Categorified groupoid-sets and their Burnside ring

Abstract

We explore the category of internal categories in the usual category of (right) group-sets, whose objects arereferred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relationbetween two categorified group-sets is given by a particular equivalence between the underlying categories. We alsoexhibit some of the differences between the old Burnside theory and the new one. Lastly, we briefly explain how toextend these new techniques and concepts to the context of groupoids, employing the categories of (right) groupoid-sets,aiming by this to give an alternative approach to the classical Burnside ring of groupoids.