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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.
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).
An étale map between smooth spaces is a local diffeomorphism, which is in particular a local homeomorphism on the underlying topological spaces.
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.
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?
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
Last revised on May 26, 2023 at 10:20:01. See the history of this page for a list of all contributions to it.