  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.
