Groupoid actions as quantale modules

TL;DR

This paper characterizes the actions of localic etale groupoids using modules over their associated quantales, providing a new algebraic perspective and representation of etendues.

Contribution

It introduces a module-theoretic description of groupoid actions, linking localic groupoids with quantale modules and offering a novel quantale-based representation of etendues.

Findings

Category of G-actions is isomorphic to O(G)-modules

Provides simple descriptions of actions including on open maps and sheaves

Establishes a new quantale-based representation of etendues

Abstract

For an arbitrary localic etale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of etendues.