Étale groupoids as germ groupoids and applications (part III)
Every etale topological groupoid gives rise to an inverse semigroup equipped with a natural representation on the space of units of . The germs of such representation can be given the structure of an etale groupoid which turns out to be isomorphic to . We extend this construction to `wide' inverse semigroups over a topological space, which allows one to effectively construct etale groupoid extensions by extending or modifying the underlying inverse semigroup. We use this machinery in order to provide a simpler construction of Paterson's universal groupoid of an inverse semigroup, and also of the \'etale groupoid that arises by applying Stone--Cech compactification to the unit space of the pair groupoid on a set, which is the translation groupoid of Skandalis, Tu, and Yu used in their study of the Novikov conjecture by coarse geometric methods.
