nLab
(n,1)-topos

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

A (Grothendieck) (n,1)(n,1)-topos is the (n,1)-category version of a Grothendieck topos: a collection of (n-1)-groupoid-valued sheaves on an (n,1)(n,1)-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 \hookrightarrow ∞Grpd is the same as an ordinary sheaf of sets.

This generalizes: every (n,1)(n,1)-topos arises as the full (∞,1)-subcategory on (n1)(n-1)-truncated objects in an (∞,1)-topos of \infty-stacks on an (n,1)-category site.

Definition

Recall that

Accordingly now,

Definition

An (n,1)(n,1)-topos 𝒳\mathcal{X} is an accessible left exact localization of the full (∞,1)-subcategory PSh n1(C)PSh (,1)(C)PSh_{\leq n-1}(C) \subset PSh_{(\infty,1)}(C) on (n1)(n-1)-truncated objects in an (∞,1)-category of (∞,1)-presheaves on a small (∞,1)-category CC:

𝒳lexPSh n1(C). \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 (n,1)Topos(n,1)Topos for the (n+1,1)-category of (n,1)(n,1)-toposes and geometreic moprphisms between these.

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

Proposition

Every (n,1)(n,1)-topos 𝒴\mathcal{Y} is the (n,1)-category of (n1)(n-1)-truncated objects in an n-localic (∞,1)-topos 𝒳 n\mathcal{X}_n

τ n1X n𝒴. \tau_{n-1} X_n \stackrel{\simeq}{\to} \mathcal{Y} \,.

This is (HTT, prop. 6.4.5.7).

Proposition

For any 0mn0 \leq m \leq n \leq \infty, (m1)(m-1)-truncation induces a localization

Topos (m,1)τ m1Topos n,1 Topos_{(m,1)} \stackrel{\overset{\tau_{m-1}}{\leftarrow}}{\hookrightarrow} Topos_{n,1}

that identifies Topos (m,1)Topos_{(m,1)} equivalently with the full subcategory of mm-localic (n,1)(n,1)-toposes.

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

Examples

(2,1)(2,1)-Toposes

If EE is a (2,1)-topos in which every object is covered by a 0-truncated object, then EE 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, EE 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)