# An Introduction to the Language of Category Theory

Textbook

Part of the Compact Textbooks in Mathematics book series (CTM)

1. Front Matter
Pages i-xii
2. Steven Roman
Pages 1-35
3. Steven Roman
Pages 37-70
4. Steven Roman
Pages 71-86
5. Steven Roman
Pages 87-117
6. Steven Roman
Pages 119-143
7. Back Matter
Pages 145-169

### Introduction

This textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible.  In writing about this challenging subject, the author has brought to bear all of the experience he has gained in authoring over 30 books in university-level mathematics.

The goal of this book is to present the five major ideas of category theory: categories, functors, natural transformations, universality, and adjoints in as friendly and relaxed a manner as possible while at the same time not sacrificing rigor. These topics are developed in a straightforward, step-by-step manner and are accompanied by numerous examples and exercises, most of which are drawn from abstract algebra.

The first chapter of the book introduces the definitions of category and functor and discusses diagrams,
duality, initial and terminal objects, special types of morphisms, and some special types of categories,
particularly comma categories and hom-set categories.  Chapter 2 is devoted to functors and natural
transformations, concluding with Yoneda's lemma.  Chapter 3 presents the concept of universality and Chapter 4 continues this discussion by exploring cones, limits, and the most common categorical constructions – products, equalizers, pullbacks and exponentials (along with their dual constructions).  The chapter concludes with a theorem on the existence of limits.  Finally, Chapter 5 covers adjoints and adjunctions.

Graduate and advanced undergraduates students in mathematics, computer science, physics, or related fields who need to know or use category theory in their work will find An Introduction to Category Theory to be a concise and accessible resource.  It will be particularly useful for those looking for a more elementary treatment of the topic before tackling more advanced texts.

### Keywords

Category Theory Category Functor Adjoints Yoneda's lemma

#### Authors and affiliations

1. 1.California State University, FullertonIrvineUSA

Steven Roman is Professor Emeritus of Mathematics at California State University Fullerton.  He is the author of numerous other mathematics textbooks, including Field Theory (2006), Advanced Linear Algebra (2008), Fundamentals of Group Theory (2012), Introduction to the Mathematics of Finance (2012), and An Introduction to Catalan Numbers (2015).

### Bibliographic information

• Book Title An Introduction to the Language of Category Theory
• Authors Steven Roman
• Series Title Compact Textbooks in Mathematics
• Series Abbreviated Title Compact Textbooks in Mathematics
• DOI https://doi.org/10.1007/978-3-319-41917-6
• Copyright Information The Author(s) 2017
• Publisher Name Birkhäuser, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-319-41916-9
• eBook ISBN 978-3-319-41917-6
• Series ISSN 2296-4568
• Series E-ISSN 2296-455X
• Edition Number 1
• Number of Pages XII, 169
• Number of Illustrations 171 b/w illustrations, 5 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

“This book offers a fast, but very complete, introduction to the basic concepts in category theory, which any reader with a basic knowledge of abstract algebra will follow easily. … The theory is very well complemented by a list of proposed exercises at the end of every chapter… . This book is appropriate, as was said previously, for a fast introduction to category theory, and could be very useful for a short introductory course on categorical methods in advanced algebra.” (Juan Antonio López-Ramos, Mathematical Reviews, July, 2017)

“This book is, as promised in this series, a compact, easy to read and useful for lecturers introduction to the basic concepts of category theory. It is very convenient for self-studying and it can be used as starting point to read more advanced book on category theory. The book includes very nice and helpful diagrams, detailed explanation of the concepts and, in every chapter, a set of exercises that will help the reader to better understanding the text.” (Blas Torrecillas, zbMATH 1360.18001, 2017)