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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 22nd 2012

It seems that only now did I come across

• Garth Warner, Homotopical topos theory (pdf)

So I went to record it in the References-section at simplicial sheaf, only to notice that this entry had never existed as a decent entry. I edited just a little for the moment.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeSep 23rd 2012
• (edited Sep 23rd 2012)

By the way, some geometers also say a simplicial sheaf for an appropriate concept of a sheaf over simplicial space or simplicial site. Is this very much against the terminology here or we should take this into consideration to mention. For example, one takes the Borel construction and simplicial sheaves over it. If the structure morphisms between $n$-th degree sheaf component and the pullback of $(n-1)$-st degree sheaf component are all iso, then we get precisely the equivariant sheaves (cf. the book of Bernstein/Lunts, smth around page 34 in my memory) as noted by Deligne as an elementary fact in his Hodge theory paper but in different terminology.

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeFeb 26th 2013

The page simplicial sheaf said

The standard model structure on simplicial presheaves restricts to the standard model structure on simplicial sheaves, this restriction is a Quillen equivalence and equipped with this model structure $SSh(C)$ is one of the standard models for ∞-stack (∞,1)-toposes for the site $C$.

I changed it to refer to the Jardine-local model structure and say that this model structure presents only the hypercomplete $(\infty,1)$-topos, which is what the page model structure on simplicial sheaves says.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 26th 2013

Thanks! I think the entry simplicial sheaf was a little orphaned…