A categorical approach to inverse semigroups, étale groupoids, and their C*-algebras

Abstract: We introduce a category of inverse semigroup actions and a category of \'etale groupoids. We show that there are three functors which send inverse semigroups to their spectral actions, inverse semigroup actions to their transformation groupoids, and \'etale groupoids to their groupoid C*-algebras, respectively. The composition of these functors is naturally equivalent to the already known functor which sends inverse semigroups to their C*-algebras. This result is a categorical version of the theorem by Paterson which states that the inverse semigroup C*-algebras coincide the groupoid C*-algebras of the universal groupoids. We also construct a functor from the category of \'etale groupoids to the category of inverse semigroup actions sending \'etale groupoids to their slice actions, and show that this functor is right adjoint to the functor sending inverse semigroup actions to their transformation groupoids.