nLab
n-localic (infinity,1)-topos

Idea

An (∞,1)-topos is $n$-localic if

• regarded as a little topos it behaves like a generalized n-groupoid;

• it behaves like the (∞,1)-category of (∞,1)-sheaves over an (∞,1)-site that is a (n,1)-category.

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

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

$$\cdots \to {\tau }_{\le 3-1}𝒳\to {\tau }_{\le 2-1}𝒳\to {\tau }_{\le 1-1}𝒳\to {\tau }_{\le 0-1}𝒳\to *$$\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 $𝒳$ corresponds a tower of $n$-localic toposes ${𝒳}_n$ such that ${\tau }_{\le n-1}𝒳\simeq {\tau }_{\le n-1}{𝒳}_n$. We may think of the $n$-localic ${𝒳}_n$ as being $n$th stage in the Postnikov tower decomposition of $𝒳$.

A 0-localic $(1,1)$-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 $𝒳$ an (∞,1)-topos we denote by

$${\tau }_{\le n-1}𝒳\hookrightarrow 𝒳$$\tau_{\leq n-1} \mathcal{X} \hookrightarrow \mathcal{X}

the (n,1)-topos of $(n-1)$-truncated objects of $𝒳$.

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

Definition

($n$-localic $(\mathrm{\infty },1)$-topos)

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

$$(\mathrm{\infty },1)\mathrm{Topos}(𝒴,𝒳)\to (n,1)\mathrm{Topos}({\tau }_{\le n-1}𝒴,{\tau }_{\le n-1}𝒴)$$(\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 $𝒳$ is $n$-localic for $0\le n\le k\le \mathrm{\infty }$ if for any other $(k,1)$-topos $𝒴$ the canonical morphism

$$(k,1)\mathrm{Topos}(𝒴,𝒳)\to (n,1)\mathrm{Topos}({\tau }_{\le n-1}𝒴,{\tau }_{\le n-1}𝒴)$$(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 $n$-localic $(\mathrm{\infty },1)$-topos is also $(n+1)$-localic and generally $k$-localic for all $k\ge n$.

Examples

Proposition

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

This is (HTT, lemma 6.4.5.6).

Remark

For $n=0$ 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 $(1,1)$-topos).

This is (LurieStructured, lemma 2.3.16).

Proposition

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

This is (StrSp, lemma 2.3.14).

Proposition

For $𝒳$ an $n$-localic $(\mathrm{\infty },1)$-topos let $U\in 𝒳$ be an object. Then the following are equivalent

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

2. the object $U$ is $n$-connected.

This is (StrSp, lemma 2.3.14).

Properties

Proposition

Every (n,1)-topos $𝒴$ is the (n,1)-category of $(n-1)$-truncated objects in an $n$-localic $(\mathrm{\infty },1)$-topos ${𝒳}_n$

$${\tau }_{n-1}{X}_n\stackrel{\simeq }{\to }𝒴.$$\tau_{n-1} X_n \stackrel{\simeq}{\to} \mathcal{Y} \,.

This is (HTT, prop. 6.4.5.7).

Let $𝒢$ be a geometry (for structured (∞,1)-toposes).

Proposition

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

This is StrSp, lemma 2.6.17

Related concepts

• localic topos

• n-localic 2-topos

References

The general noion is the topic of section 6.4.5 of

• Jacob Lurie, Higher Topos Theory

Remarks on the application of $n$-localic $(\mathrm{\infty },1)$-toposes in higher geometry are in

• Jacob Lurie, Structured Spaces