Category Theory and Applications: A Textbook for Beginners

I used to believe it was Grothendieck who should be credited with the characterization of category theory as “abstract nonsense,” but that redoubtable source of arcana, Wikipedia, notes that the appellation goes back further, to Steenrod. That said, now some three quarters of a century after the subject’s genesis at the hands of Eilenberg and Mac Lane, it is certainly clear as water that great triumphs in mathematics have come about by virtue of precisely this (less-and-less-)abstract (not-so-)nonsense. I guess it’s fair to say that, at least psychologically if not philosophically, the subject’s quality of abstraction is mitigated by its ubiquity and therefore its familiarity to more and more mathematicians.

It is certainly undeniable that we’re by no means dealing with nonsense: just look at what sub-disciplines rely heavily on category theory (and homological algebra), from algebraic topology through algebraic geometry (in the wake of Grothendieck) to knot theory. I mention the latter in light of, for example, the beautiful 2011 result by Kronheimer and Mrowka that, unlike the vaunted Jones polynomial, Khovanov homology detects the unknot — and Khovanov homology is the so-called categorification of the Jones polynomial, meaning that the Euler characteristic of the Khovanov complex is nothing less than the Jones polynomial. So we can see in this single example a more than sufficient raison d’être for any all things categorical. And there’s so much more: try to imagine sheaf cohomology without the scaffolding afforded by abelian categories (of sheaves) or doing algebraic topology without the notion of homotopy category in place.

All right, then, if there’s ample reason to study category theory, then there’s ample reason for a book like this: it’s for beginners, and it includes applications, including a lot of concrete ones. To start with the latter, yes: Grandis does right by them. His chapters 4, 5, and 6 concern, respectively, applications in algebra, in topology, and in homological algebra. These are pretty beefy applications, including, e.g. varieties of algebras, simplicial sets, and (of course!) additive and abelian categories. And then his 7th chapter is concerned with higher dimensional category theory, which is certainly the right thing to do in view of, for example, what I mentioned above: categorification fits naturally in a framework in which one passes beyond categories, i.e. one goes from sets to categories to 2-categories (cf. p. 247 ff. of Grandis’ book) to 3-categories and so on, to \(n\)-categories. So, without doubt, Grandis is quite up to date with this book: the latter material is pretty avant garde.

The book is laid out very effectively, starting with the obligatory overture dealing with categories, functors, and natural transformations, then hitting limits and colimits, and then going at adjunction and monads with some zest. Throughout, some familiar bells are ringing in the distance — well, for me they’re pretty faint. But, after all, my own exposure to this sort of stuff goes back to my reading Saunders Mac Lane’s Homology way back in the 1980s.

I guess all this attests to how very much alive this subject is, and I am very happy to have Grandis’ book in my collection. It’s well written and is peppered with sections titled, “exercises and complements.” Clearly the reader needs to take these very seriously. I personally find exercises in category theory and homological algebra very satisfying because of their architecture and their minimalist quality: it’s a lot of fun for it all to come down to dancing all over a commutative diagram or two and to draw a lot of arrows. I like the subject a great deal, and I think this is a good book to learn it from.

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