This text and reference book on Category Theory, a branch of abstract algebra, is aimed not only at students of Mathematics, but also researchers and students of Computer Science, Logic, Linguistics, Cognitive Science, Philosophy, and any of the other fields that now make use of it. 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 it assumes 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!
Author(s): Steve Awodey
Series: Oxford Logic Guides 49
Edition: 1st
Publisher: Oxford University Press
Year: 2006
Language: English
Pages: 269
CATEGORY THEORY......Page 1
Oxford Logic Guides......Page 3
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface......Page 7
Contents......Page 10
1.1 Introduction......Page 14
1.2 Functions of Sets......Page 16
1.3 Definition of a Category......Page 17
1.4 Examples of Categories......Page 18
1.5 Isomorphisms......Page 24
1.6 Constructions on Categories......Page 26
1.7 Free Categories......Page 29
1.8 Foundations: Large, Small, and Locally Small......Page 34
1.9 Exercises......Page 36
2.1 Epis and Monos......Page 38
2.2 Initial and Terminal Objects......Page 41
2.3 Generalized Elements......Page 42
2.4 Sections and Retractions......Page 46
2.5 Products......Page 47
2.6 Examples of Products......Page 49
2.7 Categories with Products......Page 54
2.8 Hom-sets......Page 55
2.9 Exercises......Page 58
3.1 The Duality Principle......Page 60
3.2 Coproducts......Page 62
3.3 Equalizers......Page 67
3.4 Coequalizers......Page 70
3.5 Exercises......Page 76
4.1 Groups in a Category......Page 78
4.2 The Category of Groups......Page 81
4.3 Groups as Categories......Page 83
4.4 Finitely Presented Categories......Page 86
4.5 Exercises......Page 87
5.1 Subobjects......Page 90
5.2 Pullbacks......Page 93
5.3 Properties of Pullbacks......Page 97
5.4 Limits......Page 102
5.5 Preservation of Limits......Page 107
5.6 Colimits......Page 108
5.7 Exercises......Page 115
6.1 Exponential in a Category......Page 118
6.2 Cartesian Closed Categories......Page 121
6.3 Heyting Algebras......Page 126
6.4 Equational Definition......Page 131
6.5 Lambda-calculus......Page 132
6.6 Exercises......Page 136
7.1 Category of Categories......Page 138
7.2 Representable Structure......Page 140
7.3 Stone Duality......Page 144
7.4 Naturality......Page 146
7.5 Examples of Natural Transformations......Page 148
7.6 Exponentials of Categories......Page 152
7.7 Functor Categories......Page 155
7.8 Equivalence of Categories......Page 159
7.9 Examples of Equivalence......Page 163
7.10 Exercises......Page 168
8.1 Set-valued Functor Categories......Page 172
8.2 The Yoneda Embedding......Page 173
8.3 The Yoneda Lemma......Page 175
8.4 Applications of the Yoneda Lemma......Page 179
8.5 Limits in Categories of Diagrams......Page 180
8.6 Colimits in Categories of Diagrams......Page 181
8.7 Exponentials in Categories of Diagrams......Page 185
8.8 Topoi......Page 187
8.9 Exercises......Page 189
9.1 Preliminary Definition......Page 192
9.2 Hom-set Definition......Page 196
9.3 Examples of Adjoints......Page 200
9.4 Order Adjoints......Page 204
9.5 Quantifiers as Adjoints......Page 206
9.6 RAPL (Right Adjoints Preserve Limits)......Page 210
9.7 Locally Cartesian Closed Categories......Page 215
9.8 Adjoint Functor Theorem......Page 223
9.9 Exercises......Page 232
10.1 The Triangle Identities......Page 236
10.2 Monads and Adjoints......Page 238
10.3 Algebras for a Monad......Page 242
10.4 Comonads and Coalgebras......Page 247
10.5 Algebras for Endofunctors......Page 249
10.6 Exercises......Page 257
References......Page 262
Index......Page 264
