Book:Steve Awodey/Category Theory

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Steve Awodey: Category Theory

Published $2006$, Oxford University Press

ISBN 0-1985-6861-4.

Subject Matter


Contents

Preface

1 Categories

1.1 Introduction
1.2 Functions of sets
1.3 Definition of a category
1.4 Examples of categories
1.5 Isomorphisms
1.6 Constructions on categories
1.7 Free categories
1.8 Foundations: large, small, and locally small
1.9 Exercises

2 Abstract structures

2.1 Epis and monos
2.2 Initial and terminal objects
2.3 Generalized elements
2.4 Sections and retractions
2.5 Products
2.6 Examples of products
2.7 Categories with products
2.8 Hom-sets
2.9 Exercises

3 Duality

3.1 The duality principle
3.2 Coproducts
3.3 Equalizers
3.4 Coequalizers
3.5 Exercises

4 Groups and categories

4.1 Groups in a category
4.2 The category of groups
4.3 Groups as categories
4.4 Finitely presented catgories
4.5 Exercises

5 Limits and colimits

5.1 Subobjects
5.2 Pullbacks
5.3 Properties of pullbacks
5.4 Limits
5.5 Preservation of limits
5.6 Colimits
5.7 Exercises

6 Exponentials

6.1 Exponential in a category
6.2 Cartesian closed categories
6.3 Heyting algebras
6.4 Equational definition
6.5 $\lambda$-calculus
6.6 Exercises

7 Functors and naturality

7.1 Category of categories
7.2 Representable structure
7.3 Stone duality
7.4 Naturality
7.5 Examples of natural transformations
7.6 Exponentials of categories
7.7 Functor categories
7.8 Equivalence of categories
7.9 Examples of equivalence
7.10 Exercises

8 Categories of diagrams

8.1 Set-valued functor categories
8.2 The Yoneda embedding
8.3 The Yoneda Lemma
8.4 Applications of the Yoneda Lemma
8.5 Limits in categories of diagrams
8.6 Colimits in categories of diagrams
8.7 Exponentials in categories of diagrams
8.8 Topoi
8.9 Exercises

9 Adjoints

9.1 Preliminary definition
9.2 Hom-set definition
9.3 Examples of adjoints
9.4 Order adjoints
9.5 Quantifiers as adjoints
9.6 RAPL
9.7 Locally cartesian closed categories
9.8 Adjoint functor theorem
9.9 Exercises

10 Monads and algebras

10.1 The triangle identities
10.2 Monads and adjoints
10.3 Algebras for a monad
10.4 Comonads and coalgebras
10.5 Algebras for endofunctors
10.6 Exercises

References

Index

Further editions