nLab étale 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

$$\pi_F : Sh(Y_F) \to Sh(X) \,.$$

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

$$\pi_F : Sh(Y_F) \stackrel{\simeq}{\to} Sh(X)/F \to Sh(X) \,.$$

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

$$\mathcal{X} \to \mathcal{Y}$$

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

Definition

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

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

$$\pi = (\pi_! \dashv \pi^* \dashv \pi_*) : \mathbf{H}_{/X} \stackrel{\overset{\pi_!}{\longrightarrow}}{\stackrel{\overset{\pi^*}{\longleftarrow}}{\underset{\pi_*}{\longrightarrow}}} \mathbf{H} \,.$$

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

For toposes

Definition

A geometric morphism $\mathbf{K} \to \mathbf{H}$ 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 $\mathbf{K} \stackrel{\simeq}{\to} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ for some $X \in \mathbf{H}$ .

For structured toposes

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

Definition

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

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

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

This is StSp, Def. 2.3.1.

Examples

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

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

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

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

This is StrSp, example 2.3.8.

Properties

Proposition

The inverse image of an étale geometric morphism is a cartesian closed functor.

See at cartesian closed functor for proof.

Therefore

Proposition

An étale geometric morphism is a cartesian Wirthmüller context.

Proposition

(recognition of étale geometric morphisms)

A geometric morphism $(f^* \dashv f_*) : \mathbf{K} \to \mathbf{H}$ 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 $\mathbf{H}$ the induced diagram

   $$\array{ f_!(f^* X \times_{f^* Y} Z) &\to& f_! Z \\ \downarrow && \downarrow \\ X &\to& Y }$$

   is a pullback diagram.

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

Proposition

(Recovering a topos from its etale overcategory)

For $\mathbf{H}$ an $(\infty,1)$-topos we have

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

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

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 .