Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?

Question

Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the system of local isomorphisms generated by the Grothendieck topology (one way to describe these is as the morphisms that become isomorphisms under sheafification, but there are other, more direct methods to deduce what they are).

Recall that given any pair $(C,W)$ of a category and a class of morphisms, we can apply any of the methods of Dwyer-Kan simplicial localization to enrich $C$ to a simplicial category such that $\pi_0$ of the function complexes are exactly the hom-sets in the classical localization of the category.

Hammock localization is probably the best choice in this case, since local isomorphisms arising from a Grothendieck topology admit a calculus of left fractions, which means that the resulting function complexes are exactly the nerves of certain cospan categories.

Anyhow, is there any interesting descent-related information contained in the higher homotopical data of the function complexes? Is there any relation between this sort of thing and cohomological descent?

Answer 1

The function complexes will have no higher homotopy: they will be weakly equivalent to discrete sets, and so the simplicial localization will be DK-equivalent, as a simplicial category, to the category of sheaves regarded as a locally discrete simplicial category. This is because

1. Every presheaf is locally isomorphic to a sheaf, so the simplicial localization will be DK-equivalent to its full subcategory on the sheaves, and
2. Between sheaves, all local isomorphisms are already isomorphisms, so there is no further localization to do.