Groupoid Quantales: a non \'etale setting

TL;DR

This paper extends the construction of unital involutive quantales from étale topological groupoids to non-étale groupoids with sober unit spaces, establishing a new algebraic framework.

Contribution

It introduces a novel class of quantales associated with non-étale groupoids and proves a representability theorem under spatiality conditions.

Findings

Constructs a canonical unital involutive quantale for non-étale groupoids.

Defines and axiomatizes the class of quantales arising from inverse semigroups of G-sets.

Proves a representability theorem linking these quantales to the groupoid structure.

Abstract

It is well known that if G is an \'etale topological groupoid then its topology can be recovered as the sup-lattice generated by G-sets, i.e. by the images of local bisections. This topology has a natural structure of unital involutive quantale. We present the analogous construction for any non \'etale groupoid with sober unit space G_0. We associate a canonical unital involutive quantale with any inverse semigroup of G-sets which is also a sheaf over G_0. We introduce axiomatically the class of quantales so obtained, and revert the construction mentioned above by proving a representability theorem for this class of quantales, under a natural spatiality condition.