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 étale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup étale over S is actually an involutive semigroup. Any left involutive semigroup étale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive.