(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The notion of $(\infty,1)$-quasitopos is the (∞,1)-topos-analog of the notion of quasitopos.
An (∞,1)-bisite is an (∞,1)-category $C$ together with two (∞,1)-Grothendieck topologies, $J$ and $K$ such that $J \subseteq K$.
Let $C$ be an (∞,1)-bisite. Say an (∞,1)-presheaf $F \in (\infty,1)PSh(C)$ is $\left(J,K\right)$-biseparated if it is an (∞,1)-sheaf for $J$ and for every $K$-covering sieve $U \to X$ in $C$ we have that the induced morphism
in ∞Grpd is a full and faithful (∞,1)-functor.
We say it is $n-\left(J,K\right)$-biseparated if
the induced morphism
is an (n-1)-truncated object in the (∞,1)-overcategory $\left(\infty-Gpd\right)/(\infty,1)PSh_C(U,F)$.
A (Grothendieck) $(\infty,1)$-quasitopos is an (∞,1)-category that is equivalent to the full sub-(∞,1)-category of some $(\infty,1)PSh_C$ on the $n-\left(J,K\right)$-biseparated $(\infty,1)$-presheaves, on some (∞,1)-bisite $\left(C,J,K\right)$.
For $\mathbf{H}$ a local (∞,1)-topos
and $C$ be a site of definition for $\mathbf{H}$, the $(\infty,1)$-quasitopos on $C$ that factors the geometric embedding $Codisc \infty Grpd \hookrightarrow \mathbf{H}$
is that of concrete objects in $\mathbf{H}$, the analog of concrete sheaves.
$(\infty,1)$-quasitopos
The definition as it stands, originated out of a discussion between Urs Schreiber and David Carchedi. The suggestion to rephrase the definition in terms of bisites came from Mike Shulman.