Basic Category Theory

Category theory is a relatively young subject whose birth is traced accurately to a single 1942 article by Saunders Mac Lane and Samuel Eilenberg. While working in algebraic topology, the duo was led to define categories, functors, and natural transformations, laying down the foundations to what they could not know would be the beginning of a marked success story as well as intense controversy.

Much can be said about the early years of category theory and I will keep it brief by considering just two anecdotes. One is the unusual title of Mac Lane’s book Categories for the Working Mathematician, which still serves as an excellent account of category theory. The need to emphasise the working mathematician (as opposed to the idly philosophising amateur mathematician) as the intended readership stems from the then quite prevalent attitude that category theory was just general abstract nonsense. The second anecdote is a personal one. When I was a graduate student at the Hebrew University and was in search of a topic to dig into, I came across Mac Lane’s book and read it with much fascination. I was seeking the advice of several professors at the time and asked one of them, a highly prominent algebraist who shall remain nameless, “what about category theory?”. The response was equally fascinating as the subject itself:

Well… it’s very abstract, and sort of applies to everything, and kind of too general, and well, are you hooked already? Because you may want to consider other areas, and well, I can’t say that I recommend it. But on the other hand, you can’t argue with success!

I certainly had no desire to argue with success and while I did not become an expert in category theory, I use it with delight all the time. Luckily, the controversy seems to have subsided and the area enjoys a very active community of researchers and numerous applications throughout mathematics. A student contemplating learning category theory today should not find it hard to identify supportive reasons. In fact, unless one has a clear answer to why not learn category theory, one probably should learn it.

There are many reasons to wish to learn category theory. One of them is that it is a convenient language in which almost all of mathematics can be discussed. This is a point of view that likens category theory to naive set theory. And indeed, rudimentary set theory is considered something everybody should know, just so that we all speak the same underlying language when discussing groups, rings, fields, functions, etc. If category theoretic terminology is making its way into the day-to-day jargon mathematicians use, it is probably a good idea to learn the basics.

Unlike set theory, category theory, even at this very fundamental level as “just” a language, has many extremely useful theorems. In a sense, the book under review, aptly titled Basic Category Theory, is an introduction to the language of category theory and to its usefulness in a very broad sense. The author spares no effort at making the book accessible to all students and explains very well how the highly abstract machinery translates into useful labour-saving devices. The reader completely unfamiliar with categories, or one with a passing knowledge of the meaning of the common terms used, will be taken through a journey equipping the reader with a wonderful tool-set for solving problems, organising one’s thoughts, and fruitfully directing one’s energies along the quest to bring order and enlightenment to complex problems.

The book is quite different than other category theory books. It aims at being a first introduction to category theory, requiring essentially no prior knowledge of any subject. This is quite a challenging expository decision, and one the author tackles masterfully. The reader will find illuminating examples carefully chosen to allow one to concentrate on the categorical ideas with minimal distraction by other detail. The order in which some of the most fundamental concepts of category theory is presented is a refreshing change from the usual route. In fact, the book could have been titled “An introduction to adjoint functors”. Adjunctions are the main focus of the book (a good choice for a target concept for such a text as the famous saying “adjunctions are everywhere” indicates the ubiquity of adjunctions). Thus, categories, functors and natural transformation are treated, and then adjunctions take the prominent role on the stage. Limits, colimits, monics, epics, etc. are treated, but later on, and all in conjunction with adjunctions.

Category theory is certainly highly abstract and sometimes a bit technical. The book does a very good job in balancing the abstract with concrete applications, and the technicalities are treated gradually and, as much as possibly, beautifully. After presenting the basic terminology the book can (somewhat inaccurately) be considered as a geodesic tour in the landscape of useful and elegant mathematics towards the general adjoint functor theorem. That does not mean that the author strives to prove that theorem as quickly as possible, but rather that the journey is designed to prepare the reader to truly appreciate and enjoy the theorem, its proof, and its consequences. Along the way, limits and colimits are treated, representability is discussed, presheaf categories are met, as well as an elegant discussion of set theory. When the going gets rough, the author skillfully eliminates the technicalities by exploiting ordered sets, before giving the full details.

To summarise, category theory is becoming increasingly more important and prevalent, even if just as a language. Being just a language should not be taken lightly (and category theory is much more than just a language). After all, group theory is just a language to speak of symmetry, the \(\varepsilon - \delta\) formalism is just a language to speak of limiting processes, and set theory is just a language to speak of the most common things all other mathematical structures are built upon. So, category theory, even if viewed just as a language, is a language to speak of virtually all mathematical structures and the relationships between them.

The fundamental concept of adjunction and some of the basic theorems laid down extremely accessibly in this well-written book, coupled with more than a few well chosen exercises, will endow the reader with a valuable ability to tackle many problems throughout mathematics and will force the reader in more than one place in the text to think, re-think, and perhaps re-mould some of her current understandings. I can strongly recommend the book to all students who cannot yet prove Yoneda’s lemma in their sleep. Students who either have no knowledge of category theory and seek a very gentle introduction, or students from any discipline who already encountered categories but found the abstractness to be an obstacle and thus seek a text that takes that it easy, will find the book invaluable. More advanced students, with existing knowledge of category theory are highly likely to enjoy the book greatly and are surely to learn a thing or two from the author’s personal perspective of the theory.

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.