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(n,1)-topos

Idea

A (Grothendieck) $(n,1)$-topos is the (n,1)-category version of a Grothendieck topos: a collection of (n-1)-groupoid-valued sheaves on an $(n,1)$-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 $(n,1)$-topos arises as the full (∞,1)-subcategory on $(n-1)$-truncated objects in an (∞,1)-topos of $\mathrm{\infty }$-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 $\mathrm{PSh}(C)$ on some small category $C$

  $$\mathrm{Sh}(C)\stackrel{\stackrel{\mathrm{lex}}{\leftarrow }}{\hookrightarrow }\mathrm{PSh}(C).$$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 ${\mathrm{PSh}}_{(\mathrm{\infty },1)}(C)$ on some small (∞,1)-category $C$:

  $${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)\stackrel{\stackrel{\mathrm{lex}}{\leftarrow }}{\hookrightarrow }{\mathrm{PSh}}_{(\mathrm{\infty },1)}(C).$$Sh_{(\infty,1)}(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} PSh_{(\infty,1)}(C) \,.

Accordingly now,

Definition

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

$$𝒳\stackrel{\stackrel{\mathrm{lex}}{\leftarrow }}{\hookrightarrow }{\mathrm{PSh}}_{\le n-1}(C).$$\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 $(n,1)\mathrm{Topos}$ for the (n+1,1)-category of $(n,1)$-toposes and geometreic moprphisms between these.

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

Proposition

Every $(n,1)$-topos $𝒴$ is the (n,1)-category of $(n-1)$-truncated objects in an n-localic (∞,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).

Proposition

For any $0\le m\le n\le \mathrm{\infty }$, $(m-1)$-truncation induces a localization

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

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

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

Examples

$(2,1)$-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

• Jacob Lurie, Higher Topos Theory