nLab (infinity,1)-quasitopos

Context

$\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

The notion of $\left(\infty ,1\right)$-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 $F\in \left(\infty ,1\right)\mathrm{PSh}\left(C\right)$ 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

$\left(\infty ,1\right){\mathrm{PSh}}_{C}\left(X,F\right)↪\left(\infty ,1\right){\mathrm{PSh}}_{C}\left(U,F\right)$(\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)

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

the induced morphism

$\left(\infty ,1\right){\mathrm{PSh}}_{C}\left(X,F\right)↪\left(\infty ,1\right){\mathrm{PSh}}_{C}\left(U,F\right)$(\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)

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

Definition

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

Examples

For $H$ a local (∞,1)-topos

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

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

$\infty \mathrm{Grpd}\stackrel{\stackrel{\Gamma }{←}}{\underset{\mathrm{Codisc}}{↪}}\mathrm{Conc}\left(H\right)\stackrel{\stackrel{\mathrm{concretization}}{←}}{\underset{}{↪}}H$\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 $H$, the analog of concrete sheaves.

• quasitopos

• $\left(\infty ,1\right)$-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.

Revised on November 17, 2010 11:57:28 by Urs Schreiber (87.212.203.135)