Etale groupoids as germ groupoids and their base extensions

Details Etale groupoids as germ groupoids and their base extensions Etale groupoids as germ groupoids and their base extensions

2008-07-24

arxiv.org/abs/0807.3916v1

Proc. Edinb. Math. Soc. 53 (2010) 765-785

Mathematics

General Topology

25 pages

Scientific paper

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.

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