Ana Agore, A First Course in Category Theory

The most recommended introductory books on category theory (at least for pure mathematicians) are probably those by Steve Awodey, Tom Leinster, and Emily Riehl. All three have very considerable virtues. But for differing reasons, each presents quite steep challenges to the beginner (especially for self-study). Having, back in the day, worked through Awodey’s book with a reading-group of super-smart Cambridge Part III (i.e. graduate) students, I can only report that we found it engaging but a much bumpier ride than the author surely intended. Leinster’s shorter book, although my favourite, is often quite compressed and I’m told that students can again find it quite tough for that reason. Riehl’s book is full of good things — her title Category Theory in Context points up that she is particularly seeking to make multiple connections across mathematics. But she goes at pace and the connections made can be distractingly/dauntingly sophisticated.

So there is certainly room on the shelf for another introductory book, especially one advertised as being “unlike traditional category theory books, which can often be overwhelming for beginners. …[It] has been carefully crafted to offer a clear and concise introduction to the subject. … the book is perfectly suited for classroom use in a first introductory course in category theory. Its clear and concise style, coupled with its detailed coverage of key concepts, makes it equally suited for self-study.” So: does Ana Agore’s recently published A First Course in Category Theory (Springer, Dec. 2023) live up to the blurb?

Here’s the very first sentence of Chapter 1: “We start by setting very briefly the set theory model that will be assumed to hold throughout.” Which is garbled English. Quite unsurprising, I’m afraid, from Springer who don’t seem to proof-read their books properly these days. And I do wonder whether Agora has run her text past enough readers including a native speaker or two. For in fact there are quite a few unEnglish sentences. Fortunately, the intended message is only occasionally obscure, at least to this reader who has the advantage of knowing what Agore should be saying. I suspect, however, that some — especially if English is not their first language — may sometimes stumble.

The Preface tells us that the book is based on lecture notes from a graduate course. And that’s how it reads. We get action-packed notes, with a lot of detail given at a relentless pace, and with really very little added motivating classroom chat. The typical approach is to plonk on the table a categorial definition without preliminary scene-setting, and then give a long (sometimes very long) list of examples. And the level of discussion sometimes seems rather misplaced — is it really helpful for the introduction of categorial ideas to be interrupted, as early as p. 7, by an unobvious argument more than a page long to show that epimorphisms in Grp are surjective?

Again as early as p. 12, we are given the categorial definition of a subobject of C as an equivalence class of monics with codomain C. What could motivate pulling that strange-seeming rabbit out of the hat? We aren’t told. Rather, we quickly find ourselves in a discussion of how the definition applies in KHaus vs Top.

Another case: on p. 24 the definition of a functor is served up ‘cold’, followed by thirty-five examples. Or more accurately, we get thirty-five numbered items, but general points (e.g. that functors compose) are jumbled in with particular examples.

All in all, this does read rather like handout-style notes expanded with more proofs written out and with multiple extra examples, but without the connecting tissue of classroom remarks which can give life and direction to it all and which the self-studying reader is surely going to miss rather badly.

What does the book cover? How is it structured?

There are three long chapters. Chapter 1 (82 pp.) is on Categories and Functors, taking us up to the Yoneda Lemma. Chapter 2 (70 pp.) is on Limits and Colimits. Chapter 3 (98 pp.) is on Adjoint Functors. There follows a welcome chapter (26 pp.) of solutions to selected exercises.

But note that although Agore tells us about subobjects early on, we don’t get round to subobject classifiers. We meet limits and colimits galore, but we don’t meet exponentials. And again as contrasted with e.g. Awodey, while of course we get to know about categories of groups and groups as categories, we don’t get to know about groups in categories, internal groups.

In a little more detail, Chapter 1 covers what you would expect, basic definitions and examples of categories, types of arrows and special objects (like initial/terminal objects), functors, natural isomorphisms and natural transformations more generally, hom-functors and representables, ending up with Yoneda. There are some oddities along the way — the idea of elements as arrows from 1 (like the idea of ‘generalized elements) is never mentioned, I think, while the idea of a universal property makes its first appearance on p. 16 but seems never to be given a categorial treatment.

Tom Leinster has written “The level of abstraction in the Yoneda Lemma means that many people find it quite bewildering.” It’s a good test for an introductory book how clear it makes the lemma (in its various forms) and now natural the relevant proofs seem. How does Agore do? Here’s her initial statement.

She then adds that the bijections here, for a start, form a natural transformation in C:

If you are reading this review you are quite likely to know what’s going on. But if you were quite new to the material, I bet — for a start — that these notational choices won’t be maximally helpful, and the ensuing pages of proofs will look significantly messier and harder work than they need to be. So I certainly wouldn’t recommend Agore’s pages 70-77 as my go-to presentation of Yoneda.

Chapter 2 on limits and colimits continues in the same style. So the first definition is of multiproducts (rather than softening us up with binary products first). There’s no initial motivation given: the definition is stated and some theorems proved before we get round to seeing examples of how the definition works out in practice in various categories. We then meet equalizers and pullbacks done in much the same spirit (I don’t suppose anyone will be led astray, by the way, but contrary to her initial definition of a commuting diagram, Agore now starts allowing fork diagrams with non-equal parallel arrows to count as commuting).

On the positive side, I do very much approve of the approach of first talking about limits over diagrams, where a diagram is initially thought of as a graph living in a category, before getting fancy and re-conceptualizing limits as being limits for functors. And if you have already met this material in a less action-packed presentation, this chapter would make useful consolidating material. But, I’d say, don’t start here.

And much the same goes for Chapter 3 on adjunctions, which gets as far as Freyd’s Adjoint Functor Theorem and the Special Adjoint Functor Theorem. This is another rather relentless chapter, but with more than the usual range of examples. Some proofs, such as the proof of RAPL, seem more opaque than they need to be. Again, I wouldn’t recommend anyone starting here: but treated as further reading it could be a useful exercise to work through (depending on your interests and preferred mathematical style).

So the take-home verdict? The book advertises itself as a ‘first course’ and as suitable for self-study. However, I do find it pretty difficult to believe it would work well as both. Yes, I can imagine a long graduate lecture course, with this book on the reading list, as potentially useful back-up reading once the key ideas have been introduced in a more friendly way, with more motivating classroom chat. But for a first encounter with category theory, flying solo? Not so much.

2 thoughts on “Ana Agore, A First Course in Category Theory”

  1. I considered this book a while back and thought that, as well as being too expensive, it didn’t even come close to living up to the blurb. It is not a “gentle approach” — “Designed for students with no previous knowledge of the subject, this book offers a gentle approach to mastering its fundamental principles” — and it is not suitable for self-study.

    After reading enough blurbs, one learns to translate. “Suitable for self-study” often means no more than “has solutions to some exercises”, rather than being suitable in any other sense. “Basic”, which isn’t used for this book, is stretched to cover quite advanced books (such as Levi’s Basic Set Theory and Jacobson’s Basic Algebra I and II). With “gentle”, though, I’m still wondering what it’s being used to mean. How on earth is this book’s approach “gentle”?

  2. I’ve been studying Category theory for a little over a year, following the study guide from this website. What surprises me is that Goldblatt’s “Topoi” is not much mentioned on the web, but it has proven invaluable to me (apart from Professor Smith’s own notes, of course), as it is much more “gentle” (i.e. gradual) than what is typically recommended out there. Even Riehl’s and Leinster’s introductory books seemed to me to go way too fast, introducing functors and natural transformations in the first couple of chapters. Whereas Goldblatt doesn’t introduce them until chapter 9, when we’ve reached a sufficient level of abstraction and ability to work with “chasing arrows”.

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