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n-localic (infinity,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

An (∞,1)-topos is nn-localic if

More precisely: if (∞,1)-geometric morphisms into it are fixed by their restriction to the underlying (n,1)-toposes of (n1)(n-1)-truncated objects.

To the tower of (n,1)-toposes of (n1)(n-1)-truncated objects

τ 31𝒳τ 21𝒳τ 11𝒳τ 01𝒳* \cdots \to \tau_{\leq 3-1} \mathcal{X} \to \tau_{\leq 2-1} \mathcal{X} \to \tau_{\leq 1-1} \mathcal{X} \to \tau_{\leq 0-1} \mathcal{X} \to *

of a given (∞,1)-topos 𝒳\mathcal{X} corresponds a tower of nn-localic toposes 𝒳 n\mathcal{X}_n such that τ n1𝒳τ n1𝒳 n\tau_{\leq n -1} \mathcal{X} \simeq \tau_{\leq n-1} \mathcal{X}_n. We may think of the nn-localic 𝒳 n\mathcal{X}_n as being nnth stage in the Postnikov tower decomposition of 𝒳\mathcal{X}.

A 0-localic (1,1)(1,1)-topos is a localic topos from ordinary topos theory.

Definition

We write (∞,1)Topos for the (∞,1)-category of (∞,1)-toposes and (∞,1)-geometric morphisms between them.

For 𝒳\mathcal{X} an (∞,1)-topos we denote by

τ n1𝒳𝒳 \tau_{\leq n-1} \mathcal{X} \hookrightarrow \mathcal{X}

the (n,1)-topos of (n1)(n-1)-truncated objects of 𝒳\mathcal{X}.

We write (n,1)Topos(n,1)Topos for the (n+1,1)-category of (n,1)-toposes and (n,1)(n,1)-geometric morphisms between them.

Definition

(nn-localic (,1)(\infty,1)-topos)

An (∞,1)-topos 𝒳\mathcal{X} is nn-localic if for any other (,1)(\infty,1)-topos 𝒴\mathcal{Y} the canonical morphism

(,1)Topos(𝒴,𝒳)(n,1)Topos(τ n1𝒴,τ n1𝒴) (\infty,1)Topos(\mathcal{Y},\mathcal{X}) \to (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{Y})

is an equivalence of (∞,1)-categories (of ∞-groupoids).

More generally,

a (k,1)-topos 𝒳\mathcal{X} is nn-localic for 0nk0 \leq n \leq k \leq \infty if for any other (k,1)(k,1)-topos 𝒴\mathcal{Y} the canonical morphism

(k,1)Topos(𝒴,𝒳)(n,1)Topos(τ n1𝒴,τ n1𝒴) (k,1)Topos(\mathcal{Y},\mathcal{X}) \to (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{Y})

is an equivalence of (∞,1)-categories (of ∞-groupoids).

This is (HTT, def. 6.4.5.8).

Remark

This implies that an nn-localic (,1)(\infty,1)-topos is also (n+1)(n+1)-localic and generally kk-localic for all knk \geq n.

Examples

Proposition

The (∞,1)-category of (∞,1)-sheaves over an (∞,1)-site CC with finite limits which is an (n,1)-category is nn-localic.

This is (HTT, lemma 6.4.5.6).

Remark

For n=0n = 0 this implies the familiar statement from ordinary topos theory: a category of sheaves over a posite=(0,1)-site is a localic topos (= 0-localic (1,1)(1,1)-topos).

This is (LurieStructured, lemma 2.3.16).

Proposition

For nn \in \mathbb{N} and 𝒳\mathcal{X} an nn-localic (,1)(\infty,1)-topos, the over-(∞,1)-topos 𝒳/U\mathcal{X}/U is nn-localic precisely if the object UU is nn-truncated.

This is (StrSp, lemma 2.3.14).

Proposition

For 𝒳\mathcal{X} an nn-localic (,1)(\infty,1)-topos let U𝒳U \in \mathcal{X} be an object. Then the following are equivalent

  1. the restriction of the inverse image U *:𝒳𝒳/UU^* : \mathcal{X} \to \mathcal{X}/U (of the etale geometric morphism from the over-(∞,1)-topos) to (n1)(n-1)-truncated objects is an equivalence of (∞,1)-categories;

  2. the object UU is nn-connected.

This is (StrSp, lemma 2.3.14).

Properties

Proposition

Every (n,1)-topos 𝒴\mathcal{Y} is the (n,1)-category of (n1)(n-1)-truncated objects in an nn-localic (,1)(\infty,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).

Let 𝒢\mathcal{G} be a geometry (for structured (∞,1)-toposes).

Proposition

If 𝒢\mathcal{G} is an (∞,n)-category then a nn-localic 𝒢\mathcal{G}-structured (∞,1)-topos is an nn-truncated object in the (∞,1)-category Topos(𝒢)Topos(\mathcal{G}).

This is StrSp, lemma 2.6.17

References

The general noion is the topic of section 6.4.5 of

Remarks on the application of nn-localic (,1)(\infty,1)-toposes in higher geometry are in

Revised on November 19, 2013 08:39:27 by Urs Schreiber (145.116.130.92)