Book:Steve Awodey/Category Theory

Subject Matter

 * Category Theory

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