Categories for the Working Mathematician

Category Theory, developed by Saunders Mac Lane and Samuel Eilenberg around 1942 in the context of algebraic topology, is a highly abstract theory which is (fondly or mockingly, depending on one’s point-of-view) referred to as general abstract nonsense. The reviewer belongs to the set of people considering general abstract nonsense positively.

The book under review is an introduction to the theory of categories which, as the title suggests, is addressed to the (no-nonsense) working mathematician, thus presenting the ideas and concepts of Category Theory in a broad context of mainstream examples (primarily from algebra). With numerous applications of Category Theory in homotopy theory, logic, computer science, and other areas of mathematics and physics, the book’s secondary goal of portraying Category Theory in a more positive light can be said to have been achieved. Among several excellent introductory level textbooks on the subject, the book remains an authoritative source on the foundations of the theory and an accessible first introduction to categories. The rest of this review touches briefly on the topics covered in each chapter, attempting to further illustrate their relevance to the working mathematician.

Chapter 1 presents the basic objects of study of Category Theory, namely categories, functors, and natural transformations. Many constructions in mathematics are preceded with the adjective “natural,” typically referring to a certain independence from non-canonical choices, but otherwise leaving the precise meaning of the naturality of the construction to the reader’s fantasy. Historically, categories were invented to be able to speak of functors, and functors were created to be able to speak of natural transformations. After reading this chapter the reader will thus have a precise definition for the naturality of a construction, one that applies to an almost endless supply of examples throughout mathematics.

Chapter 2 is an algebraic treatment of categories viewed as algebraic structures. Loosely speaking, a category is a set (or class) together with extra structure. The general paradigm of modern algebra, i.e., the construction of new objects from old ones by means of products or function spaces etc., is applied to categories.

Chapter 3 presents the pivotal concepts of universality and limits in a category. These notions give category theory its unique flavour, illustrating how numerous concepts in mathematics (e.g., cartesian products and free objects) can be captured uniformly just by diagrams of objects and arrows. Chapter 4 presents yet another fundamental notion, that of adjunctions (which are famously said to be found everywhere). Again, it is shown that a single categorical concept encompasses a vast array of constructions and arguments throughout mathematics, and in particular adjunctions unify concepts in category theory itself.

Chapter 5 is a more in-depth view of general limits and colimits and Chapter 6 presents monads. Monads can be seen as a categorical machinery that allows one to codify various algebraic structures, for instance monoids. In this way, general abstract nonsense can produce quite a lot of standard algebraic constructions. Chapter 7 then shows how monoids can be interpreted inside a category, provided the category is a monoidal category, i.e., a category equipped with a binary operation which satisfies what is known as coherence. Coherence manifests itself at almost any area of study. For instance, in set theory, the reason that nobody ever got into trouble for identifying the sets \((A\times B)\times C\) and \(A\times(B\times C)\) is because the cartesian product of sets is coherent.

Chapter 8 treats abelian categories, an important class of categories with extra structure. Chapter 9 is a more advanced view of limits, describing special constructions. Chapter 10 presents Kan extensions, whose importance is attested to by the final section’s title “All Concepts Are Kan Extensions.” Chapter 11 presents more advanced topics on coherence.

Finally, Chapter 12 is a glance at higher dimensional categories, and in particular the two-dimensional nature of the category of categories. This chapter relates nicely back to the first chapter and to the seminal idea that categories exist for the sake of functors, and functors exist for the sake of natural transformations.

This book is an excellent introduction to category theory on a rather broad spectrum. It is very well-written, with plenty of interesting discussions and stimulating exercises. The most stringent requirement of the reader is that elusive sufficient level of maturity expected after completing an undergraduate degree in mathematics. The formal requirements for following most of the examples are actually quite rudimentary.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.