An Introduction to the Language of Category Theory

1. Front Matter

   Pages i-xii

   PDF

2. Categories

   • Steven Roman

   Pages 1-35

3. Functors and Natural Transformations

   • Steven Roman

   Pages 37-70

4. Universality

   • Steven Roman

   Pages 71-86

5. Cones and Limits

   • Steven Roman

   Pages 87-117

6. Adjoints

   • Steven Roman

   Pages 119-143

7. Back Matter

   Pages 145-169

   PDF

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.

• Category Theory
• Category
• Functor
• Adjoints
• Yoneda's lemma

“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)

• California State University, Fullerton, Irvine, USA

  Steven Roman

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).

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