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nLab > Latest Changes: simplicial sheaf

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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 nn-th degree sheaf component and the pullback of (n−1)(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)SSh(C) is one of the standard models for ∞-stack (∞,1)-toposes for the site CC.

    I changed it to refer to the Jardine-local model structure and say that this model structure presents only the hypercomplete (∞,1)(\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…

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 11th 2022

    Added:

    Historically, Joyal constructed his model structure on simplicial sheaves first and Jardine later constructed his model structure on simplicial presheaves.

    References

    The original construction due to Joyal in 1984 is in

    • André Joyal, Lettre d’André Joyal à Alexandre Grothendieck, April 11, 1984, PDF.

    diff, v7, current

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