nLab
(infinity,1)-quasitopos

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

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

Definition

Definition

An (∞,1)-bisite is an (∞,1)-category CC together with two (∞,1)-Grothendieck topologies, JJ and KK such that JKJ \subseteq K.

Definition

Let CC be an (∞,1)-bisite. Say an (∞,1)-presheaf F(,1)PSh(C)F \in (\infty,1)PSh(C) is (J,K)\left(J,K\right)-biseparated if it is an (∞,1)-sheaf for JJ and for every KK-covering sieve UXU \to X in CC we have that the induced morphism

(,1)PSh C(X,F)(,1)PSh C(U,F) (\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(J,K)n-\left(J,K\right)-biseparated if

the induced morphism

(,1)PSh C(X,F)(,1)PSh C(U,F) (\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)

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

Definition

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

Examples

For H\mathbf{H} a local (∞,1)-topos

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

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

GrpdCodiscΓConc(H)concretizationH \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\mathbf{H}, the analog of concrete sheaves.

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)