Groupoid Actions on Sets, Duality and a Morita Context

TL;DR

This paper extends duality theorems from graded algebras to groupoid actions, establishing a Morita context through a new duality involving (G_{eta},K_{eta})-sets and module categories.

Contribution

It introduces the concept of (G_{eta},K_{eta})-sets and proves a duality theorem that generalizes graded algebra duality to groupoid actions, leading to a Morita context.

Findings

Established a duality theorem for groupoid actions on sets.

Constructed an isomorphism between module categories for G-graded algebras.

Developed a Morita context from the duality and module equivalences.

Abstract

Let G and K be groupoids. We present the notion of a (G_{\alpha},K_{\beta})-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 \#_{\alpha}^{G}X-modules. As an application of this isomorphism, we construct a Morita context.