An Introduction to Category Theory

by Steven Roman

The purpose of this book is to provide an introduction to the basic language of category theory. It is intended for the graduate student, advanced undergraduate student, non specialist mathematician or scientist working in a need-to-know area. The treatment is abstract in nature, with examples drawn mainly from abstract algebra.

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Free Chapter

I have made Chapter 1 of the book available as a free download. Just click here to download.

YouTube Videos

I have done six YouTube videos based on the book. They are not as comprehensive as the book, but will give you an idea of my approach to learning category theory. Here is a sampling.

Lecture 1
Lecture 2

Table of Contents

• Preface
• Motivation
• The Five Main Concepts of Category Theory
• Coverage
• Thanks
• Chapter 1: Categories
• Foundations
• The Definition
• Functors
• The Category of All Categories
• Concrete Categories
• Subcategories
• Diagrams
• Special Types of Morphisms
• Initial, Terminal and Zero Objects
• Duality
• New Categories From Old Categories
• Exercises
• Chapter 2: Functors and Natural Transformations
• Examples of Functors
• Morphisms of Functors: Natural Transformations
• Functor Categories
• The Category of Diagrams
• Natural Equivalence
• Yoneda's Embedding
• Yoneda's Lemma
• Exercises
• Chapter 3: Universality
• The Universal Mapping Property
• The Mediating Morphism Maps
• Examples
• The Importance of Universality
• Uniqueness of Universal Objects
• Couniversality
• A Look Ahead
• Exercises
• Chapter 4: Cones and Limits
• Cones and Cocones
• Cone and Cocone Categories
• Terminal Cones and Couniversality
• Any Category Is a Cone Category: Objects are One-Legged Cones
• Limits and Colimits
• Categorical Constructions
• Equalizers and Coequalizers
• Products and Coproducts
• Pullbacks and Pushouts
• Exponentials
• Existence of Limits
• Exercises
• Chapter 5: Adjoints
• Universal Families
• Left-Adjoint Structures
• Adjunctions
• Right-Adjoints
• Units and Counits
• Summary, or How to Define an Adjoint Structure
• Uniqueness of Adjoints
• Examples of Adjoints
• Adjoints and the Preservation of Limits
• The Existence of Adjoints
• Exercises
• References
• Index of Symbols
• Index

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