Hausdorff Tight Groupoids Generalised

TL;DR

This paper extends the Hausdorff tight groupoid construction to more general locally compact étale groupoids and demonstrates a duality between inverse semigroups and open bisections, broadening classical Stone duality.

Contribution

It generalizes Exel's tight groupoid construction to all Hausdorff locally compact étale groupoids and establishes a duality with inverse semigroups via pseudobases.

Findings

Extended tight groupoid construction to general Hausdorff étale groupoids

Represented inverse semigroups as pseudobases of open bisections

Established a duality encompassing extensions of Stone duality

Abstract

We extend Exel's ample tight groupoid construction to general locally compact \'etale groupoids in the Hausdorff case. Moreover, we show how inverse semigroups are represented in this way as 'pseudobases' of open bisections, thus yielding a duality which encompasses various extensions of the classic Stone duality.