(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A site being locally and globally $\infty$-connected means that it satisfies sufficient conditions such that the (∞,1)-category of (∞,1)-sheaves over it is a locally ∞-connected (∞,1)-topos and a ∞-connected (∞,1)-topos.
A a site is locally and globally $\infty$-connected over ∞Grpd if
it has a terminal object *
for every covering family $\{U_i \to U\}$ in $C$
the Cech nerve $C(\{U_i\}) \in [C^{op}, sSet]$ is degreewise a coproduct of representables;
the simplicial set obtained by replacing each copy of a representable by a point is contractible, equivalently the colimit $\lim_\to : [C^{op}, sSet] \to sSet$ of $C(\{U_i\})$ has a weak homotopy equivalence to the point
The (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(C)$ over locally and globally $\infty$-conneted site $C$, regarded as an (∞,1)-site, is a (1-localic) locally ∞-connected (∞,1)-topos and ∞-connected (∞,1)-topos, in that it comes with a triple of adjoint (∞,1)-functors
such that $\Pi$ preserves the terminal object.
To prove this, we we use the model structure on simplicial presheaves to present $Sh_{(\infty,1)}(C)$.
Write $[C^{op}, sSet]_{proj}$ for the projective global model structure and $[C^{op}, sSet]_{proj,loc}$ for its left Bousfield localization at the set of morphisms $C(\{U_i\}) \to U$ out of the Cech nerve for each covering family $\{U_i \to U\}$, and $[C^{op}, sSet]_{proj,loc}^\circ$ for the Kan complex-enriched category on the fibrant-cofibrant objects. By the discussion at model structure on simplicial presheaves we have
and the adjoint (∞,1)-functors on the left are presented by simplicial Quillen adjunctions on the right.
To establish these, we proceed by a sequence of lemmas.
The model categories
standard model structure on simplicial sets $sSet_{Quillen}$;
global model structure on simplicial presheaves $[C^{op}, sSet]_{proj,loc}$;
local model structure on simplicial presheaves $[C^{op}, sSet]_{proj,loc}$
are all left proper model categories.
The first since all objects are cofibrant. The second by general statements about the global model structure on functors, the third because left Bousfield localization preserves left propernes.
For $\{U_i \to U\}$ a covering family in the $\infty$-connected site $C$, the Cech nerve $C(\{U_i\}) \in [C^{op}, sSet]$ is a cofibrant resolution of $U$ both in the projective model structure $[C^{op}, sSet]_{proj}$ as well as in the Cech local model structure $[C^{op}, sSet]_{proj,loc}$.
By assumption on $C$ we have that $C(\{U_i\})$ is a split hypercover. By Dugger’s theorem on cofibrant objects in the projective model structure this implies that $C(U)$ is cofibrant in the global model structure. By general properties of left Bousfield localization we have that the cofibrations in the local model structure as the same as in the global one. Finally that $C(\{U_i\}) \to U$ is a weak equivalence in the local model structure holds effectively by definition (since we are localizing at these morphisms).
On a locally and globally $\infty$-connected site $C$ the global section (∞,1)-geometric morphism
is presented by the simplicial Quillen adjunction
where $\Gamma$ is the functor that evaluates on the point, $\Gamma X = X(*)$, and $Const$ is the functor that sends a simplicial set $S$ to the presheaf constant on that value, $Const S : U \mapsto S$.
We use (as described there) that adjoint (∞,1)-functors are modeled by simplicial Quillen adjunctions between the simplicial model categories that model the $(\infty,1)$-categories in question.
That we have an adjunction $(Const \dashv \Gamma)$ follows for instance by observing that since $C$ has a terminal object we may think of $\Gamma$ as being the functor $\Gamma = \lim_\leftarrow$ that takes the limit.
To see that we have a Quillen adjunction first notice that we have a Quillen adjunction
on the global model structure, since $\Gamma$ manifestly preserves fibrations and acyclic fibrations there. Since $[C^{op}, sSet]_{proj,loc}$ is left proper and has the same cofibrations as the global model structure, it follows with HTT, corollary A.3.7.2 (see the discussion of sSet-Quillen adjunctions) that for this to descend to a Quillen adjunction on the local model structure it is sufficient that $\Gamma$ preserves fibrant objects. But every fibrant object in the local structure is in particular fibrant in the global structure, hence in particular fibrant over the terminal object of $C$.
The left derived functor of $Const : sSet_{Quillen} \to [C^{op},sSet]_{proj}$ preserves homotopy limits (because (∞,1)-limits in an (∞,1)-category of (∞,1)-presheaves are computed objectwise), and ∞-stackification, the left derived functor of $Id : [C^{op}, sSet]_{proj} \to [C^{op}, sSet]_{proj,loc}$ is a left exact (∞,1)-functor, therefore the left derived functor of $Const : sSet_{Quillen} \to [C^{op}, sSet]_{proj,loc}$ preserves finite homotopy limits.
This means that our Quillen adjunction does model a (∞,1)-geometric morphism $Sh_{(\infty,1)}(C) \to \infty Grpd$. By the discussion at global section the space of these geometric morphisms to ∞Grpd is contractible, hence this is indeed a representative of the terminal geometric morphism as claimed.
By general abstract facts the sSet-functor $Const : sSet \to [C^{op}, sSet]$ given on $S \in sSet$ by $Const_S : U \mapsto S$ for all $U \in C$ has an sSet-left adjoint
naturally in $X$ and $S$, given by the colimit operation. Notice that since sSet is itself a category of presheaves (on the simplex category), these colimits are degreewise colimits in Set. Also notice that the colimit over a representable functor is the point (by a simple Yoneda lemma-style argument).
Regarded as a functor $sSet_{Quillen} \to [C^{op}, sSet]_{proj}$ the functor $Const$ manifestly preserves fibrations and acyclic fibrations and hence
is a Quillen adjunction, in particular $\Pi : [C^{op},sSet]_{proj} \to sSet_{Quillen}$ preserves cofibrations. Since by general properties of left Bousfield localization of model categories the cofibrations of $[C^{op},sSet]_{proj,loc}$ are the same, also $\Pi : [C^{op}, sSet]_{proj,loc} \to sSet_{Quillen}$ preserves cofibrations.
Since $sSet_{Quillen}$ is a left proper model category it follows as before with HTT, corollary A.3.7.2 (see the discussion of sSet-Quillen adjunctions) that for
to be a Quillen adjunction, it suffices to show that $Const$ preserves fibrant objects. That means that constant simplicial presheaves satisfy descent along covering families in the $\infty$-cohesive site $C$: for every covering family $\{U_i \to U\}$ in $C$ and every simplicial set $S$ it must be true that
is a homotopy equivalence of Kan complexes. (Here we use that $U$, being a representable, is cofibrant, that $C(U)$ is cofibrant by the above lemma and that $Const S$ is fibrant in the projective structure by the assumption that $S$ is fibrant. So the simplicial hom-complexes in the above equaltion really are the correct derived hom-spaces.)
But that this is the case follows by the condition on the $\infty$-cohesive site $C$ by which $\lim_\to C(U) \simeq *$: using this it follows that
So we have established that also
is a Quillen adjunction.
It is clear that the left derived functor of $\Pi$ preserves the terminal object: since that is representable by assumption on $C$, it is cofibrant in $[C^{op}, sSet]_{proj,loc}$, hence $\mathbb{L} \lim_\to * = \lim_\to * = *$.
This implies that ∞LieGrpd is a cohesive (∞,1)-topos. See there for details.
and
locally connected site / locally ∞-connected (∞,1)-site
connected site / ∞-connected (∞,1)-site