What is the homotopy theory of categories?

I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of his proposed ideas was a "homotopy theory of categories." What does this mean, precisely?

I know that any category corresponds to a simplicial set (its nerve), and an equivalence of categories introduces a homotopy equivalence (in the category of simplicial sets) of the associated nerves. I also know that there is a characterization of (the nerves of) categories among simplicial sets in terms of a unique filler extension condition. If this extension condition is weakened, so that one gets the notion of a quasicategory or $\infty$-category, one can obtain a model structure where the quasicategories are the fibrant objects.

But if we want to just work with ordinary categories, is there a natural model structure on simplicial sets in which they are the fibrant objects? And if so, is this Quillen equivalent to the (Quillen/Serre) model structure on topological spaces?

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Pursuing Stacks' is erroneously called a letter. The letter to Quillen' that is included in the first few pages is not the main point. It is clear that AG thought of the manuscript as a working diary on that project. Again he did not want to find find alternative models as such (see David's excellent reply below), but rather to search for the higher dimensional analogues of Covering Space theory, and to look for a good model of n-categories that would do the job. –  Tim Porter Apr 15 '11 at 6:42
Thanks Tim ;) –  David Roberts Apr 15 '11 at 7:40

I am not knowledgeable enough to have much to say I have not writen in my answer to a previous question of yours, and I think that David Roberts's answer (or, rather immodestly, my previous one) provides what you were looking for as regards your first question. Just a few additional small points:

Pursuing Stacks is not a letter. See Tim Porter's comment.

As regards Grothendieck's opinion of Thomason's model structure, I do not know. Actually, I am unsure he knew of Thomason's model structure when writing Pursuing Stacks [EDIT: see Tim Porter's comment below]. What he knew for sure was that the localization of $Cat$ with respect to classical weak equivalences (functors between small categories the nerve of which are simplicial weak equivalences) is equivalent to the classical homotopy category. The first proof is due to Quillen and Illusie "wrote the details" (his words) in his thesis. (And there is a quite simpler proof, by the way.) Model structures crop up in Pursuing Stacks at some point, but I am pretty sure the idea is not developed in the beginning, which is much more concerned with mere models for homotopy types. Here is a citation from Chapter 75: "the notion of asphericity structure — which, together with the closely related notion of contractibility structure, tentatively dealt with before, and the various "test notions" (e.g. test categories and test functors) seems to me the main payoff so far of our effort to come to a grasp of a general formalism of "homotopy models"." (Beware: these asphericity structures are not what Maltsiniotis called "asphericity structures" in his own work.)

Another fact Grothendieck knew was, of course, Quillen's Theorem A. It seems he did not write a detailed proof of the relative version, but he gave a sketch of a toposic proof of it, though, and took it as an axiom for what he called basic localizer.

As for your second question, I do not know, but it seems to me that Grothendieck was not that interested in simplicial sets and thus did not work extensively with them. In a 1991 letter to Thomason, he wrote: " D’autre part, pour moi le "paradis originel" pour l’algèbre topologique n’est nullement la sempiternelle catégorie ∆∧ semi-simpliciale, si utile soit-elle, et encore moins celle des espaces topologiques (qui l’une et l’autre s’envoient dans la 2-catégorie des topos, qui en est comme une enveloppe commune), mais bien la catégorie Cat des petites catégories, vue avec un œil de géomètre par l’ensemble d’intuition, étonnamment riche, provenant des topos. En effet, les topos ayant comme catégories des faisceaux d’ensembles les C∧ , avec C dans Cat, sont de loin les plus simples des topos connus, et c’est pour l’avoir senti que j’insiste tant sur l’exemple de ces topos ("catégoriques") dans SGA 4 IV". (See here.)

To conclude, let me mention that, if one takes Grothendieck's viewpoint of homotopical algebra, there should exist not only a homotopy theory of categories, but a homotopy theory of $n$-categories. In this respect, there should be a "relative Theorem A" for every $n$, which should allow one to define a workable notion of "basic $n$-localizer". (Actually, this is already done for $n=2$: see this paper by Bullejos and Cegarra for Theorem A.) And then one should work out a theory of test $n$-categories, whose $(n-1)-Cat$-valued presheaves should be models for homotopy types, and so on. To sum up, what Grothendieck wanted to do amounts to giving new foundations for homotopical algebra, and this is still a work in progress.

David Roberts gives the two most useful available references in his answer. If you want to read Grothendieck's words (and in English), just wait for the upcoming annotated version of Pursuing Stacks.

EDIT (2013/10/29): Rereading this answer, I realize that I should add something of which I was not aware at the time of my writing, still regarding Grothendieck's knowledge of Thomason's model category structure (see also Tim Porter's comment and David Roberts's answer). An annotated version of section 69 of Pursuing Stacks is available at http://www.math.jussieu.fr/~maltsin/groth/ps/ps-69.pdf. On page 4, Grothendieck writes that "it appears very doubtful still that (Cat) is a “model category” in Quillen’s sense, in any reasonable way (with W of course as the set of “weak equivalences”". Thus, he was not aware of the existence of Thomason's structure then. See also note 6 on that same page: Grothendieck has learnt of the existence of Thomason's model structure between the writing of Sections 69 and 87.

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Jonathan: you say: 'As regards Grothendieck's opinion of Thomason's model structure, I do not know.' The point was made in a letter to me in the middle of the time when he was writing PS. (The letter was to have been included in Maltsinitios' retyping of the PS correspondence but that was stopped by Grothendieck. He does mention Thomason's work (in fact, he learnt of that at least partially from me and was surprised by the result.) I think he makes some comments about the cofibrations not being that intuitive or geometric. –  Tim Porter Apr 15 '11 at 9:14
Dear Jonathan: thanks! I hadn't heard of Thomason's model structure, and was only really aware of the concrete example of simplicial sets as being a good model for classical homotopy theory. (I'll have to look up Quillen's Theorem A and B, as well.) –  Akhil Mathew Apr 16 '11 at 1:08
Dear Tim: Thanks a lot for your comment. I have edited my answer accordingly. –  Jonathan Chiche Apr 17 '11 at 8:26
Dear Akhil: You're welcome. I had trouble inserting the diagram to state Theorem A, but the relative case is one of the axioms for basic localizers, so you'll find it in Maltsiniotis's and Cisinski's books. –  Jonathan Chiche Apr 17 '11 at 8:28

The homotopy theory of categories is not quite as you envisage it. Really Grothendieck is thinking of the Thomason model structure on $Cat$ (the category of small categories), which is Quillen equivalent to the Quillen model structure on $sSet$ via the nerve functor. Then Grothedieck considered pairs $(Cat,W)$ where $W$ is a class of functors which acted as weak equivalences. This he called a basic localizer (nLab). Grothendieck conjectured, and Cisinski proved, that the class of weak equivalences in the Thomason model structure was the smallest basic localizer.

From there Grothendieck moved to considering pairs $(C,W)$ for any category $C$ and class $W$ of arrows such that $C[W^{-1}]$ was equivalent to the homotopy category of CW-complexes, or even the homotopy category of some basic localizer, and in particular he was interested in when $C = Pre(S) = Cat(S^{op},Set)$, presheaves on some small category $S$. In particular, we know that $S=\Delta$ can be used to recover the homotopy theory of CW-complexes. The question was to characterise those $S$ such that $(Pre(S),W')$, where $W'$ was inherited from a basic localizer (consult Cisinski's or Maltsiniotis' work for details), can be used to model the same homotopy types as $Cat$. Such categories $S$ were called [weak/strict] test categories.

D. Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Astérisque 308 (2006)

and

G. Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque, 301 (2005)

are central resources in this area.

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All the relevant papers can be found on Maltsiniotis' homepage: math.jussieu.fr/~maltsin/ps.html –  Theo Buehler Apr 15 '11 at 5:47
Thanks, Theo. I knew they were available electronically somewhere - just not from Cisinski's home page! –  David Roberts Apr 15 '11 at 6:32
Point of Info: Grothendieck actually did not like Thomason's structure as the cofibrations worried him. –  Tim Porter Apr 15 '11 at 6:44
Ah, that would explain the emphasis on presheaf models, where the cofibrations are taken to be monomorphisms. –  David Roberts Apr 15 '11 at 6:48
I think the emphasis on presheaf models is due to the difficulty of the general case! This is explained in the introduction of Maltsiniotis's book. –  Jonathan Chiche Apr 15 '11 at 8:19
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