Category Theory,9780198568612

Category Theory

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ISBN13: 9780198568612
ISBN10: 0198568614
Format: Hardcover
Pub. Date: 2006-07-27
Publisher(s): Oxford University Press
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Summary

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. Containingclear 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 itassumes 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 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 an Associate 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 vi
1 Categories
1(24)
1.1 Introduction
1(2)
1.2 Functions of sets
3(1)
1.3 Definition of a category
4(1)
1.4 Examples of categories
5(6)
1.5 Isomorphisms
11(2)
1.6 Constructions on categories
13(3)
1.7 Free categories
16(5)
1.8 Foundations: large, small, and locally small
21(2)
1.9 Exercises
23(2)
2 Abstract structures
25(22)
2.1 Epis and monos
25(3)
2.2 Initial and terminal objects
28(1)
2.3 Generalized elements
29(4)
2.4 Sections and retractions
33(1)
2.5 Products
34(2)
2.6 Examples of products
36(5)
2.7 Categories with products
41(1)
2.8 Horn-sets
42(3)
2.9 Exercises
45(2)
3 Duality
47(18)
3.1 The duality principle
47(2)
3.2 Coproducts
49(5)
3.3 Equalizers
54(3)
3.4 Coequalizers
57(6)
3.5 Exercises
63(2)
4 Groups and categories
65(12)
4.1 Groups in a category
65(3)
4.2 The category of groups
68(2)
4.3 Groups as categories
70(3)
4.4 Finitely presented categories
73(1)
4.5 Exercises
74(3)
5 Limits and colimits
77(28)
5.1 Subobjects
77(3)
5.2 Pullbacks
80(4)
5.3 Properties of pullbacks
84(5)
5.4 Limits
89(5)
5.5 Preservation of limits
94(1)
5.6 Colimits
95(7)
5.7 Exercises
102(3)
6 Exponentials
105(20)
6.1 Exponential in a category
105(3)
6.2 Cartesian closed categories
108(5)
6.3 Heyting algebras
113(5)
6.4 Equational definition
118(1)
6.5 A-calculus
119(4)
6.6 Exercises
123(2)
7 Functors and naturality
125(34)
7.1 Category of categories
125(2)
7.2 Representable structure
127(4)
7.3 Stone duality
131(2)
7.4 Naturality
133(2)
7.5 Examples of natural transformations
135(4)
7.6 Exponentials of categories
139(3)
7.7 Functor categories
142(4)
7.8 Equivalence of categories
146(4)
7.9 Examples of equivalence
150(5)
7.10 Exercises
155(4)
8 Categories of diagrams
159(20)
8.1 Set-valued functor categories
159(1)
8.2 The Yoneda embedding
160(2)
8.3 The Yoneda Lemma
162(4)
8.4 Applications of the Yoneda Lemma
166(1)
8.5 Limits in categories of diagrams
167(1)
8.6 Colimits in categories of diagrams
168(4)
8.7 Exponentials in categories of diagrams
172(2)
8.8 Topoi
174(2)
8.9 Exercises
176(3)
9 Adjoints
179(44)
9.1 Preliminary definition
179(4)
9.2 Horn-set definition
183(4)
9.3 Examples of adjoints
187(4)
9.4 Order adjoints
191(2)
9.5 Quantifiers as adjoints
193(4)
9.6 RAPL
197(5)
9.7 Locally Cartesian closed categories
202(8)
9.8 Adjoint functor theorem
210(9)
9.9 Exercises
219(4)
10 Monads and algebras 223(26)
10.1 The triangle identities
223(2)
10.2 Monads and adjoints
225(4)
10.3 Algebras for a monad
229(5)
10.4 Comonads and coalgebras
234(2)
10.5 Algebras for endofunctors
236(8)
10.6 Exercises
244(5)
References 249(2)
Index 251

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