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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.

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    $\begingroup$ Since $C/X$ always has a final object $X \xrightarrow{1} X$, its nerve is contractible. As described, all objects of $C$ will be weakly equivalent. In the definition of a test category one considers presheaves $[C^{op}, Set]$ and overcategories only of the form $C/X$. I assume you're asking what is the homotopy theory of presheaves on $C$ for a general $C$ via the above construction. $\endgroup$ – Anton Fetisov Mar 31 '17 at 18:35
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    $\begingroup$ If I'd take a guess I'd say that for moderately good $C$'s the corresponding theory would be the subcategory of spaces generated by spaces of the form $P\otimes N(C/\cdot)$, where $N$ is the nerve, $P: [C^{op}, Set]$, $\otimes$ means the homotopy coend (which can be described just as a homotopy colimit over a different category) and $C/\cdot$ is the covariant functor of overcategory. The problems is that 1). I see no reason why all maps between spaces would be generated this way, in fact often they won't; 2). objects of $C$ may have equivalent nerves, but not connected by a chain of equivs. $\endgroup$ – Anton Fetisov Mar 31 '17 at 19:08
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    $\begingroup$ In fact, if you just consider presheaves of sets on a discrete category $C$, then the described construction is tautological: you get again $[C^{op}, Set]$, which is purely discrete and 1-categorical. I'm not sure how one can formulate a theorem for a general $C$ that would nicely interpolate between a purely homotopy-theoretic and a purely discrete world. $\endgroup$ – Anton Fetisov Mar 31 '17 at 19:10
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    $\begingroup$ @ArturJackson That's what I mean -- or perhaps a refinement along the lines of Bousfield-Friedlander. My terminology is probably nonstandard. I'm pretty sure this is the same as the shape of the topos regarded as a locally discrete $\infty$-topos. $\endgroup$ – Tim Campion Apr 4 '17 at 1:34
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    $\begingroup$ What Cisinski proves in his book Les préfaisceaux comme modèles des types d'homotopie is 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 16:44
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Here is an example.

Consider the truncated category $\Delta_{\le n+1}$. Then presheaves on this with the model structure presents $n$-types.

EDIT: Andrea points out that the relevant Quillen equivalences+proofs can be found in Corollary 9.2.7 of Cisinski's book Les préfaisceaux comme modèles des types d'homotopie.

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  • $\begingroup$ I'm curious if this fits with Anton's comments! $\endgroup$ – Artur Jackson Apr 1 '17 at 5:07
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    $\begingroup$ I don't think I believe this. In general there is a conservative functor (of ∞-categories) $\mathcal C[\mathcal W^{-1}] \to Spaces$ whose image consists of all spaces that are homotopy colimits of $C^{op}$-diagrams of sets. When $C=\Delta_{\leq n}$, these are the spaces of homotopy dimension $\leq n$, and it seems unlikely that there is a conservative functor from $n$-types to spaces of homotopy dimension $\leq n$. What is your reasoning? $\endgroup$ – Marc Hoyois Apr 1 '17 at 15:31
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    $\begingroup$ It is instead $\Delta_{\leqslant n+1}$. Indeed, both the pairs $(i^*, i_*)$ and $(i_!, i^*)$ are Quillen equivalences for the model structures of $n$-equivalences, where here $i\colon \Delta_{\leqslant n+1} \to \Delta$ is the canonical embedding. This is proven, for instance, in Corollary 9.2.7 of Cisinski's book Les préfaisceaux comme modèles des types d'homotopie . $\endgroup$ – Andrea Gagna Apr 4 '17 at 16:56
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    $\begingroup$ I should add that the class of functors you pullback to presheaves categories in this case is not the class of Thomason weak equivalences described in the OP, but instead a further (left Boudfield) localisation. $\endgroup$ – Andrea Gagna Apr 4 '17 at 17:06
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    $\begingroup$ This is interesting, but I'm not sure how enlightening it is. As Andreas points out, in order to get this result, you need to change the basic localizer on $\mathsf{Cat}$ you're using. In fact, you need to change it to the localizer $\mathcal{W}_n$ which presents $n$-types on $\mathsf{Cat}$. With this localizer, I presume that the induced Cisinski model structure for the test category $\Delta$ also presents $n$-types, so it's not surprising that the model structure for the local test category $\Delta_{\leq n+1}$ does too. $\endgroup$ – Tim Campion Apr 5 '17 at 6:36

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