In mathematics, the étale topos of a scheme ''X'' is the category of all étale sheaves on ''X''. An étale

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper s ...

is a sheaf on the étale site of ''X''.

Definition

Let ''X'' be a scheme. An ''étale covering'' of ''X'' is a family

\_

, where each

\varphi_i

is an étale morphism of schemes, such that the family is jointly surjective that is

X = \bigcup_ \varphi_i(U_i)

. 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 In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is c ...

, the ''étale topology'' on ''X''. The category together with the étale topology on it is called the ''étale site'' on ''X''. The ''étale topos''

X^\text

of a scheme ''X'' is then the category of all sheaves of sets on the site Ét(''X''). Such sheaves are called étale sheaves on ''X''. In other words, an étale sheaf

\mathcal F

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 :

0 \to \mathcal F(U) \to \prod_ \mathcal F(U_i)   \prod_ \mathcal F(U_)

is exact, where

U_ = U_i \times_U U_j

. {{DEFAULTSORT:Etale topos Topos theory Sheaf theory