I bought Steve Awodey’s book *Category Theory* (Oxford Logic Guides, Clarendon Press, 2006) when it first came out. Awodey says that his book is aimed, inter alia, at “researchers and students” in philosophy; I’d been impressed and intrigued by a couple of his lucid contributions to *Philosophia Mathematica*, and had hoped for an equally approachable book. But, whatever *Category Theory*‘s virtues, easy approachability isn’t one of them, and after reading a fair bit of it, I had to put the book aside for when I had enough time to work through it again more slowly. At last, I’ve got back to it, and I’ll give some reactions here.

I have to say immediately (as in fact I said here before) that I can’t imagine that there are many philosophers who would be equipped to dive straight in and cope with this book. Meeting Cayley’s Theorem (about representing groups as permutations on sets) at p. 11 or free monoids at p. 16 is going to be quite a challenge to those without a background in mathematics. It isn’t that those ideas are intrinsically very difficult; but you surely won’t grasp their *point* or feel comfortable with the ideas just from their brisk presentations here. Likewise, I bet no one will understand Remark 1.7 (p. 12) on concrete categories who hasn’t already met the idea of “test objects” from elsewhere. By the time the reader gets to the first example of a “universal mapping property” at pp. 17-18, most philosophers surely will be floundering: Awodey’s explanations of what is going on are too terse to help the not-so-mathematical. And things seem only to get worse as the book progresses. I’m pretty sure, then, that this book wouldn’t work as a first introduction to category theory e.g. for philosophy graduate students interested in logic and the philosophy of maths (unless they have an unusually strong background in pure maths already). Although Awodey says in the preface that, if Mac Lane’s book is for mathematicians, his is for ‘everyone else’, in fact *Category Theory* is actually orientated to students who are, as they say, ‘mathematically mature’.

So, from now on, I’ll be taking the book as in fact operating at (so to speak) a level up from the one Awodey says that it is designed for, i.e. as a follow-up text for mathematically ept readers, to read after mastering e.g Lawvere and Rosebrugh’s *Sets for Mathematicians* — a follow-up which starts again from scratch to consolidate some basic ideas and then pushes things on deeper and further.

How does the introductory first chapter work on *this* level? Well, to be frank, still not entirely brilliantly. For example, the whys and wherefores of the first example of a universal mapping property are not really explained that well (nor why we should be particularly interested in free categories). However, on the other side, I like the way that the idea of a functor between categories is introduced early; and some of the illustrative examples of categories and functors between categories in the chapter are illuminating. And the idea of “forgetful functors” comes across nicely.

Bryan HannHave you looked at the book Conceptual Mathematics by Lawvere and Schanuel? It is ostensibly aimed at the High School level.

Peter SmithYes, and I mention it here

Andrew L.I agree Awodey’s book is probably far too difficult for most philosophy students unless they specialize in analytic philosophy and have the corresponding logic and/or mathematics background. That being said-I think it’s the absolutely perfect introduction for undergraduate science majors,especially undergraduates in mathematics,physics or computer science. MacLane’s book would shell-shock all but the most gifted undergraduates and for a long time, we’ve really needed a book like this to soften the blow.

Peter SmithActually, I had occasion a few weeks ago to re-read the opening chapters of Awodey and I was wondering if my earlier reaction was too harsh …

Paolo G. GiarrussoI also felt that the book is not written for the described target. I’d classify myself as a mathematically mature computer scientist, and still struggled following quite a few steps. Some of the notation is not even really introduced (in Ch. 2, he starts writing the composition of f and g as fg — he only introduced that earlier for free monoids, where it has a different meaning).

Having said that, Awodey’s book was the first one where I understood the point of many definitions. I had tried before Pierce’s book (bad) and Barr&Wells (better, but still not as vivid). What’s the point, say, of doing representation theory (e.g. with representable functors) (one of the applications of category theory, with the Yoneda embedding)? It’s doing something like Cayley’s theorem, only more general. (I had never seen Cayley’s theorem but I am familiar with quite some group theory).

A disclaimer: I’ve heard some commenters write that category theory only clicks after reading on it a few times/on a few different books, and this does sound plausible to me.

Peter SmithI’m planning next term to run a reading group on Awodey’s book, for people who have already done the Part III Category Theory course this term here in Cambridge (last year’s excellent course notes by Julia Goedecke are here and give a good idea of this year’s coverage too). I think that reading Awodey should really help understanding of the basics on a second pass through some of the material. But I still think it isn’t the place to

start.Peter SmithBoth Lawvere/Rosebrugh and Goldblatt certainly develop enough general category theoretic ideas to make very good introductions.

AnonymousThanks for your recommendation. I’d ruled out Lawvere and Rosebrugh’s

Sets for Mathematics(SfM) and Goldblatt’sTopoibecause they both appear to beappliedcategory theory, in the following sense. SfM appears to cover category theory only as applied to or limited to sets, and Topoi appears similarly limited to logic. I gather that category theory is much more general. However, maybe category theory proper is so abstract that the best way to learn it is to first learn it as applied to sets or logic. Is that your impression? Indeed, is that one reason why you recommend them?Peter SmithI agree on all counts. The Oxford Logic Guides are wildly overpriced (and some recent volumes are hardly “Guides”). And all praise to Dover. Which reminds me that last year I was trying to encourage them to reprint Gentzen’s Collected Papers. I must find out what, if anything, is happening about that.

a cartesianMany of the Oxford Logic Guides are described as much easier entry points than they are. I can recall a blurb for Shelah’s

Cardinal Arithmeticthat made is sound suitable for someone with only a basic knowledge of set theory. (!)They’re also absurdly overpriced. Awodey’s book lists for $140 in the US!

Dover is very good at producing affordable editions of interesting books, though one has to wait. I think Cohen’s book on forcing and the continuum hypothesis, and Sacks’s

Saturated Model Theory,are appearing later this year. If they eventually get to Drake’sLarge Cardinals, I will be very pleased.Peter SmithI’d recommend Lawvere/Rosebrugh and Goldblatt’s

Topoi(a cheap Dover pbk too).AnonymousHaving said that the book probably wouldn’t work as a first introduction to category theory unless the reader has an unusually strong background in pure maths already, I wonder if you might know of anything you

couldrecommend for that first introduction. I’m looking for something that does for category theory what your Godel book does for his theorem. Everything I have attempted to read was either entirely to superficial or mostly followed the “theorem statement, theorem proof” formula.