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