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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2011
    • (edited Jan 20th 2011)

    expanded and polished the entry model structure on simplicial sheaves (to be distinguished from the one of simplicial pre-sheaves!)

    Made explicit the little corollary that for DC a dense sub-site, the corresponding hypercompleted -sheaf -toposes are equivalent.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJan 20th 2011

    Is that corollary not true without hypercompletion? That would be surprising to me.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2011
    • (edited Jan 20th 2011)

    That would be surprising to me.

    True. I just went the way of least resistance.

    So let’s see. Let f:DC be a dense sub-site. Then by standard arguments ( I have now typed that out here) we have that restriction and right Kan extension constitute a simplicial Quillen adjunction between the Cech-local injective model structures on simplicial presheaves

    (f*f*):[Dop,sSet]inj,loc[Cop,sSet]inj,loc

    Is this a Quillen equivalence?

    Since the (f*f*)-counit is the identity, it would be sufficient to check that for all fibrant A[Cop,sSet]inj,loc the unit

    Af*f*A

    is a weak equivalence (we don’t even need to throw in an extra fibrant replacement, since the left adjoint f* preserves fibrant objects in our case). That’s the analog statement of the last paragraph on p. 547 of the Elephant .

    Now, that looks like it ought to be easy, but I may nevertheless be stuck.

    By Dugger-Hollander-isaksen it would be sufficient to show that for every XC and morphism Xf*f*A there is a cover {UiX} and local lifts

    UiAXf*f*A.

    How do I deduce this? I will need to use that by fibrancy of A we have that the morphisms

    [Cop,sSet](X,f*f*A)[Cop,sSet](S({Ui}),f*f*A)

    are acyclic fibrations, for S({Ui})X the sieve generated by the {UiX}.

    Hm…