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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 20th 2011
   • (edited Jan 20th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexexpanded 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 $D \to C$ a dense sub-site, the corresponding hypercompleted $\infty$-sheaf $\infty$-toposes are equivalent.

   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 $D \to C$ a dense sub-site, the corresponding hypercompleted $\infty$-sheaf $\infty$-toposes are equivalent.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 20th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs that corollary not true without hypercompletion? That would be surprising to me.

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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 20th 2011
   • (edited Jan 20th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> That would be surprising to me. True. I just went the way of least resistance. So let's see. Let $f : D \to C$ be a dense sub-site. Then by standard arguments ( I have now typed that out <a href="http://nlab.mathforge.org/nlab/show/model+structure+on+simplicial+presheaves#DependencyOnSite">here</a>) we have that restriction and right Kan extension constitute a simplicial Quillen adjunction between the Cech-local injective model structures on simplicial presheaves $$ (f^* \dashv f_*) : [D^{op}, sSet]_{inj,loc} \stackrel{\leftarrow}{\to} [C^{op}, sSet]_{inj,loc} $$ Is this a Quillen equivalence? Since the $(f^* \dashv f_*)$-counit is the identity, it would be sufficient to check that for all fibrant $A \in [C^{op}, sSet]_{inj,loc}$ the unit $$ A \to f_* f^* A $$ is a weak equivalence (we don't even need to throw in an extra fibrant replacement, since the left adjoint $f^*$ 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 $X \in C$ and morphism $X \to f_* f^* A$ there is a cover $\{U_i \to X\}$ and local lifts $$ \array{ U_i &\to& A \\ \downarrow && \downarrow \\ X &\to& f_* f^* A } \,. $$ How do I deduce this? I will need to use that by fibrancy of $A$ we have that the morphisms $$ [C^{op}, sSet](X, f_*f^* A) \to [C^{op}, sSet](S(\{U_i\}), f_*f^* A) $$ are acyclic fibrations, for $S(\{U_i\}) \to X$ the sieve generated by the $\{U_i \to X\}$. Hm...

   That would be surprising to me.

   True. I just went the way of least resistance.

   So let’s see. Let $f : D \to C$ 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

   $$(f^* \dashv f_*) : [D^{op}, sSet]_{inj,loc} \stackrel{\leftarrow}{\to} [C^{op}, sSet]_{inj,loc}$$

   Is this a Quillen equivalence?

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

   $$A \to f_* f^* A$$

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

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

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

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

   are acyclic fibrations, for $S(\{U_i\}) \to X$ the sieve generated by the $\{U_i \to X\}$.

   Hm…