(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An (∞,1)-site is locally $\infty$-connected if it has properties that ensure that the hypercompletion of the (∞,1)-category of (∞,1)-sheaves over it is a locally ∞-connected (∞,1)-topos
Call an (∞,1)-site $C$ locally contractible if every constant (∞,1)-presheaf on it is an (∞,1)-sheaf in the hypercomplete (∞,1)-topos over $C$.
By the general notion of (∞,1)-colimit the constant $(\infty,1)$-presheaf functor has a left adjoint (∞,1)-functor given by taking colimits
Since the (∞,1)-category of (∞,1)-sheaves sits by a full and faithful (∞,1)-functor inside presheaves and by assumption that every constant $(\infty,1)$-presheaf is an $(\infty,1)$-sheaf, this implies that we have also natural equivalences
Let $C$ be an 1-site such that every object $U$ has a split hypercover $Y \to U$ such that contracting all representables to points yields a weak equivalence. Equivalently, if the colimit functor $\lim_\to : [C^{op}, sSet] \to sSet$ sends this to a weak equivalence
Then $C$ is locally $\infty$-connected.
We may present $Sh_{(\infty,1)}(C)$ by the projective model structure on simplicial presheaves $[C^{op}, sSet]_{proj}$ left Bousfield localized at the Cech nerve projections $C(\coprod_i U_i) \to U$ for each covering family $\{U_i \to U\}$ in $C$.
It is immediate that we have a Quillen adjunction $(\underset{\rightarrow}{\lim} \dashv const)$ 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 $\infty$-connected site. This is what we do now.
By the discussion of cofibrant resolution at model structure on simplicial presheaves we have that a split hypercover $Y \to U$ is a cofibrant resolution in $[C^{op}, sSet]_{proj, loc}$ of $U$.
For $S \in sSet$ a Kan complex let $Const S : C^{op} \to sSet$ the corresponding constant simplicial presheaf. This is fibrant in $[C^{op}, sSet]_{proj}$. Since every split hypercover is cofibrant, it follows that $Const S$ is an $\infty$-sheaf precisely if for all $U \in C$ and some split hypercover $Y \to U$ we have that the morphism on derived hom-spaces
is a weak equivalence (of Kan complexes, necessatily). But we have
and
so that the condition is that
is a weak equivalence. This is the case for all $S$ precisely if $\lim_\to S$ is contractible, which is precisely our assumption on $Y$.
Let $X$ be a locally contractible topological space. Then $\hat Sh_{(\infty,1)}(C)$ is a locally ∞-connected (∞,1)-topos.
The category of open subsets $Op(X)$ is not in general a locally $\infty$-connected site according to the above definition. But there is another site of definition for $\hat Sh_{(\infty,1)}(X)$ which is: the full subcategory $cOp(X) \hookrightarrow Op(X)$ on the contractible open subsets.
and
locally connected site / locally ∞-connected (∞,1)-site