structures in a cohesive (∞,1)-topos
Since the (∞,1)-category of (∞,1)-sheaves sits by a full and faithful (∞,1)-functor inside presheaves and by assumption that every constant -presheaf is an -sheaf, this implies that we have also natural equivalences
Let be an 1-site such that every object has a split hypercover such that contracting all representables to points yields a weak equivalence. Equivalently, if the colimit functor sends this to a weak equivalence
Then is locally -connected.
It is immediate that we have a Quillen adjunction for the global model structure on simplicial presheaves on both sides. Now by the recognition theorem for simplicial Quillen adjunctions for this to descend to a Quillen adjunction on the local model structure it is sufficient that the left adjoint preserves the cofibrations of the local model structure and (already) that the right adjoint preserves the fibration objects. Since left Bousfield localization of model categories does not change the cofibrations, the first of these is immediate.
This means that to establish the claim it is now sufficient to show that constant simplicial presheaves already satisfy descent for a locally -connected site. This is what we do now.
For a Kan complex let the corresponding constant simplicial presheaf. This is fibrant in . Since every split hypercover is cofibrant, it follows that is an -sheaf precisely if for all and some split hypercover we have that the morphism on derived hom-spaces
is a weak equivalence (of Kan complexes, necessatily). But we have
so that the condition is that
is a weak equivalence. This is the case for all precisely if is contractible, which is precisely our assumption on .
The category of open subsets is not in general a locally -connected site according to the above definition. But there is another site of definition for which is: the full subcategory on the contractible open subsets.
locally connected site / locally ∞-connected (∞,1)-site