Pseudogroups of symmetries and Morita equivalences

TL;DR

This paper revisits classical pseudogroup theory using modern Lie groupoid concepts, introducing Pfaffian Morita equivalences and exploring their applications to symmetry pseudogroups.

Contribution

It introduces the notion of Pfaffian Morita equivalence for Lie groupoids with multiplicative forms, connecting classical pseudogroup theory with modern geometric frameworks.

Findings

Defines Pfaffian Morita equivalence and relates it to gauge constructions.

Analyzes interactions between principal actions and Morita equivalences.

Provides examples and applications to symmetry pseudogroups.

Abstract

This work is a spin-off of an on-going programme which aims at revisiting the original studies of Lie and Cartan on pseudogroups and geometric structures from a modern perspective. Within the framework of Lie groupoids equipped with a special multiplicative form - called Pfaffian groupoids - we focus on principal bibundles and Morita equivalences. In particular, we discuss in details the notion of Pfaffian Morita equivalence, its relation to the gauge construction in the Pfaffian setting, and its interactions with principal actions. We briefly present some examples and applications to transitive pseudogroups of symmetries, which we explored in great detail in arXiv:2211.16639.