nLab
(infinity,1)Topos

Idea

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

Definition

$(\mathrm{\infty },1)\mathrm{Topos}$ 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 $(\mathrm{\infty },2)$-categories is not well developed, this point of view is not often taken yet.

Properties

Existence of sites of definition

(…)

Existence of limits and colimits

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

Proposition

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

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

preserves these.

This is HTT, prop. 6.3.2.3.

Proposition

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

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

sends (∞,1)-limits to (∞,1)-limits.

Propoisition

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

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

preserves these.

This is HTT, prop. 6.3.3.1.

Propoisition

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

This is HTT, prop. 6.3.4.7.

Remark

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

Proposition

Let

$$\begin{array}{ccc}& & 𝒳\\ & & {\downarrow }^{(g^{*}⊣g_{*})}\\ 𝒴& \stackrel{(f^{*}⊣f_{*})}{\to }& 𝒵\end{array}$$\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 $\widetilde{𝒟}$, $𝒟$, and $𝒞$ with finite (∞,1)-limit and morphisms of sites

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

such that

$$\left(\begin{array}{ccc}& & 𝒳\\ & & {\downarrow }^{(g^{*}⊣g_{*})}\\ 𝒴& \stackrel{(f^{*}⊣f_{*})}{\to }& 𝒵\end{array}\right)\simeq \left(\begin{array}{ccc}& & {\mathrm{Sh}}_{(\mathrm{\infty },1)}(𝒟)\\ & & {\downarrow }^{({\mathrm{Lan}}_g⊣(-)\circ g)}\\ {\mathrm{Sh}}_{(\mathrm{\infty },1)}(\widetilde{𝒟})& \stackrel{({\mathrm{Lan}}_f⊣(-)\circ f)}{\to }& {\mathrm{Sh}}_{(\mathrm{\infty },1)}(𝒞)\end{array}\right).$$\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

$$\begin{array}{ccc}\widetilde{𝒟}\coprod _{𝒞}𝒟& \stackrel{f\prime }{\leftarrow }& 𝒟\\ {}^{g\prime }\uparrow & {⇙}_{\simeq }& {\uparrow }^g\\ \widetilde{𝒟}& \stackrel{f}{\leftarrow }& 𝒞\end{array}\in (\mathrm{\infty },1){\mathrm{Cat}}^{\mathrm{lex}}$$\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${}^{\mathrm{lex}}\subset (\mathrm{\infty },1)\mathrm{Cat}$ of $(\mathrm{\infty },1)$-categories with finite $(\mathrm{\infty },1)$-limits.

Let moreover

$${\mathrm{Sh}}_{(\mathrm{\infty },1)}(\widetilde{𝒟}\coprod _{𝒞}𝒟)\hookrightarrow {\mathrm{PSh}}_{(\mathrm{\infty },1)}(\widetilde{𝒟}\coprod _{𝒞}𝒟)$$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 ${\mathrm{Lan}}_{f\prime }(-)$ of the coverings of $𝒟$ and the inverse image ${\mathrm{Lan}}_{g\prime }(-)$ of the coverings of $\widetilde{𝒟}$.

Then

$$\begin{array}{ccc}{\mathrm{Sh}}_{(\mathrm{\infty },1)}(\widetilde{𝒟}\coprod _{𝒞}𝒟)& \to & 𝒳\\ \downarrow & {⇙}_{\simeq }& {\downarrow }^{(g^{*}⊣g_{*})}\\ 𝒴& \stackrel{(f^{*}⊣f_{*})}{\to }& 𝒵\end{array}$$\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.

Related concepts

• Topos

• $(\mathrm{\infty },1)$Topos

References

section 6.3 in

• Jacob Lurie, Higher Topos Theory

category: category