Groupoid Actions on Sets, Duality and a Morita Context
Let G and K be groupoids. We present the notion of a (G_α,K_β)-set and we prove a duality theorem in this context, which extends the duality theorem for graded algebras by groups. For A a unital G-graded algebra and X a finite split G-set, we show that there is an isomorphism between the category of the left A-modules X-graded and the category of the left A \#_α^{G}X-modules. As an application of this isomorphism, we construct a Morita context.
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