# Book:Steve Awodey/Category Theory

## 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**