In mathematics, the étale topos of a scheme X is the category of all étale sheaves on X.
An étale sheaf is a sheaf on the étale site of X.
An étale covering of X is a family
is an étale morphism of schemes, such that the family is jointly surjective that is
The category Ét(X) is the category of all étale schemes over X.
The collection of all étale coverings of a étale scheme U over X i.e. an object in Ét(X) defines a Grothendieck pretopology on Ét(X) which in turn induces a Grothendieck topology, the étale topology on X.
The category together with the étale topology on it is called the étale site on X.
of a scheme X is then the category of all sheaves of sets on the site Ét(X).
In other words, an étale sheaf
is a (contravariant) functor from the category Ét(X) to the category of sets satisfying the following sheaf axiom: For each étale U over X and each étale covering
of U the sequence is exact, where