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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

Due to the special nature of $Y_F$, the topos on the left is equivalent to the overcategory 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 moro 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). This is sometimes called the petit topos associated to $X \in \mathbf{H}$.

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 $X \to *$.

For toposes

For structured toposes

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

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

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

This is StrSp, example 2.3.8.

Properties

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

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 .