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We extend the notion of the enveloping semigroup of a locally compact group to
the enveloping semigroupoid of a locally compact groupopid and show that there is a
universal enveloping semigroupoid which is unique up to isomorphism. As in the group
case, we associate the Ellis semigroupoid to an action of a locally compact groupoid
on a fibrewise compact space. We define the notion of proximality for groupoid actions
and characterize it in terms of Ellis semigroupoid.
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