higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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 diffeomorphisms, 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.
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