Diffeological Morita Equivalence

Abstract: We introduce a new notion of Morita equivalence for diffeological groupoids, generalising the original notion for Lie groupoids. For this we develop a theory of diffeological groupoid actions, -bundles and -bibundles. We define a notion of principality for these bundles, which uses the notion of a subduction, generalising the notion of a Lie group(oid) principal bundle. We say two diffeological groupoids are Morita equivalent if and only if there exists a biprincipal bibundle between them. Using a Hilsum-Skandalis tensor product, we further define a composition of diffeological bibundles, and obtain a bicategory DiffBiBund. Our main result is the following: a bibundle is biprincipal if and only if it is weakly invertible in this bicategory. This generalises a well known theorem from the Lie groupoid theory. As an application of the framework, we prove that the orbit spaces of two Morita equivalent diffeological groupoids are diffeomorphic. We also show that the property of a diffeological groupoid to be fibrating, and its category of actions, are Morita invariants.