A colimit construction for groupoids

Abstract

We consider Ore monoid actions in a certain bicategory of étale groupoids Gr_prop. Examples of such actions include self-similar groups, higher rank graphs and actions of Ore monoids on spaces by topological correspondences. We prove that every Ore monoid action in Gr_prop has a colimit. We construct a functor from Gr_prop to the bicategory of C*-correspondences Corr. We prove that this functor preserves colimits of Ore monoid actions. We write the colimit of an Ore monoid action concretely, and in doing so provide a groupoid model for the Cuntz--Pimsner algebra of the product system associated with the action. In the second part of this thesis, we study colimit equivalence in the bicategories Corr and Gr. We show that under certain assumptions on a diagram, cofinal subdiagrams have equivalent colimits. This generalises the notions of shift equivalences of graphs and C*-correspondences.