Definition

An (∞,1)-bisite is an (∞,1)-category $C$ together with two (∞,1)-Grothendieck topologies, $J$ and $K$ such that $J\subseteq K$.

Definition

Let $C$ be an (∞,1)-bisite. Say an (∞,1)-presheaf $F\in (\mathrm{\infty },1)\mathrm{PSh}(C)$ is $(J,K)$-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 $n-(J,K)$-biseparated if

the induced morphism

is an (n-1)-truncated object in the (∞,1)-overcategory $(\mathrm{\infty }-\mathrm{Gpd})/(\mathrm{\infty },1){\mathrm{PSh}}_C(U,F)$.

Definition

A (Grothendieck) $(\mathrm{\infty },1)$-quasitopos is an (∞,1)-category that is equivalent to the full sub-(∞,1)-category of some $(\mathrm{\infty },1){\mathrm{PSh}}_C$ on the $n-(J,K)$-biseparated $(\mathrm{\infty },1)$-presheaves, on some (∞,1)-bisite $(C,J,K)$.