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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 DCD \to C a dense sub-site, the corresponding hypercompleted \infty-sheaf \infty-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:DCf : D \to C 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 *):[D op,sSet] inj,loc[C op,sSet] inj,loc (f^* \dashv f_*) : [D^{op}, sSet]_{inj,loc} \stackrel{\leftarrow}{\to} [C^{op}, sSet]_{inj,loc}

    Is this a Quillen equivalence?

    Since the (f *f *)(f^* \dashv f_*)-counit is the identity, it would be sufficient to check that for all fibrant A[C op,sSet] inj,locA \in [C^{op}, sSet]_{inj,loc} the unit

    Af *f *A A \to f_* f^* A

    is a weak equivalence (we don’t even need to throw in an extra fibrant replacement, since the left adjoint f *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 XCX \in C and morphism Xf *f *AX \to f_* f^* A there is a cover {U iX}\{U_i \to X\} and local lifts

    U i A X f *f *A. \array{ U_i &\to& A \\ \downarrow && \downarrow \\ X &\to& f_* f^* A } \,.

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

    [C op,sSet](X,f *f *A)[C op,sSet](S({U i}),f *f *A) [C^{op}, sSet](X, f_*f^* A) \to [C^{op}, sSet](S(\{U_i\}), f_*f^* A)

    are acyclic fibrations, for S({U i})XS(\{U_i\}) \to X the sieve generated by the {U iX}\{U_i \to X\}.

    Hm…

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