Actions of \'etale-covered groupoids

Abstract

By restricting to a class of localic open groupoids $G$ which, similarly to Lie groupoids, possess appropriate covers $\widehat G\to G$ by \'etale groupoids, we extend results about groupoid actions and quantales that were previously proved for \'etale groupoids but do not seem to work for arbitrary open groupoids. In particular we obtain a characterization of the category of $G$-actions as a category of quantale modules on $\mathcal O(\widehat G)$ that satisfy a condition related to the quantale $\mathcal O(G)$. This leads to a simple description of $G$-sheaves and the classifying topos of $G$ in terms of Hilbert $\mathcal O(\widehat G)$-modules. The bicategory whose 1-cells are the groupoid bi-actions is bi-equivalent to a corresponding bicategory of quantales and bimodules.