Category theory, in its modern sense (well, there’s really no older sense, properly speaking, as it’s a quintessentially young theory: Eilenberg, MacLane, then Grothendieck, Lawvere, &c.), is at one and the same time an autonomous part of mathematics and a big part of the arsenal of half a dozen other mathematical specialties. So it is, for example, that on the one hand there are true professional category theorists, just like there are group theorists and ring theorists. On the other hand, Grothendieck’s own Homeric contribution to the field occurred in the larger context of his homological algebra and algebraic geometry, and this very readily spilled over into algebraic topology (replete with a complementary back-flow), and, transitively, as it were, into such disparate areas as arithmetic geometry and quantum field theory —just witness what Alain Connes has wrought, for example, or consider that huge and idiosyncratic work, Quantum Fields and Strings, a Course for Mathematicians, whose editors include both Pierre Deligne and Ed Witten: the book’s first part deals with quantum fields and super symmetry, but, contained therein, its opening chapter, titled “Multilinear Algebra,” has its second section titled “Categorical Approach.” There is no doubt about it: what Grothendieck himself used to call “abstract nonsense” is everywhere, even in avant garde quantum physics.
This said, perhaps the most natural place for almost everyone to meet category theory for the first time is a course in algebraic topology, although for number theorists like me it’s more often than not algebraic geometry, but I guess we are a minority. The point is that the categorical method most naturally fits into a topological context; indeed category theory’s pioneers were generally algebraic topologists such as Saunders Mac Lane, Henri Cartan and Samuel Eilenberg, and so its original habitat, and still a favored dwelling place, is topology: e.g., the objects are topological spaces and the morphisms are continuous mappings. However, category theory is, by design, much more abstract than that, of course, and one manifestation of this great degree of abstraction is Grothendieck’s notion of a topos, the focus of the book under review, and a category theoretical notion par excellence.
Well, what is a topos? Here is a fabulous and definitive answer, written for the non-expert by Luc Illusie, one of Grothendieck’s pupils: http://www.ams.org/notices/200409/what-is-illusie.pdf (and see the article http://www.ams.org/notices/201009/rtx100901106p.pdf for more about the environment in which all this marvelous business was transacted). In a shorter phrasing, and going to the book we are concerned with, here is what Johnstone says about topoi (= plural of “topos”) on p. xii of his book: “Giraud … showed that the categories of generalized sheaves which arise [in order to develop a framework for étale cohomology] can be completely characterized by exactness properties and size conditions; in light of this result, it became apparent that these categories of sheaves were a more important subject of study than the sites (= categories + topologies) which gave rise to them. In view of this, and because a category with a topology was seen as a “generalized topological space,” the (slightly unfortunate) name topos was given to any category satisfying Giraud’s axioms. So there.
But where in Johnstone’s book do we find Giraud’s axioms? Although the name, Jean Giraud, does appear in the book’s index of names, the topic of his axioms does not even appear in the index of definitions. What’s going on? Well, if we read further in the book’s introduction, we encounter the following: “… the full impact of the dictum that ‘the topos is more important than the site’ seems never to have been appropriated by the Grothendieck school … It was, therefore, necessary that a second line of development should provide the impetus for the elementary theory of toposes [= topoi, I guess — see below] … The starting point for this second line is generally taken to be F. W. Lawvere’s pioneering 1964 paper on the elementary theory of the category of sets. However I [= Johnstone] believe that it is necessary to go back a little further, to the proof of the Lubkin-Heron-Freyd-Mitchell embedding theorem for abelian categories …” Again, so there — or maybe: “Say what?!”.
Well, you get the idea: Johnstone’s Topos Theory is categorically not for the faint-hearted. It is truly insider stuff written by an expert. Johnstone is additionally a topology student of J. Frank Adams and this latter point resonates in many frequencies, including that of humor: Adams was known for his mots justes, and so is Johnstone; for example, his titanic compendium on topos theory is titled Sketches of an Elephant and on p. ix of this opus Johnstone himself quotes an anonymous referee as saying that it is “far too hard to read and not for the faint-hearted.” I guess one concludes that topos theory per se is not for the faint-hearted.
And that is really the by-word for the current book, which is also unquestionably excellent and thorough, even if it’s not composed on the same scale as its pachydermatic relative. It is still encyclopaedic in its own way, as the table of contents reveals. I must admit that any book that sports the chapters “topologies and sheaves” (Chapter 3) and “cohomology” (Chapter 8) has little trouble winning my heart. But there is so much more there, that it should be smash with many others (not too faint-hearted, of course): Deligne appears in Chapter 7, the continuum hypothesis in Chapter 9, and so on.
Furthermore, as already indicated, the scholarship is outstanding, and there’s British humour, too. Consider, e.g., the closing section of the Introduction: “The reader will already have observed that I use the English plural [toposes]: I do so because (in its mathematical sense) the word topos is not a direct derivative of its Greek root, but rather a back-formation from topology. I have nothing further to say on the matter, except to ask those toposophers [!] who persist in talking about topoi whether, when they go out for a ramble on a cold day, they carry supplies of hot tea with them in thermoi.”
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.