Quantales of open groupoids

TL;DR

This paper extends the relationship between inverse semigroups and étale groupoids to a broader class of open groupoids using open quantal frames, with implications for operator algebras and geometry.

Contribution

It introduces open quantal frames as a new class of quantales to generalize the inverse semigroup-groupoid correspondence beyond étale cases.

Findings

Open quantal frames generalize inverse quantal frames.

A new correspondence between quantales and open groupoids is established.

The approach relies on inverse semigroups of local bisections.

Abstract

It is well known that inverse semigroups are closely related to \'etale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic \'etale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially ``the same'' as inverse quantal frames, and in this paper we exploit this fact in order to find a suitable replacement for inverse semigroups in the context of open groupoids that are not necessarily \'etale. The interest of such a generalization lies in the importance and ubiquity of open groupoids in areas such as operator algebras, differential geometry and topos theory, and we achieve it by means of a class of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail. The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning \'etale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved.