Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists!This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study.
Author(s): Steve Awodey
Series: Oxford Logic Guides 52
Edition: 2nd
Publisher: Oxford University Press
Year: 2010
Language: English
Pages: 328
Tags: Математика;Общая алгебра;Теория категорий;
Contents......Page 14
Preface to the second edition......Page 8
Preface......Page 10
1.1 Introduction......Page 18
1.2 Functions of sets......Page 20
1.3 Definition of a category......Page 21
1.4 Examples of categories......Page 22
1.5 Isomorphisms......Page 29
1.6 Constructions on categories......Page 31
1.7 Free categories......Page 35
1.8 Foundations: large, small, and locally small......Page 40
1.9 Exercises......Page 42
2.1 Epis and monos......Page 46
2.2 Initial and terminal objects......Page 50
2.3 Generalized elements......Page 52
2.4 Products......Page 55
2.5 Examples of products......Page 58
2.6 Categories with products......Page 63
2.7 Hom-sets......Page 65
2.8 Exercises......Page 67
3.1 The duality principle......Page 70
3.2 Coproducts......Page 72
3.3 Equalizers......Page 79
3.4 Coequalizers......Page 82
3.5 Exercises......Page 88
4.1 Groups in a category......Page 92
4.2 The category of groups......Page 97
4.3 Groups as categories......Page 100
4.4 Finitely presented categories......Page 102
4.5 Exercises......Page 104
5.1 Subobjects......Page 106
5.2 Pullbacks......Page 108
5.3 Properties of pullbacks......Page 112
5.4 Limits......Page 117
5.5 Preservation of limits......Page 122
5.6 Colimits......Page 125
5.7 Exercises......Page 131
6.1 Exponential in a category......Page 136
6.2 Cartesian closed categories......Page 139
6.3 Heyting algebras......Page 146
6.4 Propositional calculus......Page 148
6.5 Equational definition of CCC......Page 151
6.6 λ-calculus......Page 152
6.7 Variable sets......Page 157
6.8 Exercises......Page 161
7.1 Category of categories......Page 164
7.2 Representable structure......Page 166
7.3 Stone duality......Page 170
7.4 Naturality......Page 172
7.5 Examples of natural transformations......Page 174
7.6 Exponentials of categories......Page 178
7.7 Functor categories......Page 181
7.8 Monoidal categories......Page 185
7.9 Equivalence of categories......Page 188
7.10 Examples of equivalence......Page 192
7.11 Exercises......Page 198
8.1 Set-valued functor categories......Page 202
8.2 The Yoneda embedding......Page 204
8.3 The Yoneda lemma......Page 205
8.4 Applications of the Yoneda lemma......Page 210
8.5 Limits in categories of diagrams......Page 211
8.6 Colimits in categories of diagrams......Page 212
8.7 Exponentials in categories of diagrams......Page 216
8.8 Topoi......Page 218
8.9 Exercises......Page 220
9.1 Preliminary definition......Page 224
9.2 Hom-set definition......Page 228
9.3 Examples of adjoints......Page 232
9.4 Order adjoints......Page 236
9.5 Quantifiers as adjoints......Page 238
9.6 RAPL......Page 242
9.7 Locally cartesian closed categories......Page 248
9.8 Adjoint functor theorem......Page 256
9.9 Exercises......Page 265
10.1 The triangle identities......Page 270
10.2 Monads and adjoints......Page 272
10.3 Algebras for a monad......Page 276
10.4 Comonads and coalgebras......Page 281
10.5 Algebras for endofunctors......Page 283
10.6 Exercises......Page 291
Solutions to selected exercises......Page 296
References......Page 320
