Principal bundles of open groupoids

TL;DR

This paper establishes a bijective correspondence between principal bundles of open groupoids and principal locales of their quantales, generalizing supported modules for groupoid quantales, thus deepening the understanding of their structural relationship.

Contribution

It introduces a new correspondence linking principal bundles of open groupoids with principal locales of their quantales, extending the theory through a generalization of supported modules.

Findings

Established a bijective correspondence between principal G-bundles and principal locales.

Generalized supported modules for groupoid quantales.

Connected the structure of open groupoids with their quantale representations.

Abstract

Taking into account the correspondence between open groupoids and their quantales, we establish a bijective correspondence between the principal $G$-bundles whose left projection is an open surjection and the principal $\opens(G)$-locales for any open groupoid $G$ and its groupoid quantale $\opens(G)$, the latter requires a generalization of the supported modules for groupoid quantales.