Category Theory

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 other fields in which the ideas are being applied. Containing clear definitions of 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 and methods of category theory understandable to a broad readership.

Steve Awodey studied Mathematics and Philosophy at the University of Marburg (Germany) and the University of Chicago, earning his Ph.D. from Chicago under Saunders Mac Lane in 1997. He is now a Professor in the Department of Philosophy at Carnegie Mellon University. He is an active researcher in Category Theory and Logic, and has authored and co-authored numerous journal articles.

Preface to the second edition	p. vii
Preface	p. ix
Categories	p. 1
Introduction	p. 1
Functions of sets	p. 3
Definition of a category	p. 4
Examples of categories	p. 5
Isomorphisms	p. 12
Constructions on categories	p. 14
Free categories	p. 18
Foundations: large, small, and locally small	p. 23
Exercises	p. 25
Abstract structures	p. 29
Epis and monos	p. 29
Initial and terminal objects	p. 33
Generalized elements	p. 35
Products	p. 38
Examples of products	p. 41
Categories with products	p. 46
Hom-sets	p. 48
Exercises	p. 50
Duality	p. 53
The duality principle	p. 53
Coproducts	p. 55
Equalizers	p. 62
Coequalizers	p. 65
Exercises	p. 71
Groups and categories	p. 75
Groups in a category	p. 75
The category of groups	p. 80
Groups as categories	p. 83
Finitely presented categories	p. 85
Exercises	p. 87
Limits and colimits	p. 89
Subobjects	p. 89
Pullbacks	p. 91
Properties of pullbacks	p. 95
Limits	p. 100
Preservation of limits	p. 105
Colimits	p. 108
Exercises	p. 114
Exponentials	p. 119
Exponential in a category	p. 119
Cartesian closed categories	p. 122
Heyting algebras	p. 129
Propositional calculus	p. 131
Equational definition of CCC	p. 134
¿-calculus	p. 135
Variable sets	p. 140
Exercises	p. 144
Naturality	p. 147
Category of categories	p. 147
Representable structure	p. 149
Stone duality	p. 153
Naturality	p. 155
Examples of natural transformations	p. 157
Exponentials of categories	p. 161
Functor categories	p. 164
Monoidal categories	p. 168
Equivalence of categories	p. 171
Examples of equivalence	p. 175
Exercises	p. 181
Categories of diagrams	p. 185
Set-valued functor categories	p. 185
The Yoneda embedding	p. 187
The Yoneda lemma	p. 188
Applications of the Yoneda lemma	p. 193
Limits in categories of diagrams	p. 194
Colimits in categories of diagrams	p. 195
Exponentials in categories of diagrams	p. 199
Topoi	p. 201
Exercises	p. 203
Adjoints	p. 207
Preliminary definition	p. 207
Hom-set Definition	p. 211
Examples of adjoints	p. 215
Order adjoints	p. 219
Quantifiers as adjoints	p. 221
RAPL	p. 225
Locally cartesian closed categories	p. 231
Adjoint functor theorem	p. 239
Exercises	p. 248
Monads and algebras	p. 253
The triangle identities	p. 253
Monads and adjoints	p. 255
Algebras for a monad	p. 259
Comonads and coalgebras	p. 264
Algebras for endofunctors	p. 266
Exercises	p. 274
Solutions to selected exercises	p. 279
References	p. 303
Index	p. 305
Table of Contents provided by Ingram. All Rights Reserved.

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