The Bicategory of Lie Groupoids within Diffeological Groupoids

Abstract

We consider the localisation of the 2-category of diffeological groupoids at weak equivalences from the perspective of anafunctors, and with this language, prove that the localisation of the 2-category of Lie groupoids is an essentially full sub-bicategory of that of diffeological groupoids. In particular, we solve the open problem affirmatively of whether two Lie groupoids that are diffeologically Morita equivalent are Morita equivalent in the usual Lie sense.