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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:

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:

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.

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. 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 "ashericity 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 realtive 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.

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, 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.

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.

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.