Abstract
To an inverse semigroup, we associate an \'etale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn \'etale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.
Highlights
Étale topological groupoids are closely related to actions of inverse semigroups on topological spaces by partial homeomorphisms
In order to construct an étale topological groupoid out of an inverse semigroup, we first need it to act on some topological space
Our non-Hausdorff topology has the following crucial feature: if Gr(S) is the étale groupoid of germs for the action of S on E, the category of actions of S on topological spaces is equivalent to the category of actions of Gr(S) on topological spaces
Summary
Used, in particular, to study actions of étale topological groupoids on C∗-algebras and their crossed products. The map S → Gr(S) is functorial, and left adjoint to the functor Bis that maps an étale topological groupoid G to its inverse semigroup of bisections Bis(G), provided we turn étale groupoids into a category in an unusual way, using a notion of morphism due to Zakrzewski (see [2]): an algebraic morphism from G to H, denoted G H, is an action of G on the arrow space of H that commutes with the right translation action of H. Groupoid C∗-algebras are functorial for algebraic morphisms, but not for continuous functors (see [1, 2]); the orbit space is functorial for continuous functors, but not for algebraic morphisms
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