## 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.

I am offering a PDF version of the book for \$29.95 (plus tax if you live in California). You can pay through PayPal by clicking the link below. AFTER PAYING, PLEASE CLICK 'RETURN TO MERCHANT' ON THE PAYPAL WEB SITE AND YOU WILL BE TAKEN TO A PAGE WHERE YOU CAN DOWNLOAD THE BOOK.

### Free Chapter

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

• 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
• 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
• Universal Families