(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A site is $\infty$-local if it satisfies sufficient conditions for the (∞,1)-sheaf (∞,1)-topos over it to be a local (∞,1)-topos.
A site $C$ is $\infty$-local if
it has a terminal object $*$;
the limit-functor $\lim_\leftarrow : [C^{op}, sSet] \to$ sSet sends Cech nerve projections $C(U) \to X$ over covering families $\{U_i \to X\}$ to weak homotopy equivalences:
If $C$ is also a strongly ∞-connected site then it is an ∞-cohesive site.
For $C$ an $\infty$-local site, the (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(C)$ over it is a local (∞,1)-topos, in that the global section (∞,1)-geometric morphism has a further right adjoint (∞,1)-functor
We may present the (∞,1)-sheaf (∞,1)-topos by the local model structure on simplicial presheaves
For the notation see the details of the analagous proof at ∞-connected site. As discussed there, the functor $\Gamma$ is given by evaliation on the terminal object. At the level of simplicial presheaves the sSet-enriched right adjoint to $\Gamma$ is given by
as confirmed by the following end/coend calculus computation:
where in the second but last step we used the co-Yoneda lemma.
It is clear that
is a Quillen adjunction, since $\nabla$ manifestly preserves fibrations and acyclic fibrations. Since $[C^{op}, sSet]_{proj,loc}$ is a left proper model category to see that this descends to a Quillen adjunction on the local model structure on simplicial presheaves it is sufficient to check that $\nabla : sSet_{Quillen} \to [C^{op}, sSet]_{proj,loc}$ preserves fibrant objects, in that for $S$ a Kan complex we have that $\nabla S$ satisfies descent along Cech nerves of covering families.
This follows from the second defining condition on the $\infty$-local site $C$, that $Hom_C(*,C(U)) \simeq Hom_C(*,U)$. Using this we have for fibrant $S \in sSet_{Quillen}$ the descent weak equivalence
where we use in the middle step that $sSet_{Quillen}$ is a simplicial model category so that homming the weak equivalence between cofibrant objects into the fibrant object $S$ indeed yields a weak equivalence (using the factorization lemma).
and
local site / ∞-local site