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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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 $D \to C$ 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 $f : D \to C$ 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…