# Is there a local projective model structure on simplicial sheaves? What are its fibrant objects?

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds). The category of simplicial presheaves SPSh(S) on S can be equipped with the local projective model structure given by the left Bousfield localization of the global projective model structure (weak equivalences and fibrations are componentwise) with respect to all hypercovers (for hypercomplete sites Čech covers are sufficient).

Fibrant objects in the resulting model structure are precisely those objects that are fibrant in the global projective model structure (i.e., componentwise fibrant) and satisfy descent with respect to all hypercovers (Čech covers for hypercomplete sites).

Now consider the full subcategory SSh(S) of the category SPSh(S) consisting of simplicial sheaves on S, i.e., simplicial objects in the category of sheaves of sets on S. The local injective model structure restricts to SSh(S), as explained, for example, by Theorem 5 of Jardine's lectures on simplicial presheaves, and yields a Quillen equivalent model category. Theorem 6 in the same lecture notes contains a similar statement for the local projective model structure on simplicial presheaves, but doesn't say anything about simplicial sheaves.

Does the local projective model structure on simplicial presheaves restrict to simplicial sheaves?

The other part of my question concerns the fibrancy condition for the local projective model structure on simplicial presheaves (or sheaves, if the answer to the above question is positive). Assume we have a simplicial sheaf that is fibrant in the global projective structure.

Can we exploit the fact that individual simplicial components of a simplicial sheaf are sheaves (and not merely presheaves) to simplify the fibrancy condition for the local projective structure, or is it just as complicated as for arbitrary presheaves?

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This paper (math.uiuc.edu/K-theory/0462/combination2.pdf) by Benjamin Blander seems to imply that the answer to your first question is 'yes,' and you again get a Quillen equivalent model category, with sheafifying and forgetting being a Quillen equivalence. –  Paul VanKoughnett Apr 28 '13 at 23:35
@Paul: Thanks a lot, the article does answer the first question. Mysteriously enough I missed it while searching, even though it pops up as the first link after nForum/nLab pages when searching for “local projective model structure on simplicial sheaves”. –  Dmitri Pavlov Apr 29 '13 at 8:41
I guess, the answer to your second question should be no. "Homotopical" descent and point-set descent should be unrelated. Indeed, I find the fibrancy condition (apart from possible hypercovers) not very complicated and very natural. Furthermore, I think a simplification of the fibrancy condition in the case of sheaves would have been noticed before; using sheaves instead of presheaves is, indeed, also the way Morel and Voevodsky work (although with the injective model structure). –  Lennart Meier Apr 30 '13 at 19:15