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Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield localisation of the injective (= Heller) model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ with respect to the Čech hypercovers of $(\mathcal{C}, J)$, or equivalently, the $J$-covering sieves. This is a presentation of the $(\infty, 1)$-category of sheaves of $\infty$-groupoids on $(\mathcal{C}, J)$.

Question. Is there an explicit description of the Čech-local equivalences of (simplicial pre)sheaves on $(\mathcal{C}, J)$, i.e. the weak equivalences in the Čech model structure?

The theory of left Bousfield localisation says that a morphism $f : X \to Y$ is a Čech-local equivalence if and only if the induced morphism $$f^* : \underline{\mathrm{Hom}}(Y, Z) \to \underline{\mathrm{Hom}}(X, Z)$$ is a weak homotopy equivalence of simplicial sets for all Čech-local injective-fibrant simplicial presheaves on $(\mathcal{C}, J)$. In particular, a Čech-local equivalence between Čech-fibrant sheaves on $(\mathcal{C}, J)$ is just a componentwise weak homotopy equivalence.

Although the above is elegant in some sense, I would like to understand what is going on at the level of simplicial presheaves. For contrast, consider the Jardine model structure: there, the weak equivalences are easily described as the morphisms that induce isomorphisms of sheaves of homotopy groups – so this is a notion with eminent geometric meaning. Unfortunately, not every weak equivalence in the Jardine model structure is a Čech-local equivalence. So where is the difference?

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I attacked a related problem over the summer while visiting Carles Casacuberta in Barcelona. We were also working in motivic spectra, but much of our work applies to motivic spaces as well. Instead of the Cech model structure we were looking at the results of localizing the maps seen to be isomorphisms by various motivic homology theories. This led us to studying the difference between maps which are seen to be isomorphisms by the whole homotopy sheaf vs. just by the homotopy sheaf evaluated at the point. –  David White Dec 19 '13 at 15:49
    
(continued) It's related to the question of whether $Hom(f^*,Z)$ is a weak equivalence for $Hom$ the internal hom of the monoidal model category vs. whether $Map(f^*,Z)$ is a weak equivalence for the simplicial mapping space. Some details are in my research statement (which I remember you read earlier this fall), and we hope to have a preprint soon. We'll both be at the MSRI conference next month and my goal is to be done by then. –  David White Dec 19 '13 at 15:50
    
What reference are you using for studying the Cech model structure? For the Jardine? How do you know that not every Jardine weak equivalence is a Cech weak equivalence? What about the converse? I keep thinking the Cech weak equivalences should be the maps which induce isomorphisms on subsheaves or something, but if that was true then there would likely be a Bousfield localization relationship between the Jardine and Cech model structures. –  David White Dec 20 '13 at 18:46
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Dugger, Hollander, and Isaksen in their paper “Hypercovers and simplicial presheaves” state before Corollary A.7 that “It would be interesting to know more about sPre(C)_Č, for instance to have an explicit characterization of the weak equivalences.”. So it seems that such an explicit description was an open problem at the time their paper was published (2004). There are, of course, several examples of nonhypercomplete sites, for example, the above paper has one in Example A.10. –  Dmitri Pavlov Dec 20 '13 at 19:59
    
@DavidWhite The Jardine model structure is indeed a left Bousfield localisation of the Čech model structure. –  Zhen Lin Dec 20 '13 at 22:43
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