Could not include topos theory - contents
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
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 $Sh(X)/F \to Sh(X)$
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
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.
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:
This is the base change geometric morphism for the terminal morphism $X \to *$.
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 $X \in \mathbf{H}$ .
If the (∞,1)-toposes in question are structured (∞,1)-toposes, then this is refined to the following
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
the underlying morphism of $(\infty,1)$-toposes is an étale geometric morphism;
the induced map $f^* \mathcal{O}_\mathcal{Y} \to \mathcal{O}_\mathcal{X}$ is an equivalence.
This is StSp, Def. 2.3.1.
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
is an étale geometric morphism of structured $(\infty,1)$-toposes.
This is StrSp, example 2.3.8.
The inverse image of an étale geometric morphism is a cartesian closed functor.
See at cartesian closed functor for proof.
Therefore
An étale geometric morphism is a cartesian Wirthmüller context.
(recognition of étale geometric morphisms)
A geometric morphism $(f^* \dashv f_*) : \mathbf{K} \to \mathbf{H}$ is étale precisely if
it is essential;
$f_!$ is a conservative functor;
For every diagram $X \to Y \leftarrow f_! Z$ in $\mathbf{H}$ the induced diagram
is a pullback diagram.
For (∞,1)-toposes this is HTT, prop. 6.3.5.11.
(Recovering a topos from its etale overcategory)
For $\mathbf{H}$ an $(\infty,1)$-topos we have
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.
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