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(infinity,1)Topos

Idea

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

Definition

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

Properties

In the following, $(\infty,1)Cat$ will refer to the (superlarge) $(\infty,1)$ category of large $(\infty, 1)$-categories.

Existence of sites of definition

(…)

Existence of limits and colimits

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

Proposition

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

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

preserves small limits.

This is HTT, prop. 6.3.2.3. In the notation there, $LTop$ is the $(\infty,1)$-category of toposes whose arrows are the inverse image morphisms, and thus opposite to $(\infty,1)Topos$.

Proposition

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

$$(\infty,1)Topos \to (\infty,1)Cat$$

preserves these.

This is HTT, prop. 6.3.3.1.

Proposition

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

This is HTT, prop. 6.3.4.7.

Remark

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

This is HTT, remark 6.3.4.10.

Proposition

The terminal object in (∞,1)Topos is ∞Groupoids.

This is HTT, Prop. 6.3.4.1.

Computation of limits and colimits

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

Proposition

Let

$$\array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }$$

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 $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with finite (∞,1)-limit and morphisms of sites

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

such that

$$\left( \array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} } \right) \,\,\, \simeq \,\,\, \left( \array{ && Sh_{(\infty,1)}(\mathcal{D}) \\ && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}} \\ Sh_{(\infty,1)}(\tilde \mathcal{D}) &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}& Sh_{(\infty,1)}(\mathcal{C}) } \right) \,.$$

Let then

$$\array{ \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D} \\ {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} } \,\,\,\,\, \in (\infty,1)Cat^{lex}$$

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

Let moreover

$$Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) \hookrightarrow PSh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})$$

be the reflective sub-(∞,1)-category obtained by localization at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the coverings of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.

Then

$$\array{ Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) &\to& \mathcal{X} \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} }$$

is an (∞,1)-pullback square.

This is HTT, prop. 6.3.4.6.

Colimits of over-toposes

Proposition

For any $(\infty,1)$ topos $\mathbf{H}$, there is a colimit preserving functor $\mathbf{H} \to (\infty,1)Topos$ sending an object $X$ to its over-topos $\mathbf{H}_{/X}$, and sending an arrow $f : X \to Y$ to the essential geometric morphism $f_* : \mathbf{H}_{/X} \to \mathbf{H}_{/Y}$.

Proof

HTT, Prop. 6.1.3.9 implies the “terminal vertex” Cartesian fibration $\mathbf{H}^{[1]} \to \mathbf{H}$ is classified by a limit preserving functor $\mathbf{H}^{op} \to Pr^L$, the $(\infty,1)$-category of locally presentable $(\infty,1)$-categories and colimit-preserving functors between them.

This functor factors through the subcategory $(\infty,1)Topos^{op} \subseteq Pr^L$ that sends a geometric morphism to its inverse image part. By [HTT, Prop. 6.3.2.3] and [HTT, Prop. 5.5.1.13], it follows that this is also a limit preserving functor.

The opposite category is then formed by taking right adjoints.

Related concepts

• Topos

• $(\infty,1)$Topos

References

section 6.3 in

• Jacob Lurie, Higher Topos Theory

category: category