Characterizations of Morita equivalent inverse semigroups

TL;DR

This paper establishes the equivalence of four different notions of Morita equivalence for inverse semigroups and explores the categorical structure of their actions, linking semigroup theory with topos and $C^*$-algebra perspectives.

Contribution

It demonstrates the equivalence of multiple Morita notions for inverse semigroups and characterizes their action categories as monadic and equivalent to presheaf categories.

Findings

Four notions of Morita equivalence are shown to be equivalent.

The category of unitary actions is monadic over étale actions.

Categories of actions are equivalent to presheaf categories.

Abstract

We prove that four different notions of Morita equivalence for inverse semigroups motivated by, respectively, $C^{\ast}$-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of \'etale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.