Functoriality of groupoid quantales. II

TL;DR

This paper explores the functorial properties of groupoid quantales, focusing on principal bundles, Hilsum-Skandalis maps, and Morita equivalence through the lens of inverse quantal frames and modules, extending the algebraic framework of groupoids.

Contribution

It introduces a module-theoretic approach to principal bundles and Morita equivalence for étale groupoids using inverse quantal frames, advancing the algebraic understanding of these structures.

Findings

Characterization of principal bundles via modules on inverse quantal frames

Formulation of Morita equivalence using bimodules resembling C*-algebra imprimitivity bimodules

Extension of quantale-theoretic methods to study groupoid equivalences

Abstract

Taking advantage of the quantale-theoretic description of \'etale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms of modules on inverse quantal frames. The Hilbert module description of quantale sheaves leads naturally to a formulation of Morita equivalence in terms of bimodules that resemble imprimitivity bimodules of C*-algebras.