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An Introduction to Category Theory

Harold Simmons
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on

Category theory is one of my favorite things. I guess that, even at this stage of its history, it’s still a lot like Wagner’s operas: you either adore them or you can’t stand them. Well, both regarding category theory and Wagner, I am a big fan, and am happy to have a chance, now, to comment on Simmons’ Introduction to Category Theory. It’s a very good place to get off the ground with this material, which is increasingly present in the development of a host of different branches of modern mathematics, both as a means whereby to phrase things more organically and as a vehicle toward new discoveries. The book under review is unquestionably an introductory text, but this is quite relative in the sense that what was rather exotic in the field a generation ago is today all but everyday fare, and this evolution and subsequent steady state is reflected in the work at hand.

One particularly telling case in point concerns the notions of sheaf and presheaf. Let’s take Saunders Mac Lane’s late 1960s classic Categories for the Working Mathematician as our foil, so to speak. That book, certainly pitched at a higher level than Simmons’s, treats the notion of sheaf in the pre-Grothendieckian style, focusing entirely on sheaves of germs of continuous functions on a topological space as an example of a contravariant functor. By contrast, Simmons’ approach to this material includes treatments of preasheaves, first on posets and then on any category. On the other hand, true to his goal of keeping things at an introductory level, Simmons does not go at sheaves themselves, what with the indicated additional topological trimmings going beyond the intended introductory level.

Introduction to Category Theory certainly squarely addresses in an entirely contemporary idiom a spectrum of needs for this material which none other than Grothendieck himself termed “abstract nonsense.” The story starts where it must, i.e. with the opening chapter titled “Categories,” replete with a caveat in the form of a section headed “An arrow need not be a function,” a pedagogically sound move. But then, and happily, before Simmons gets to “Functors and natural transformations” in his third chapter, he presents, in Chapter 2, something called “Basic gadgetry.” And this is truly a wonderful move: we encounter things like diagram chasing, limits and colimits (part one), initial and final objects, products and coproducts, equalizers and coequalizers, and pullbacks and pushouts, all before the rubber truly hits the road. Incredibly useful in the present context, of course. And then Simmons ends his “gadgetry” with a how-to-section: “Using the opposite category.”

After this setting of the stage and the aforementioned needed discussion on functors in the third chapter, we get to “Limits and colimits in general,” “Adjuctions,” and finally “Posets and monoid sets,” all in pretty short order. The topic of adjunctions is of particular importance vis à vis, e.g., algebraic geometry and algebraic topology (in all their modern ubiquity), and it is worthwhile to highlight some of Simmons’ prose in this connection to bring out his fetching style:

There is a lot going on in adjunctions, and you will probably get confused more than once. You might get things mixed up, forget which way an arrow is supposed to go, forget how to spell cotafurious, and so on. Don’t worry, I’ve been at it for over 40 years and I still can’t remember some of the details. In fact, I don’t try to. You should get yourself to the position where you can recognize that perhaps there is an adjunction somewhere around, but you may not be quite sure where. You can then look up the details. If you ever have to use adjunctions every day, then the details will become second nature to you.

Clearly these are words from an insider and, to use Mac Lane’s phrase, a working mathematician: in this bit of homespun wisdom there resides some serious pedagogical advice, applicable in a far more general context. After all, mathematics is not memorization, it’s a question, instead, of Fingerspitzengefühl, i.e., developing a feel for how a set of themes behaves, and one acquires this only by getting one’s fingers dirty.

Well, what of the nuts and bolts of the book? Simply put, it’s all very well done, and, as already demonstrated, with a suitably light touch. Simmons says that “[t]he material could be developed in 50 pages or so, but here it takes some 220 pages … because there are many examples … [and] over 220 exercises.” Furthermore, “perhaps even more importantly, solutions to these exercises are available online.”

Additionally, Introduction to Category Theory aims at beginning graduate students and is designed to function both as a classroom text, a text for small groups, and even a text for solo study. Given that my own experience of category theory entails self study almost exclusively (something characteristic to my generation, I guess), I find the latter option most evocative: yes, this is a fine book for this purpose: it’s easy to read, detailed, and carefully written by a true veteran. But I’m sure it will do well indeed in a classroom setting, where Simmons has of course market tested the material, especially in a “[s]everal short courses of about 10 hours … using some of the material.” To be sure, it should do a lot of solid service.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

1. Categories
2. Basic gadgetry
3. Functors and natural transformations
4. Limits and colimits in general
5. Adjunctions
6. Posets and monoid sets