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 diffeomorphisms, 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 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.

Between toposes

• étale geometric morphism

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)