Pseudogroups of symmetries and Morita equivalences

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.