The Joy of Abstraction, Eugenia Cheng, 2023

It's hard to give this book an overall score. It could be a 4 or a 5 out of 5 as an introductory maths text book on category theory, but it fails in a couple of other directions that might make it a 2/5.

As a first introduction to category theory (apart from How to Bake Pi, on which see below) it seems very good. As someone interested in CT I am in the target market, though that might mean I am not qualified to judge.

The book is nicely-written, with clear explanations & motivations, and lots of worked examples. I will definitely be using it as a reference and helper as I journey deeper into CT.

So far, so 4-5/5.

The book's subtitle is "An Exploration of Math, Category Theory, and Life", and it opens almost as a continuation of How to Bake Pi, which had all kinds of links between higher mathematics and everyday life. HtBP is a complete success in this (and is an easy 5/5). However, in JoA, this kind of thing gradually peters out and from about halfway in is gone, apart from occasional references to intersectionality dogma. For most of the book, JoA is purely and simply a (good, nicely-written, etc.) maths text book, aimed at undergraduate students. A lot of non-students buying this book on the back of HtBP will be disappointed.

Almost every time partially-ordered sets are mentioned, illustrative examples are drawn from intersectionality dogma (eg intersecting sets of privilege). The repeated use of CT to formalise this dogma provides a very good example of how formalism can be used to lend credibility, respectability, a kind of veneer of depth, to what is extremely superficial analysis.

Formalism is very much the handmaid to dogma here, and no room is allowed for rational critique (instead we have things like "some poor white men are particularly angry about the theory of privilege").

These two aspects detract from the pleasure of the book overall, but don't detract from its usefulness as a text book.

JoA is about 400 pages, plus appendices etc., and it's in three parts.

The first part, "Building up to categories" (eight chapters, about 100 pages), has as very similar feel to HtBP, discussing relationships, abstraction, patterns, the uses of formalism. It brings us to categories thamselves, with a formal definition.

The second part is "Interlude: a tour of math" (five chapters, about 50 pages). This reviews the familiar ideas and structures we saw in part one, now from a categorial perspective.

JoA covers monoids at pages 125-8, while Lawvere & Schaneul's Conceptual Mathematics doesn't get to them until pages 160-70 (they are similar sized books), so JoA is exploring the space differently, not just more gently.

The third part, "Part two: doing category theory" (eleven chapters, about 230 pages), gets down to the business of being a text book. You'll see from the toc linked above what is covered, but for example functors as mappings between categories are covered and then natural transformations as mappings between functors.

In the last couple of chapters the author gives an overview of her own research interests in higher-dimensional category theory and rounds of the whole book nicely.