Morita equivalence of pseudogroups

TL;DR

This paper establishes a new notion of Morita equivalence for pseudogroups by leveraging their correspondence with inverse quantal frames and localic étale groupoids, drawing analogies with $C^{ ext{*}}$-algebra theory.

Contribution

It introduces Morita equivalence for pseudogroups using biprincipal bisheaves and explores its implications and applications in the context of localic groupoids.

Findings

Morita equivalence for pseudogroups is characterized via localic étale groupoids.

The new definition aligns with classical Morita equivalence in $C^{ ext{*}}$-algebra theory.

The approach provides a framework for analyzing pseudogroups through inverse quantal frames.

Abstract

We take advantage of the correspondence between pseudogroups and inverse quantal frames, and of the recent description of Morita equivalence for inverse quantal frames in terms of biprincipal bisheaves, to define Morita equivalence for pseudogroups and to investigate its applications. In particular, two pseudogroups are Morita equivalent if and only if their corresponding localic \'etale groupoids are. We explore the clear analogies between our definition of Morita equivalence for pseudogroups and the usual notion of strong Morita equivalence for $C^{\ast}$-algebras and these lead to a number of concrete results.