Groupoid quantales: A non-etale setting

We establish a bijective correspondence involving a class of unital involutive quantales and a class of groupoids whose space of units is a sober space. This class includes equivalence relations that arise from group actions. The resulting axiomatization of the class of quantales, as well as the correspondence defined here, extend the theory of etale groupoids and their quantales (Resende (2007) [10]) to a point-set, non-etale setting. (C) 2010 Elsevier BM. All rights reserved.