Category Theory
by Awodey, SteveEdition: 2nd
Format: Hardcover
Pub. Date: 2010-08-13
Publisher(s): Oxford University Press
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Summary
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, philosoph
Author Biography
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.
Table of Contents
| 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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