The Bicategory of Lie Groupoids within Diffeological Groupoids

TL;DR

The paper develops a localisation of the $2$-category of diffeological groupoids at weak equivalences using anafunctors and demonstrates that the localisation of Lie groupoids embeds as an essentially full sub-bicategory of the localisation for diffeological groupoids. It proves that a Morita equivalence between Lie groupoids, realized as a biprincipal diffeological bibundle, is already a Lie Morita equivalence, thereby affirmatively answering van der Schaaf's open question. The authors further connect the anafunctor localisation with the bibundle localisation, establishing an equivalence of bicategories ${f DGpoid}_{ ext{ana}} o {f DBiBund}$ and showing that Morita equivalences are preserved under this transition. They also illustrate the framework with applications to orbit spaces, inertia groupoids, and principal bundles, highlighting Morita-invariant constructions in the difféologique setting and their connections to Čech cohomology in the abelian case.

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.

Theorems & Definitions (65)

• Definition 2.1: Diffeology
• Definition 2.3: Inductions & Subductions
• Definition 2.5: Diffeological Groupoid
• Remark 2.6
• Definition 3.1: Weak Equivalence
• Remark 3.2
• Definition 3.3: Singleton Strict Pretopology
• Lemma 3.4
• proof
• Proposition 3.5: $(\mathbf{Diffeol},{\mathcal{J}})$ is a $2$-Site