nLab étale map

Idea

The notion of étale map is an abstraction of that of local homeomorphism in topology. The concept is usually found in places with a geometric or topological flavour.

Examples

Between topological spaces

An étale map between topological spaces is a local homeomorphism; see étalé space (which is the total space of such a map viewed as a bundle).

Between smooth manifolds

An étale map between smooth spaces is a local diffeomorphism, which is in particular a local homeomorphism on the underlying topological spaces.

Between schemes (affine schemes / rings)

For an étale map between schemes see étale morphism of schemes.

Restricted to affine schemes, this yields, dually, a notion of étale morphisms between rings. Étale maps between noncommutative rings have also been considered.

Between $E_\infty$-rings

• étale morphism of E-∞ rings

Between analytic spaces

• Étale maps between analytic spaces are closely related to étale maps between schemes, while also (when the spaces are smooth) a special case of an étale map between smooth spaces.

Zoran: I do not understand this statement. Analytic spaces have a different structure sheaf; in general nilpotent elements are allowed. This is additional structure not present in theory of smooth spaces.

Toby: Is it correct now?

Between toposes

• étale geometric morphism

Between anabelioids

• finite étale morphism of anabelioids

Between presheaves

Due to (Joyal-Moerdijk 1994): an étale map of presheaves is one which is cartesian qua natural transformation.

Axiomatization

The idea of étale morphisms can be axiomatized in any topos. This idea goes back to lectures by André Joyal in the 1970s. See (Joyal-Moerdijk 1994) and (Dubuc 2000).

Related concepts

• open morphism, closed morphism

• separated morphism

• formally étale morphism

• log-étale morphism

References

Axiomatizations of the notion of étale maps in general toposes are discussed in

• André Joyal, Ieke Moerdijk, A completeness theorem for open maps, Annals of Pure and Applied Logic 70, Issue 1, 18 November 1994, p. 51-86, MR95j:03104, doi

• Eduardo Dubuc, Axiomatic etal maps and a theory of spectrum, Journal of pure and applied algebra 149 (2000) 15–45