topos theory

# Contents

## Idea

A sheaf $F$ on (the site of open subsets of) a topological space $X$ corresponds to an étalé space ${\pi }_{F}:{Y}_{F}\to X$. This space ${Y}_{F}$ has itself a sheaf topos associated to it, and the map ${Y}_{F}\to X$ induces a geometric morphism of sheaf toposes

${\pi }_{F}:\mathrm{Sh}\left({Y}_{F}\right)\to \mathrm{Sh}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_F : Sh(Y_F) \to Sh(X) \,.

Due to the special nature of ${Y}_{F}$, the topos on the left is equivalent to the slice topos $\mathrm{Sh}\left(X\right)/F$, and the projection morphism above factors through a canonical standard geometric morphism $\mathrm{Sh}\left(X\right)/F\to \mathrm{Sh}\left(X\right)$

${\pi }_{F}:\mathrm{Sh}\left({Y}_{F}\right)\stackrel{\simeq }{\to }\mathrm{Sh}\left(X\right)/F\to \mathrm{Sh}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_F : Sh(Y_F) \stackrel{\simeq}{\to} Sh(X)/F \to Sh(X) \,.

And conversely, every local homeomorphism $Y\to X$ of topological spaces corresponds to a geometric morphism of sheaf toposes of this form.

This motivates calling a geometric morphism

$𝒳\to 𝒴$\mathcal{X} \to \mathcal{Y}

a local homeomorphism of toposes or étale geometric morphism if it factors as an equivalence followed by a projection out of an overcategory topos.

If the topos is a locally ringed topos, or moro generally a structured (∞,1)-topos, it makes sense to require additionally that the local homeomorphism is compatible with the extra structure.

## Definition

For $H$ a topos (or (∞,1)-topos, etc.) and for $X\in H$ an object, the overcategory ${H}_{/X}$ is also a topos ($\left(\infty ,1\right)$-topos, etc), the slice topos (slice (∞,1)-topos, …).

The canonical projection ${\pi }_{!}:{H}_{/X}\to H$ is part of an essential (in fact, locally connected/ locally ∞-connected) geometric morphism:

$\pi =\left({\pi }_{!}⊣{\pi }^{*}⊣{\pi }_{*}\right):{H}_{/X}\stackrel{\stackrel{{\pi }_{!}}{\to }}{\stackrel{\stackrel{{\pi }^{*}}{←}}{\underset{{\pi }_{*}}{\to }}}H\phantom{\rule{thinmathspace}{0ex}}.$\pi = (\pi_! \dashv \pi^* \dashv \pi_*) : \mathbf{H}_{/X} \stackrel{\overset{\pi_!}{\to}}{\stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}} \mathbf{H} \,.

This is the base change geometric morphism for the terminal morphism $X\to *$.

### For toposes

###### Definition

A geometric morphism $K\to H$ is called a local homeomorphism of toposes, or an étale geometric morphism, if it is equivalent to such a projection— in other words, if it factors by geometric morphisms as $K\stackrel{\simeq }{\to }{H}_{/X}\stackrel{\pi }{\to }H$ for some $X\in H$ .

### For structured toposes

If the (∞,1)-toposes in question are structured (∞,1)-toposes, then this is refined to the following

###### Definition

A morphism $f:\left(𝒳,{𝒪}_{𝒳}\right)\to \left(𝒴,{𝒪}_{𝒴}\right)$ of structured (∞,1)-toposes is an étale morphism if

1. the underlying morphism of $\left(\infty ,1\right)$-toposes is an étale geometric morphism;

2. the induced map ${f}^{*}{𝒪}_{𝒴}\to {𝒪}_{𝒳}$ is an equivalence.

This is StSp, Def. 2.3.1.

## Examples

If $H$ is a localic topos $\mathrm{Sh}\left(S\right)$ over a topological space $S$ we have that $X\in \mathrm{Sh}\left(S\right)$ corresponds to an étalé space over $X$ and $H/X\to H$ to an étale map.

If $𝒢$ is a geometry (for structured (∞,1)-toposes) then for $f:U\to X$ an admissible morphism in $𝒢$, the induced morphism of structured (∞,1)-toposes

${\mathrm{Spec}}^{𝒢}U\to {\mathrm{Spec}}^{𝒢}X$Spec^\mathcal{G} U \to Spec^{\mathcal{G}} X

is an étale geometric morphism of structured $\left(\infty ,1\right)$-toposes.

This is StrSp, example 2.3.8.

## Properties

###### Proposition

(recognition of étale geometric morphisms)

A geometric morphism $\left({f}^{*}⊣{f}_{*}\right):K\to H$ is étale precisely if

1. it is essential;

2. ${f}_{!}$ is a conservative functor;

3. For every diagram $X\to Y←{f}_{!}Z$ in $H$ the induced diagram

$\begin{array}{ccc}{f}_{!}\left({f}^{*}X{×}_{{f}^{*}Y}Z\right)& \to & {f}_{!}Z\\ ↓& & ↓\\ X& \to & Y\end{array}$\array{ f_!(f^* X \times_{f^* Y} Z) &\to& f_! Z \\ \downarrow && \downarrow \\ X &\to& Y }

is a pullback diagram.

For (∞,1)-toposes this is HTT, prop. 6.3.5.11.

###### Proposition

(Recovering a topos from its etale overcategory)

For $H$ an $\left(\infty ,1\right)$-topos we have

$H\simeq \left(\left(\infty ,1\right)\mathrm{Topos}/H{\right)}_{\mathrm{et}}\phantom{\rule{thinmathspace}{0ex}},$\mathbf{H} \simeq ((\infty,1)Topos/\mathbf{H})_{et} \,,

where $\left(\left(\infty ,1\right)\mathrm{Topos}/H{\right)}_{\mathrm{et}}\subset \left(\infty ,1\right)\mathrm{Topos}/H$ is the full sub-(∞,1)-category of the over-(∞,1)-category on the etale geometric morphisms $K\to H$.

This is HTT, remark 6.3.5.10.

## References

The notion of local homeomorphisms of toposes is page 651 (chapter C3.3) of

The notion of étale geometric morphisms between (∞,1)-toposes is introduced in section 6.3.5 of

Discussion of the refinement to structured (∞,1)-toposes is in section 2.3 of

Revised on November 15, 2012 17:58:35 by Urs Schreiber (131.211.231.129)