nLab (n,1)-topos

Context

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

(∞,1)-topos theory

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

A (Grothendieck) $\left(n,1\right)$-topos is the (n,1)-category version of a Grothendieck topos: a collection of (n-1)-groupoid-valued sheaves on an $\left(n,1\right)$-categorical site.

Notice that an ∞-stack on an ordinary (1-categorical) site that takes values in ∞-groupoids which happen to by 0-truncated, i.e. which happen to take values just in Set $↪$ ∞Grpd is the same as an ordinary sheaf of sets.

This generalizes: every $\left(n,1\right)$-topos arises as the full (∞,1)-subcategory on $\left(n-1\right)$-truncated objects in an (∞,1)-topos of $\infty$-stacks on an (n,1)-category site.

Definition

Recall that

• a 1-Grothendieck topos is precisely a accessible geometric embedding into a category of presheaves $\mathrm{PSh}\left(C\right)$ on some small category $C$

$\mathrm{Sh}\left(C\right)\stackrel{\stackrel{\mathrm{lex}}{←}}{↪}\mathrm{PSh}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$Sh(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh(C) \,.
• a (∞,1)-topos (of ∞-stacks/(∞,1)-sheaves) is precisely a accessible geometric embedding into a (∞,1)-category of (∞,1)-presheaves ${\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)$ on some small (∞,1)-category $C$:

${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)\stackrel{\stackrel{\mathrm{lex}}{←}}{↪}{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$Sh_{(\infty,1)}(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.

Accordingly now,

Definition

An $\left(n,1\right)$-topos $𝒳$ is an accessible left exact localization of the full (∞,1)-subcategory ${\mathrm{PSh}}_{\le n-1}\left(C\right)\subset {\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)$ on $\left(n-1\right)$-truncated objects in an (∞,1)-category of (∞,1)-presheaves on a small (∞,1)-category $C$:

$𝒳\stackrel{\stackrel{\mathrm{lex}}{←}}{↪}{\mathrm{PSh}}_{\le n-1}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathcal{X} \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{\leq n-1}(C) \,.

This appears as HTT, def. 6.4.1.1.

Properties

Write (∞,1)-Topos for the (∞,1)-category of (∞,1)-topos and (∞,1)-geometric morphisms. Write $\left(n,1\right)\mathrm{Topos}$ for the (n+1,1)-category of $\left(n,1\right)$-toposes and geometreic moprphisms between these.

The following proposition asserts that when passing to the $\left(n,1\right)$-topos of an (∞,1)-topos $𝒳$, only the n-localic “Postnikiov stage” of $𝒳$ matters.

Proposition

Every $\left(n,1\right)$-topos $𝒴$ is the (n,1)-category of $\left(n-1\right)$-truncated objects in an n-localic (∞,1)-topos ${𝒳}_{n}$

${\tau }_{n-1}{X}_{n}\stackrel{\simeq }{\to }𝒴\phantom{\rule{thinmathspace}{0ex}}.$\tau_{n-1} X_n \stackrel{\simeq}{\to} \mathcal{Y} \,.

This is (HTT, prop. 6.4.5.7).

Proposition

For any $0\le m\le n\le \infty$, $\left(m-1\right)$-truncation induces a localization

${\mathrm{Topos}}_{\left(m,1\right)}\stackrel{\stackrel{{\tau }_{m-1}}{←}}{↪}{\mathrm{Topos}}_{n,1}$Topos_{(m,1)} \stackrel{\overset{\tau_{m-1}}{\leftarrow}}{\hookrightarrow} Topos_{n,1}

that identifies ${\mathrm{Topos}}_{\left(m,1\right)}$ equivalently with the full subcategory of $m$-localic $\left(n,1\right)$-toposes.

(This is 6.4.5.7 in view of the following remarks.)

Examples

$\left(2,1\right)$-Toposes

If $E$ is a (2,1)-topos in which every object is covered by a 0-truncated object, then $E$ is equivalent to the category of (2,1)-sheaves on a 1-site (rather than merely a (2,1)-site, as is the case for general (2,1)-topoi), and is thus canonically associated to a 1-topos, namely the category of 1-sheaves on that same 1-site. And in fact, $E$ can be recovered from this 1-topos as the category of (2,1)-sheaves for its canonical topology.

See truncated 2-topos for more.

References

Section 6.4 of

Revised on January 31, 2011 17:17:11 by Mike Shulman (71.136.232.119)