Principal bundles of open groupoids

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.