Equivariant E-theory for groupoids acting on C∗-algebras

We define equivariant asymptotic morphisms between C0(X)-algebras and construct the category of homotopy classes of such morphisms. Based on this category we construct the category of homotopy classes of asymptotic morphisms which are equivariant with respect to an action of a groupoid. We then define and study the corresponding equivariant E-theory.