Inverse semigroup actions as groupoid actions

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.