Dmitry Matsnev*
Instituto Superior Técnico, Lisbon, Portugal
Étale groupoids as germ groupoids and their extensions
Every étale topological groupoid G gives rise to an inverse
semigroup (collection of open local sections of G). This semigroup
is naturally equipped with a full representation on the space of units of G. The germs of such representation can be given the structure of an étale groupoid which turns out to be isomorphic to G.
We extend this construction to unital representations of inverse
monoids and, more generally, to `wide' inverse semigroups over a
topological space, which allows one to effectively construct éetale
groupoid extensions by extending or modifying the underying inverse semigroup.
Two applications of such technique are of particular interest: the
Stone-Čech compactification of the unit space of the given groupoid,
which resembles some elements of the translation groupoid of Skandalis,
Tu, and Yu (but without restriction to the discrete case) and the patch topology
extension, which imitates Paterson's universal groupoid without irrelevant
restrictions pertaining to the unit space.
* This is a joint work with Pedro Resende.
