Discussing "Categories for the Working Mathematician"

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Author: Saunders Mac Lane (University of Chicago)

Reference: Mac Lane, Saunders. Categories for the working mathematician. Second Edition. Graduate Texts in Mathematics Vol. 5. Springer-Verlag, 1998 (or for the First Edition, 1971).

Why this paper? Cited by Under Lock and Key: A Proof System for a Multimodal Logic, Groupoid-Valued Presheaf Models of Univalent Type Theory, and KZ-monads and Kan Injectivity

When I started my research career with a Master’s degree at Victoria University of Wellington under the supervision of Rob Goldblatt, this book was not quite where I got started with the fascinating world of category theory (that was Rob’s own wonderful, if somewhat idiosyncratic, Topoi), but was the working reference book I quickly moved to. When Rob and I published my first ever paper (a nice, if not exactly earth-shattering, exploration of collections of coalgebras) we cited this book for some standard definitions and results in category theory we were relying on. This was and remains a typical way to use this book – it has such canonical status that if something in one’s paper relies on something in the book, one can always safely leave its definition to a citation to Mac Lane on the understanding that reviewers and readers should have access to it, preferably in a prominent place on their bookshelf. While I have bought and acquired various reference books over the years, it remains the only one I have ever requested to receive as a Christmas present (thank you to my sister-in-law!). Its iconic status can also be seen from the fact that it is the only abstract mathematics book that I am aware of to have launched a meme, via the quote “All told, a monad in X is just a monoid in the category of endofunctors of X”:
 

Sources: 1 2 3. The word ‘just’ does a lot of work in making this quote funny. Indeed the full exposition about monoids in a category doesn’t come until the chapter after the infamous quote, so the reader could be forgiven for being a little perplexed, let alone the poor functional programmer who just wants to deal with side effects.

Mac Lane was one of the inventors (or discoverers, depending on how one feels about the ontological status of mathematics) of category theory, along with Samuel Eilenberg in 1942. There is no particular rule that the inventor of a discipline will be its best communicator, but surely it doesn’t hurt that Mac Lane maintained an overview of the field from its infancy. The original work was motivated by an attempt to better understand the notion of ‘limit’ in algebraic topology. This led them to formalise the previously informal notion of natural transformation; to get to this, one must first define functor, and to get to that, one must define category. Even though category theory can enter deep and difficult waters, and has a reputation for being hard, the definition of category is startlingly simple and general – a collection of objects, and a collection of arrows between them, obeying identity and associativity rules.

Category theory was “initially… used chiefly as a language” in Mac Lane’s telling, that is a formal language of objects and arrows, from which one can translate back and forth to the sets and elements of ‘normal’ mathematics. It still has that feeling to some extent, as learning category theory after being taught mathematics via sets can indeed feel like learning a second language as an adult. It is often claimed, correctly I think, that a second language can enrich your thinking beyond one’s ability to communicate with a different group of people, because it helps you to organise your thoughts in different ways and hence see the world in different ways. So it certainly is with category theory; armed with the basic definitions, and the knowledge that much of mathematics can be naturally viewed via objects and arrows (typically, some notion of homomorphism or structure-preserving function) one can develop category theory as a topic in its own right, bringing forward new fundamental concepts such as adjunction that do not obviously arise as fundamental when one is thinking in a different mathematical language, but once discovered can be widely and fruitfully applied. The analogy with human languages should not be overdone, of course, because for the most part human languages must be compatible with a variety of viewpoints and personalities, whereas a mathematical language can more strongly bias one to emphasise certain ways of looking at the world. With category theory, one is invited to view the mathematical world in terms of arrows (indeed, Mac Lane shows that one can formulate it without reference to objects at all), so transformations, rather than static elements, become the basis of thought and practice.

This might not be the best book to learn category theory from scratch (I haven’t done a survey of all options, but many speak highly of Awodey’s Category Theory), but once one has the basics down, it gives a clear and readable exposition of many important concepts. It is not impressive in its size and weight in the same manner as Johnstone’s Elephant, but the judicious selection of material for a relatively slim volume is impressive in its own way. It certainly still lives up to its billing as a valuable tool for the working mathematician (or, indeed, theoretical computer scientist) to consult when authors deal with some fundamental piece of category theory by citation.