nLab
etale geometric morphism

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

$${\pi }_F:\mathrm{Sh}(Y_F)\to \mathrm{Sh}(X).$$\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 $\mathrm{Sh}(X)/F$, and the projection morphism above factors through a canonical standard geometric morphism $\mathrm{Sh}(X)/F\to \mathrm{Sh}(X)$

$${\pi }_F:\mathrm{Sh}(Y_F)\stackrel{\simeq }{\to }\mathrm{Sh}(X)/F\to \mathrm{Sh}(X).$$\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 ($(\mathrm{\infty },1)$-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:

$$\pi =({\pi }_{!}⊣{\pi }^{*}⊣{\pi }_{*}):H_{/X}\stackrel{\stackrel{{\pi }_{!}}{\to }}{\stackrel{\stackrel{{\pi }^{*}}{\leftarrow }}{\underset{{\pi }_{*}}{\to }}}H.$$\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 $f:(𝒳,{𝒪}_{𝒳})\to (𝒴,{𝒪}_{𝒴})$ of structured (∞,1)-toposes is an étale morphism if

1. the underlying morphism of $(\mathrm{\infty },1)$-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 $\mathrm{Sh}(S)$ over a topological space $S$ we have that $X\in \mathrm{Sh}(S)$ 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 $(\mathrm{\infty },1)$-toposes.

This is StrSp, example 2.3.8.

Properties

Proposition

(recognition of étale geometric morphisms)

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

1. it is essential;

2. $f_{!}$ is a conservative functor;

3. For every diagram $X\to Y\leftarrow f_{!}Z$ in $H$ the induced diagram

   $$\begin{array}{ccc}{f}_{!}(f^{*}X{\times }_{f^{*}Y}Z)& \to & f_{!}Z\\ \downarrow & & \downarrow \\ X& \to & Y\end{array}$$\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 $(\mathrm{\infty },1)$-topos we have

$$H\simeq ((\mathrm{\infty },1)\mathrm{Topos}/H{)}_{\mathrm{et}},$$\mathbf{H} \simeq ((\infty,1)Topos/\mathbf{H})_{et} \,,

where $((\mathrm{\infty },1)\mathrm{Topos}/H{)}_{\mathrm{et}}\subset (\mathrm{\infty },1)\mathrm{Topos}/H$ 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.

Related concepts

• slice topos, slice (∞,1)-topos

References

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

• Peter Johnstone, Sketches of an Elephant .

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

• Jacob Lurie, Higher Topos Theory .

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

• Jacob Lurie, Structured Spaces .