étale map



Étale morphisms




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.


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 E_\infty-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


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).


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

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

Last revised on January 7, 2020 at 01:17:35. See the history of this page for a list of all contributions to it.