# nLab (infinity,1)Topos

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

By $\left(\infty ,1\right)\mathrm{Topos}$ is denoted the collection of all (∞,1)-toposes. This is the (∞,1)-category-theory analog of Topos.

## Definition

$\left(\infty ,1\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{Topos}$ is the (non-full) sub-(∞,1)-category of (∞,1)Cat on (∞,1)-toposes and (∞,1)-geometric morphisms between them.

Morally, this should actually be an (∞,2)-category, just as Topos is a 2-category, but since the technology of $\left(\infty ,2\right)$-categories is not well developed, this point of view is not often taken yet.

## Properties

(…)

### Existence of limits and colimits

We discuss existence of (∞,1)-limits and (∞,1)-colimits in $\left(\infty ,1\right)\mathrm{Topos}$.

###### Proposition

The $\left(\infty ,1\right)$-category $\left(\infty ,1\right)\mathrm{Topos}$ has all small $\left(\infty ,1\right)$-colimits and functor

$\left(\infty ,1\right)\mathrm{Topos}\to \left(\infty ,1\right)\mathrm{Cat}$(\infty,1)Topos \to (\infty,1)Cat

preserves these.

This is HTT, prop. 6.3.2.3.

###### Proposition

The $\left(\infty ,1\right)$-category $\left(\infty ,1\right)\mathrm{Topos}$ has all small $\left(\infty ,1\right)$-colimits and the inclusion

$\left(\infty ,1\right){\mathrm{Topos}}^{\mathrm{op}}\to \left(\infty ,1\right)\mathrm{Cat}$(\infty,1)Topos^{op} \to (\infty,1)Cat

sends (∞,1)-limits to (∞,1)-limits.

###### Propoisition

The $\left(\infty ,1\right)$-category $\left(\infty ,1\right)\mathrm{Topos}$ has filtered (∞,1)-limits and the inclusion

$\left(\infty ,1\right)\mathrm{Topos}\to \left(\infty ,1\right)\mathrm{Cat}$(\infty,1)Topos \to (\infty,1)Cat

preserves these.

This is HTT, prop. 6.3.3.1.

###### Propoisition

The $\left(\infty ,1\right)$-category $\left(\infty ,1\right)\mathrm{Topos}$ has all small (∞,1)-limits.

This is HTT, prop. 6.3.4.7.

###### Remark

The $\left(\infty ,1\right)$-limits in $\left(\infty ,1\right)\mathrm{Topos}$ coincide actually with the proper $\left(\infty ,2\right)$-limits.

This is HTT, remark 6.3.4.10.

### Computation of limits and colimits

We discuss more or less explicit descriptions of (∞,1)-limits and (∞,1)-colimits in $\left(\infty ,1\right)\mathrm{Topos}$.

###### Proposition

Let

$\begin{array}{ccc}& & 𝒳\\ & & {↓}^{\left({g}^{*}⊣{g}_{*}\right)}\\ 𝒴& \stackrel{\left({f}^{*}⊣{f}_{*}\right)}{\to }& 𝒵\end{array}$\array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }

be a diagram of (∞,1)-toposes. Then its (∞,1)-pullback is computed, roughly, by the (∞,1)-pushout of their (∞,1)-sites of definition (see above).

More in detail: there exist (∞,1)-sites $\stackrel{˜}{𝒟}$, $𝒟$, and $𝒞$ with finite (∞,1)-limit and morphisms of sites

$\begin{array}{ccc}& & 𝒟\\ & & {↑}^{g}\\ \stackrel{˜}{𝒟}& \stackrel{f}{←}& 𝒞\end{array}$\array{ && \mathcal{D} \\ && \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} }

such that

$\left(\begin{array}{ccc}& & 𝒳\\ & & {↓}^{\left({g}^{*}⊣{g}_{*}\right)}\\ 𝒴& \stackrel{\left({f}^{*}⊣{f}_{*}\right)}{\to }& 𝒵\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\simeq \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\begin{array}{ccc}& & {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(𝒟\right)\\ & & {↓}^{\left({\mathrm{Lan}}_{g}⊣\left(-\right)\circ g\right)}\\ {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\stackrel{˜}{𝒟}\right)& \stackrel{\left({\mathrm{Lan}}_{f}⊣\left(-\right)\circ f\right)}{\to }& {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(𝒞\right)\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\left( \array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} } \right) \,\,\, \simeq \,\,\, \left( \array{ && Sh_{(\infty,1)}(\mathcal{D}) \\ && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}} \\ Sh_{(\infty,1)}(\tilde \mathcal{D}) &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}& Sh_{(\infty,1)}(\mathcal{C}) } \right) \,.

Let then

$\begin{array}{ccc}\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟& \stackrel{f\prime }{←}& 𝒟\\ {}^{g\prime }↑& {⇙}_{\simeq }& {↑}^{g}\\ \stackrel{˜}{𝒟}& \stackrel{f}{←}& 𝒞\end{array}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\in \left(\infty ,1\right){\mathrm{Cat}}^{\mathrm{lex}}$\array{ \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D} \\ {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} } \,\,\,\,\, \in (\infty,1)Cat^{lex}

be the (∞,1)-pushout of the underlying (∞,1)-categories in the full sub-(∞,1)-category (∞,1)Cat${}^{\mathrm{lex}}\subset \left(\infty ,1\right)\mathrm{Cat}$ of $\left(\infty ,1\right)$-categories with finite $\left(\infty ,1\right)$-limits.

Let moreover

${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟\right)↪{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟\right)$Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) \hookrightarrow PSh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})

be the reflective sub-(∞,1)-category obtained by localization at the class of morphisms generated by the inverse image ${\mathrm{Lan}}_{f\prime }\left(-\right)$ of the coverings of $𝒟$ and the inverse image ${\mathrm{Lan}}_{g\prime }\left(-\right)$ of the coverings of $\stackrel{˜}{𝒟}$.

Then

$\begin{array}{ccc}{\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\stackrel{˜}{𝒟}\coprod _{𝒞}𝒟\right)& \to & 𝒳\\ ↓& {⇙}_{\simeq }& {↓}^{\left({g}^{*}⊣{g}_{*}\right)}\\ 𝒴& \stackrel{\left({f}^{*}⊣{f}_{*}\right)}{\to }& 𝒵\end{array}$\array{ Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) &\to& \mathcal{X} \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }

is an (∞,1)-pullback square.

This is HTT, prop. 6.3.4.6.

• Topos

• $\left(\infty ,1\right)$Topos

## References

section 6.3 in

category: category

Revised on October 31, 2012 22:18:17 by Urs Schreiber (82.169.65.155)