augmented simplicial object, 179 Axiom of choice, 291 Barycentric coordinates, 178 Based category, 184 Base point, 12 Basic - arrows, 166 - graphs, 166 Beck's theorem, 151 Bicategory, 209, 281, 283 - of rings, 283 Bifunctor, 37 Bilinear composition, 28 Bimodule, 283 Binary - relation, 26 -words, 165 Biproduct, 194 Boolean algebra, 123 Bound greatest lower -, 114, 126 least upper -, 114, 126 - variable, 31, 223 Boundary - homomorphism, 179 - of tetrahedron, 178 Braid,260 Braiding, 251, 252 Box Product, 262 fT Cancellable (left or right), 19 Canonical - arrow, 73, 74 - map, 73, 169, 215, 253 Index - presentation, 155 Cartesian - closed category, 106 -product, Category, 1, 10 Ab-, 28 abelian -, 198 abstract -, 31 additive -, 196 ambient -, 267 based -,184 bi -,281 cartesian closed -, 106 closed -, 184 comma-,45 complete -, 109, 110 concrete -,26 connected -, 90 Ex co-well powered -, 180 discrete -, 11 double-,44 dual-,31 empty -,233 enriched -,29,276 equivalence of -, 93 fibered -, 35 filtered -, 211 free -,49 functor -, 40 horizontal-,277 image -,247 internal-, 267, 285 isomorphism of -,92 large -, 12, 23, 24 locally small-, 131 monoidal-, 251 opposite -, 33 preadditive -,28 product -, 36 pseudo-filtered -,216 Ex relative -, 187 simplicial-, 12 single set -,279 skeletal, 93 strict monoidal-, 161,257 subdivision -,224 Index super-comma -, 115 Ex symmetric -, 251 symmetric monoidal-, 184,251 tensor -, 252 -,104,272 two-dimensional-, 44, 272 well-powered -, 180 vertical-,273 - of adjunctions, 99 - of algebras, 128 - of arrows, 40 - of diagrams, 52 - object, 267 - of small sets, 12, 24 Chain complex, 202 Character group, 17 Characteristic function, 105 Chase, diagram, 75, 204 Class, 23 equational-, 124 Closed, 141 cartesian - categories, 97 - category, 184 Closure operation, 141 Cochain complex, 183 Cocomponents of a map, 74 Codense functor, 246 Codensity monad, 250 Codomain, 7, Coend,226 Coequalizers, 64 absolute -, 149 creation of -, 151 split -, 149 Cogenerating set, 127 Cogenerator, 127 Coherence theorem -,263 Cohomology, 13 Coimage, 200 Cokeme1, 193,206 - pair, 66 Colimit,67 filtered -, 212 reflection of -, 154 Comma - category, 46 305 super - category, 115 Ex Commutative diagram, 3, Commutator, 14 - subgroup, 14 Comonad, 139 Comonoid, 181 Compact Hausdorff space, 131 Compactification, 92, 131 Compactly generated spaces, 185, 190 Compact-open topology, 185 Comparison - functor, 142, 151, 153 - theorem, 142 Complete category, 109, 113 Component, 16 connected -,90 Ex matrix of -, 196 - of natural transformation, 218 - of wedge, 223 Composable pair, 200 Composite, horizontal -, 43 vertical-,43 - function, - functor, 14, 42 - of paths, 20 - of transformations, 40 Composition, 7, 9, 279 Comprehension principle, 21 Concrete category, 26 Cone, 67, 68 canonical-,246 colimiting -, 214 limiting -,67,113 universal -, 67 Congruence, 52 Conjugate natural transformation, 102, 104 Conjugation, 18 Ex., 20 Connected - category, 88, 90 Ex - component, 90 Ex - groupoid, 20 - sequence of functors, 242 - space, 26 Connecting homomorphism, 206, 242 Index 306 Connection, Galois, 95 Continuous - functor, 116 - hom-functor, 183 -map, 157 Contractible, 150 Contravariant, 17, 33 Coordinates, barycentric, 178 Copowers, 63 Coproduct, 62 denumerable -, 172 finite -, 212 infinite -, 212 injections of -,63 - diagram, 62 - object, 63 Coreflective, 91 Counit of adjunction, 83, 87 Covariant - functor, 34 - hom-functor, 34 - power-set functor, 142 Ex Co-well-powered category, 130 Creation, 122 - of coequalizers, 153 - of ends, 225 - oflimits, 111, 112 Crossed module -,285 CTT-Crude trip1eability theorem, 154 Degeneracy, 179 Dense - functor, 246 - subcategory, 245 Derived - functor, 242 - operator, 124 Determinant, 16 Diagonal - arrows, 16 - functor, 58, 66, 119, 233 - map, 73, 196 Diagram, 2, 3, 4, 51, 71 biproduct -, 194 category of -, 52 commutative -,3,8, 165 coproduct -, 63, 64 limit -,69 product -, 69 - chase, 75, 204 - scheme, 48 Difference, 70 - kernel, 70 - member, 208 Dinatural transformation, 218 Direct - product, 69 - sum, 195 Directed - graph, 10 - preorder, 211 - set, 211 Disjoint - hom-sets, 27 -union, 63 Domain, Double - category, 44 - dual, 17 - end, 230 Dual,31 - category, 32 - statement, 33 Duality principle, 32, 266 Dummy, 219 Eilenberg-Moore category of a monad,139 Element, universal, 57, 58 Elementary particles, 266 Embedding, 15 Empty - category, 10, 233 - functor, 233 - string, 258 End,222 creation of -,225 double -, 230 iterated -,230 interchange of -,241 Index preservation of -,225 - of natural transformation, 228 Ending wedge, 223 Endofunctor, 137 Enriched category, 276 Epi,19 split -,19 - monic factorization, 194, 199 Equalizer, 70 Equational class, 124 Equivalence, 16 adjoint -,93 natura1-, 16 - of categories, 18, 92, 93 ETAC,31 Euclidean vector spaces, 220 Evaluation, 98 Evaluation map, 220 Exact left - functor, 201 right - sequence, 201 short - sequence, 200 - functor, 197,201 Exponential-, 98 Ex Extensions, 233 absolute Kan - , 249 Kan -, 233, 236 left-,240 right-, 236 Exterior Algebra, 89 Ex Extranatural transformation, 219, 220 Face operator, 182 Factor-commutator, 14 Factorization of arrows, 199 Faithful functor, 15 Fibered - category, 35 - product, 71 -sum, 66 Fiber map, 71 Fields of quotients, 56, 87 Filtered, 216 Ex - category, 211 307 - colirnit, 212 - set, 211 Final - functor, 2, 17 - subcategory, 217 Finite -limit, 113 - product, 72 Five Lemma, 202, 205 Forgetful functor, 212 Fork, split, 149 Formal criteria - for existence of adjoint, 234 - for representability, 235 Ex - for a universal arrow, 235 Ex Free - category, 49, 56 -monoid,50 - monoidal category, 166 - product, 214 - T-algebra, 140 Freyd - adjoint functor theorem, 120, 121 - existence theorem for an initial object, 235 Fubinic Theorem, 230 Full functor, 14 - subcategory, 15 Function, arrow-,13 characteristic -, 105 composite -,9 identity -,9 inclusion -, insertion -,9 monotone -,175 object-,13 order preserving -,95,96 - set, 40 - space, 185 Functor,2, 13 additive -, 29, 85, 197 adjoint -, 80 category -,40 codense -, 246 Index 308 Functor (cont.) comparison -, 144 composite - , 14 continuous -, 116 contravariant - , 33 covariant -, 33 dense -,246 derived -,242 diagonal-, 233 empty -,233 exact -,201 faithful-, 15 final-, 217, 238 forgetful-, 14, 87, 120, 144, 157, 212 full-, 14 identity -, 14 inclusion -, 15 left adequate -, 250 left adjoint left inverse -, 94 left exact -, 201 monoidal-, 255 morphism of -, 16 power-set -, 13, 33 representable -, 60 underlying -, 14 Yoneda -,62 - category, 40 Fundamental groupoid, 20 Galois connections, 95, 96 General linear group, 14 Generating - object, 125 - set, 125, 127 Generators of a category, 52 Geometric realization, 227 Godel-Bemays axioms, 23 Graded set, 124 Graph, 10, 48, 80 Greatest Lower Bound, 126 Group, 11 fundamental-,20 small-,22 - actions, 141 - in a category, 75 Groupoid, 20, 51 Hausdorff spaces, 25, 135 compact -, 125, 157 compactly generated -, 185 Hom-functor, 27 contravariant -, 34 covariant -, 34 Hom-object, 184 Homology, 13, 179, 184 singular -, 179 Homomorphism, 1; see also Morphisms boundary -, 179 connecting -,206,242 crossed -, 285 Homotopy class - of maps, 12, 25 - of paths, 20 Hom-sets, 10, 27 disjoint -, 27 Horizontal composite, 42, 273, 277 - category, 277 Idempotent, 20 split -,20 Identities (for algebras), 124 Identity, 7, 8, 10 triangular -, 85 - arrow, - function, - functor, 14, 43 - natural transformation, 43 Image, 200 Inclusion - function, - functor, 15 Induced map, 34 Infinite - coproduct, 64 - product, 69 Index Initial functor, 218 Initial object, 128 existence of - , 120 Injection, 15, 19 - of coproduct, 63, 73 Injective, 15 - monotone function, 176 - object, 118 Insertion function, Integral, 228 double -, 230 iterated -,230 Interchange, 214 - of ends (Fubini), 231 -law, 43 Internal - category, 267 - functor, 269 - hom-functor, 269 Intersection of subobjects, 126 Intertwining operator, 41 Inverse, 14 left adjoint-left -, 94 left or right -, 19 two-sided -, 14 -limit, 68 Invertible arrow, 19 Isomorphic, 19 Isomorphism, 14 natural-, 16 reflection of -, 154 - of categories, 14, 92 - of objects, 19 Iterated integral, 230 Join, 14, 126 Kan extensions, 236 absolute -,249 additive -,242 left - as coends, 240 pointwise -, 237, 243, 245 right -,236 309 Kelleyfication, 186 Kelley spaces, 185 Ker-Coker sequence, 206 Kernel, 191 difference -, 70 - pair, 71 Kleisli category of a monad, 147 Large category, 12,23,24 Least upper bound, 126 Left - action, 174 - adequate functor, 250 - adjoint, 38, 81 - adjoint - inverse, 94 - adjunct, 79, 81 - cancellable, 19 - exact functor, 201 - inverse, 19 - Kan extensions, 240, 248 - regular representations, 174 Lemma Five-, 205 Short five -,202 snake-, 206 Yoneda -,61 Length of words, 165 Limit, 68, 78, 112, 233 creation of -, 112, 117 direct -,67 filtered -, 212 finite -, 113 inductive -,67 interchange of -, 214 inverse -, 68 pointwise -, 237 preservation of -, 117 projective -,68 - of a natural transformation, 228 - object, 68 Limiting cone, 67, 68, 69 Linear order, 11 Locally small category, 131 Loop space, 189 Index 310 Map, see also Arrows canonical, 169, 215, 246 continuous -, 157, 185 diagonal-,73,196 evaluation -, 98 fiber -,71 homotopic -, 13 structure - of algebras, 140 - of adjunctions, 99 Matrices, 11, 74, 196 Matrix multiplication, 196 Meet, 114, 126 Member, 204 Metacategory, 7, 8, Metagraph, Middle four exchange, 275 Modification, 278 Modules, 141 crossed -, 285 Monad, 137, 138, 180 codensity -,250 Ex free group - , 139 multiplication of - , 138 unit of -, 138 - defined by adjunction, 139 Monadic, 143 Monic, 19 split -, 19 - arrow, 19 Monoid, 2, 11, 50, 75, 170 free -,50, 172 universal -, 161 Monoidal categories, 161, 162 strict -, 151 symmetric -, 184 - functors, 255 Monotone function, 15 Ex., 176 Morphisms, - of arrows, - of categories, 13 - of functors, 16 - of graphs, 48 - of monoidal categories, 164 relaxed -, 164 strict -, 164 - of short exact sequences, 202 - of simplicial objects, 178 - of T-algebras, 140 Multiplication - in a monad, 138 - of monoidaI categories, 162 Natural, 2, 16 components of - transformation, 218 conjugate - transformation, 102, 104 universal - transformation, 39 - bijection, - equivalence, 16 - isomorphism, 16 - number object, 291 - transformation, 16 Nerve, 271 Null object, 20, 191, 194 Number, ordinal, 11 Object, 3, 7, 10 coproduct -, 63 free -,147 homology, 202 initial-, 128 injective -, 118 limit -,68 null-, 20, 191, 194 of objects, 267 projective -, 118 quotient -, 126 simplicial -, 178 terminal-, 20, 72 - function, 13, 202 - over, 45 -under, 454 O-graph,48 Operator derived -, 124 intertwining -,41 Opposite category, 33 Order linear - 11 Index partial - , 11 - preserving function, 93 Ordinal finite - , 12 - addition, 175 - number, 11, 17, 175, 178 P-adic - integers, 11 - solenoid, III Pair adjoint -,95 cokernel - , 66 composable - , 9, 10, 13, 49 conjugate - , 100 ff equalizer of - , 70 kernel-,71 parallel -, 11 Parameter adjunction with a - , 102 - theorem, 229 Partial order, 11 Path, 50 directed - , 166 Pointed - set, 26 - topological space, 26, 188 Pointwise - Kan extensions, 237, 243, 245 -limit, 116 Power, 70, 290 - set, 13,21,290 - set functor, 13, 33 Preadditive category, 28 Precategory,48 Precise tripleability theorem (Beck), 154 Preorder, 11, 92 directed -, 211 Presentation, canonical, 153 Preservation of - coproduct, 172 - end of functor, 225 -limit, 116 - right Kan extension, 243 311 Presheaf, 77 Product, 36, 69 amalgamated -,66 cartesian -, direct -,69 fibered - , 71 free - , 128 Ex infinite -, 69 iterated -, 176 projections of - , smash -,189 tensor - , 128, 161 - category, 36 -diagram, Projections - of comma category, 47 - of product, 36 - of product category, 69 Projective object, 118 Proper class, 23 Pseudo-filtered category, 215 PTT-Beck, 154 Pullback, 71 - square, 71 Pushout, 65 Quantifiers, 97 Quasi-inverse, 85 Quotient field of - , 56 - object, 126,202 - topology, 133 Rank of word, 166 Reflection, 89 - of colimits, 154 - of isomorphisms, 154 Reflective subcategory, 91 Reflector, 91 Relations, 26, 254 Relative category, 184 Relaxed morphism, 164 Replacement axiom, 23 Index 312 Representability, 122 formal criterion for -,234 - theorem, 122 Representation, 60 left regular -, 174 Resolution, 181 Retraction, 19 Right - adjoint, 81 - adjoint - inverse, 133 - adjunct, 79, 81 - cancellable, 19 - exact sequence, 201 - inverse, 19 - Kan extension, 236 Ring, small, 46 Root, 78 SAFT,130 Satisfaction of identities, 124 Scheme, diagram, 48 Section (= right inverse), 19 Semigroup, 144 free -,144 Sentence, 31 Sequence right exact - , 201 short exact - , 200 Sets based -,26 category of small-, 62 cogenerating -, 127 directed -, 211 filtered -, 211 function -,40 generating -, 127 graded -, 124 linearly ordered -, 180 metacategory of -, pointed -, 26 simplicial-, 12, 174 small-,22 solution -, 120 underlying -, 124 Sheaf,35 Simplex affine -,178 singular -, 180 Simplicial - category, 175 - object, 181, 178 - set, 12, 174 Single set category, 279 Singular - chain complex, 180 - homology, 180 - simplex, 180 Skeleton (of a category), 93 Small - complete category, 109, 113, 115 - group, 22 - pointed set, 26 - ring, 25 - set, 22 - topological space, 25 Smash product, 189 Snake Lemma, 206 Solenoid, p-adic, 111 Solution set (condition), 120 Source, 7, 279 Space compact Hausdorff -, 125, 157 compactly generated -, 185 Euclidean vector -,220 function -, 185 Hausdorff -, 25, 135 Kelley -, 185 loop -, 189 metric, 56 path -, 190 Ex topological-, 12,25 vector -,25, 56 Span,283 Span an object, 127 Split - coequalizers, 149 - epi, 19 - fork, 149 - idempotent, 20 -monic, 19 Index Square adjoint - , 103 cartesian -, 71 cocartesian - , 66 pullback - , 71 Standard categories, 292 Statement, atomic, 31 Stone-Cech compactification, 125 Strict functor, 257 Strict monoida1 category, 161, 257 Strings, 257 Strong functor, 258 Structure map of algebras, 143 Subcategory, 15 codense - , 246 dense -,245 final-, 217 full-, 15 reflective - , 91 Subdivision category, 224 Subobject classifier, 105 Sum direct - , 195 fibered - , 66 Super-comma category, 115 Ex Supernatural transformation, 219 Surjective monotone function, 117 Suspension, 189 Symmetric monoidal category, 184, 251,253 System, algebraic, 124 T-algebras, 140 Tangles, 266 Target, 7, 279 Tensor category, 252 Tensor product, 128 Terminal object, 73 Terminology, table of, 293 Theorem Beck's - characterizing algebras, 151 comparison - for algebras, 142 construction of free monoids, 172 313 formal criterion for existence of adjoints, 234 Freyd's adjoint functor - , 120, 121 Fubini - , 230 Kan extensions as a coend, 240 Kan extensions as a pointwise limit -,237,243,245 parameter - for ends and limits, 229 representability -, 122 special adjoint functor -, 129 special initial object -, 128 Topological spaces, 25 category of - , 51 compactly generated - , 185, 190 small-, 132 Topology algebraic -, 13 compact open - , 185 Hausdorff -, 25, 135 identification -, 133 quotient - , 133 subspace - , 132 Topos, 24, 106, 107,289 Transformations, 16 components of -, 16 composite -,40 conjugate natural - , 102, 104 dinatural -, 218 extranatural, 219 natural, 16 supernatural-, 219 Triad, 138 Triangular identities, 85 Triple, 138 Tripleable (= monadic), 143, 155 Truth, 105, 289 Two-dimensional category, 104 Two-sided inverse, 14 Underlying - functor, 14 - sets, 26 Union, 21, 126 314 Unit, 83 - of adjunction, 87 - ofKan extension, 241 -law, - of monad, 138 Universal, 1, 36, 59 weak - arrow, 235 well-powered, 130 - arrow, 55, 58, 61 -cone, 67 - element, 57, 58, 61 - monoid, 161 - natural transformation, 39 - property, 55, 57 -wedge, 223 Universality, 59 - of Kan extensions, 249 Universe, 12, 22 Urysohn Lemma, 128 Variable bound-,31 dummy -, 219, 228 free -,31 - of integration, 223 Variety of algebras, 124 Index Vector spaces, 56 Vertical category, 274 Vertical composite, 273 VTT, 154 Watt's Theorem, 131 Weak universal arrow, 235 Wedge, 219 universal-, 223 Well-powered category, 130 Whisker, 275 Word, 144, 253 binary -, 105 Yoneda - embedding, 247 -lemma, 59, 61, 75 Zermelo, 291 Zermelo-Fraenkel axioms, 289 Zero - arrow, 20, 74, 194 - morphism, 192 Zigzag, 206 Graduate Texts in Mathematics (continued from page if) 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory FARKAS/KRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 3rd ed HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras IITAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BURRIS/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FORSTER Lectures on Riemann Surfaces BOTT/Tu Differential Fonns in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELAND/ROSEN A Classical Introduction to Modem Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERCE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BR0NDSTED An Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups DIESTEL Sequences and Series in Banach Spaces DUBROVIN/FoMENKO!NOVIKOV Modern Geometry-Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups SHIRYAEV Probability 2nd ed CONWAY A Course in Functional Analysis 2nd ed KOBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed BROCKERIToM DIECK Representations of Compact Lie Groups 99 GROVEIBENSON Finite Reflection Groups 2nd ed 100 BERG/CHRISTENSEN/REsSEL Hannonic Analysis on Semigroups: Theory of Positive Definite and Related Functions 101 EDWARDS Galois Theory 102 VARADARAJAN Lie Groups, Lie Algebras and Their Representations 103 LANG Complex Analysis 3rd ed 104 DUBROVIN/FoMENKO!NOVIKOV Modem Geometry-Methods and Applications Part II 105 LANG SL,(R) 106 SILVERMAN The Arithmetic of Elliptic Curves 107 OLVER Applications of Lie Groups to Differential Equations 2nd ed 108 RANGE Holomorphic Functions and Integral Representations in Several Complex Variables 109 LEHTO Univalent Functions and Teichmuller Spaces 110 LANG Algebraic Number Theory III HUSEMOLLER Elliptic Curves 112 LANG Elliptic Functions 113 KARATZAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed 114 KOBLITZ A Course in Number Theory and Cryptography 2nd ed 115 BERGERIGOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces 116 KELLEy/SRINIVASAN Measure and Integral Vol I 117 SERRE Algebraic Groups and Class Fields 118 PEDERSEN Analysis Now 119 ROTMAN An Introduction to Algebraic Topology 120 ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAUs/HERMES et al Numbers Readings in Mathematics 124 DUBROVIN/FoMENKO!NOVIKOV Modern Geometry-Methods and Applications Part III 125 BERENSTEIN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 2nd ed 127 MASSEY A Basic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FULTON/HARRIS Representation Theory: A First Course Readings in Mathematics 130 DODSON/POSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings 132 BEARDON Iteration of Rational Functions 133 HARRIs Algebraic Geometry: A First Course 134 ROMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKINsIWEINTRAUB Algebra: An Approach via Module Theory 137 AXLERIBoURDON/RAMEY Harmonic Function Theory 2nd ed 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERIWEISPFENNING/KREDEL Griibner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNIS/FARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K- Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 EISENBUD Commutative Algebra with a View Toward Algebraic Geometry 151 SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWN/PEARCY An Introduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAVIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEIN/ERDELYJ Polynomials and Polynomial Inequalities 162 ALPERINIBELL Groups and Representations 163 DIXON/MoRTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 2nd ed 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An Introduction to Knot Theory 176 LEE Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 CLARKEILEDY AEV/STERN/WOLENSKJ Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KREss Numerical Analysis 182 WALTER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOLLOBAS Modem Graph Theory 185 COX/LITTLElO'SHEA Using Algebraic Geometry 186 RAMAKRlSHNANNALENZA Fourier Analysis on Number Fields 187 HARRIS/MORRISON Moduli of Curves 188 GOLDBLATT Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 ESMONDEIMURTY Problems in Algebraic Number Theory 191 LANG Fundamentals of Differential Geometry 192 HIRSCH/LACOMBE Elements of Functional Analysis 193 COHEN Advanced Topics in Computational Number Theory 194 ENGELINAGEL One-Parameter Semigroups for Linear Evolution Equations 195 NATHANSON Elementary Methods in Number Theory 196 OSBORNE Basic Homological Algebra 197 EISENBUD/HARRIS The Geometry of Schemes 198 ROBERT A Course inp-adic Analysis 199 HEDENMALMIKoRENBLUM/ZHU Theory of Bergman Spaces 200 BAO/CHERN/SHEN An Introduction to Riemann-Finsler Geometry 201 HINDRY/SILVERMAN Diophantine Geometry: An Introduction 202 LEE Introduction to Topological Manifolds 203 SAGAN The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Function 2nd ed 204 ESCOFIER Galois Theory 205 FELIx/HALPERIN/THOMAS Rational Homotopy Theory 206 Murty Problems in Analytic Number Theory Readings in Mathematics ... Monoidal Categories The Braid Groups Bn and the Braid Category Braided Coherence Perspectives 17 0 17 4 17 5 18 0 18 4 18 5 18 8 19 1 19 1 19 4 19 8 202 211 211 214 217 218 222 226... for a Monad Split Coequalizers Beck's Theorem Algebras Are T-Algebras Compact Hausdorff Spaces 72 75 76 79 79 86 90 92 95 97 99 10 3 10 5 10 6 10 9 10 9 11 2 11 5 11 6 11 8 12 0 12 6 12 8 13 2 13 7... 11 8 12 0 12 6 12 8 13 2 13 7 13 7 13 9 14 2 14 4 14 7 14 9 15 1 15 6 15 7 VII Monoids 16 1 Monoidal Categories Coherence 16 1 16 5 Contents xi Monoids Actions The Simplicial Category