Inverse semigroup actions as groupoid actions

TL;DR

This paper establishes a correspondence between inverse semigroup actions and étale groupoid actions, providing a functorial construction that links these algebraic and topological structures, even in non-Hausdorff cases.

Contribution

It introduces a functorial association of étale groupoids to inverse semigroups, showing their actions are equivalent and enabling recovery of groupoids from inverse semigroups.

Findings

Constructs an étale groupoid from an inverse semigroup.

Shows the equivalence of actions on topological spaces.

Provides a method to recover a groupoid from its inverse semigroup.

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.