Etale groupoids as germ groupoids and their base extensions

TL;DR

This paper develops a new framework connecting inverse semigroups, etale groupoids, and their representations, providing a unified description and characterizations that extend to quantales and localic groupoids.

Contribution

It introduces wide representations of inverse semigroups and establishes a correspondence between etale groupoids and complete, infinitely distributive inverse monoids.

Findings

Space of germs forms an etale groupoid with suitable topology

Characterization of inverse semigroups arising from groupoids

Extension of the correspondence to quantales and localic groupoids

Abstract

We introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation which has the structure of an etale groupoid. This gives an elegant description of Paterson's universal groupoid and of the translation groupoid of Skandalis, Tu, and Yu. In addition we characterize the inverse semigroups that arise from groupoids, leading to a precise bijection between the class of etale groupoids and the class of complete and infinitely distributive inverse monoids equipped with suitable representations, and we explain the sense in which quantales and localic groupoids carry a generalization of this correspondence.