generally in differential cohesion
A topological groupoid or Lie groupoid is called an étale groupoid if the source-map is a local homeomorphism or local diffeomorphism, respectively, and hence exhibits the space of morphisms as an étale space over the space of objects.
This means that an étale groupoid is equivalently an internal groupoid in the category whose objects are topological spaces/smooth manifolds and whose morphisms are local homeomorphisms/diffeomorphisms.
Definition 1 is not invariant under the general notion of equivalence of Lie groupoids, the equivalence between them regarded as smooth groupoids, specifically as differentiable stacks (“Morita equivalence”).
But it does make sense to take an étale smooth groupoid to be a smooth groupoid/differentiable stack which is equivalent, as such, to, hence is presented by an étale Lie groupoid as in def. 1. This notion has been called folitation groupoid in (Crainic-Moerdijk 00).
The following characterizes foliation groupoids
For a Lie groupoid the following are equivalent
This is (Crainic-Moerdijk 00, theorem 1).
In the literature one finds, roughly speaking, two different approaches to the study of étale groupoids. One approach is based on the construction of the convolution algebras associated to an étale groupoid, in the spirit of Connes’ noncommutative geometry, and involves the study of cyclic and Hochschild homology and cohomology of these algebras. The other approach uses methods of algebraic topology such as the construction of the classifying space of an étale groupoid and its (sheaf) cohomology groups.
For an étale groupoid, there is a canonical morphism
(e.g. Carchedi 12, section 2.2)
The groupoid convolution algebra of a Lie groupoid with its canonical atlas remembered has the structure of a Hopf algebroid. In (Mrčun 99, Kališnik-Mrčun 07) étale Lie groupoids are characterized dually by their Hopf algebroids (a refinement of Gelfand duality to noncommutative topology).
Every topological space may be regarded as an étale groupoid with only identity morphisms.
A standard textbook account is section 5.5. of
The relation between étale groupoid and foliations is analyzed in detail in
See also at orbifold for basic and introductory literature.
Further discussion of étale groupoids and their properties includes