Groupoid actions on sets, duality and a morita context

Abstract

Let and be groupoids. We present the notion of a -set and we prove a duality theorem in this context, which extends the duality theorem for graded algebras by groups. For A a unital -graded algebra and X a finite split -set, we show that there is an isomorphism between the category of the X-graded left A-modules and the category of the left -modules. As an application of this isomorphism, we construct a Morita context.