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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 7th 2013

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds). The category of simplicial presheaves 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 structures 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 assume we are working in the category of simplicial sheaves, i.e., simplicial objects in the category of sheaves of sets on S.

Presumably the global projective model structure restricts to simplicial sheaves. Is there a reference for this claim? I was only able to find references for the injective local model structure (i.e., the Joyal-Jardine structure), but nothing for the projective case.

My main question concerns descent condition for the case of simplicial sheaves. Assume we have a simplicial sheaf that is globally fibrant. Can we somehow exploit the fact that individual simplicial components are sheaves (and not merely presheaves) to simplify the general descent condition? Or is it just as complicated as the descent for presheaves? (Again we can assume the site to be hypercomplete if it helps.)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 7th 2013

Yeah, this is a natural question. But I am not aware of a projective analog of the injective-style Joyal model structure on simplicial sheaves.

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 28th 2013

I guess what confused me is the claim

“The local model structures on simplicial sheaves are just the restrictions of those on simplicial presheaves.”

It talks about local model structures on simplicial sheaves, even though apparently we only have the local injective model structure on simplicial sheaves.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 29th 2013

I have added the word “injective” in that sentence.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 29th 2013

As it was pointed out to me on MathOverflow http://mathoverflow.net/questions/129002, the projective structure does indeed restrict to sheaves, as explained in this paper: http://math.uiuc.edu/K-theory/0462/combination2.pdf (Theorem 2.1), and the componentwise sheafification functor is a Quillen equivalence (Theorem 2.2).

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 29th 2013
• (edited Apr 29th 2013)

Ah, thanks. I had forgotten about that. (Maybe when I drew the big table at model structure on simplicial presheaves I still knew it.) Have made a corresponding note there now to say this more explicitly.