Jonathon Funk | Searcharxiv

arXiv2026

Locally involutive semigroups

We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical ESN-correspondence between inverse semigroups and inductive groupoids. An important subcategory of locally involutive semigroups is formed by left involutive semigroups because the classifying topos of an inverse semigroup S is equivalent to the category of left involutive semigroups \'etale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup \'etale over S is actually an involutive semigroup. Any left involutive semigroup \'etale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive.

math.GR↗

arXiv2010

Characterizations of Morita equivalent inverse semigroups

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 étale 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.

math.CT↗