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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→C a dense sub-site, the corresponding hypercompleted ∞-sheaf ∞-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→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*):[Dop,sSet]inj,loc←→[Cop,sSet]inj,locIs 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
A→f*f*Ais 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∈C and morphism X→f*f*A there is a cover {Ui→X} and local lifts
Ui→A↓↓X→f*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 {Ui→X}.
Hm…
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