Let $\mathcal{C}$ be a category with pullbacks, and consider the functor $i_\mathcal{C}: \mathcal{C}^\mathrm{op} \to \mathsf{Cat}$, $X \mapsto \mathcal{C}/X$. We can equip $\mathcal{C}$ with a class $\mathcal{W}$ of "weak equivalences" by setting $\mathcal{W} = i_\mathcal{C}^{-1}(\mathcal{W}_\mathsf{Cat})$, where $\mathcal{W}_\mathsf{Cat}$ is the class of functors that induce weak equivalences of nerves. Cisinski shows that if $\mathcal{C}$ is a (Grothendieck) topos, then there is a model structure on $\mathcal{C}$ with weak equivalences $\mathcal{W}$ and cofibrations the monomorphisms. Moreover, Cisinski proves a conjecture of Grothendieck, saying that if $\mathcal{C}$ is the category of presheaves on a so-called test category, then the homotopy theory of $(\mathcal{C},\mathcal{W})$ is equivalent to the homotopy theory of spaces (for example, the model structure is Quillen equivalent to the usual Quillen model structure on simplicial sets).

But what about Grothendieck toposes which are not presheaf categories on test categories? Cisinski's theory gives us a way to associate a homotopy theory to any Grothendieck topos -- what is that homotopy theory?

I'm basically wondering what is known about the homotopy theory of $(\mathcal{C}, \mathcal{W})$. For example, is it related to the etale homotopy type of $\mathcal{C}$? I might guess that it is some sort of slice category over the etale homotopy type.

Les préfaisceaux comme modèles des types d'homotopieis that the category of presheaves of a small category $A$ is equipped with a model structure such that cofibrations are monomorphisms and the weak equivalences are the morphisms you describe if and only if $A$ is local test (and it always model homotopy types). It is not true in general: think about the category of semi-simplices $\Delta'$,i.e.the subcategory of simplices where you consider only non-decreasing maps. The category of semi-simplicial presheaves do not allow a model structure of that kind. $\endgroup$ – Andrea Gagna Apr 4 '17 at 16:445more comments