nLab (infinity,1)-quasitopos

Idea

The notion of $(\infty,1)$-quasitopos is the (∞,1)-topos-analog of the notion of quasitopos.

Definition

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 $F \in (\infty,1)PSh(C)$ 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

$$(\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)$$

in ∞Grpd is a full and faithful (∞,1)-functor.

We say it is $n-\left(J,K\right)$-biseparated if

the induced morphism

$$(\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)$$

is an (n-1)-truncated object in the (∞,1)-overcategory $\left(\infty-Gpd\right)/(\infty,1)PSh_C(U,F)$.

Definition

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 $\left(C,J,K\right)$.

Examples

For $\mathbf{H}$ a local (∞,1)-topos

$$\mathbf{H} \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}} \infty Grpd$$

and $C$ be a site of definition for $\mathbf{H}$, the $(\infty,1)$-quasitopos on $C$ that factors the geometric embedding $Codisc \infty Grpd \hookrightarrow \mathbf{H}$

$$\infty Grpd \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} Conc(\mathbf{H}) \stackrel{\overset{concretization}{\leftarrow}}{\underset{}{\hookrightarrow}} \mathbf{H}$$

is that of concrete objects in $\mathbf{H}$, the analog of concrete sheaves.

Related concepts

• quasitopos

• $(\infty,1)$-quasitopos

References

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.