Inverse semigroup actions as groupoid actions

Abstract

To an inverse semigroup, we associate an étale 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 étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.

Corrected a typo in Lemma 2.14 in the published version

References in corpus (1)

• C*-Algebras over Topological Spaces: The Bootstrap Class

Cited by in corpus (9)

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• Actions of étale-covered groupoids
• Functoriality of groupoid quantales. II
• Relating ample and biample topological categories with Boolean restriction and range semigroups
• Étale inverse semigroupoids - the fundamentals