The Bicategory of Lie Groupoids within Diffeological Groupoids

TL;DR

This paper demonstrates that the bicategory of Lie groupoids is essentially a full sub-bicategory of diffeological groupoids when localised at weak equivalences, resolving an open problem about Morita equivalences.

Contribution

It proves that diffeological Morita equivalence implies classical Lie Morita equivalence for Lie groupoids.

Findings

The localisation of Lie groupoids is an essentially full sub-bicategory of diffeological groupoids.

Diffeological Morita equivalence implies Lie Morita equivalence for Lie groupoids.

The open problem about Morita equivalences is affirmatively solved.

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.