On zigzag maps and the path category of an inverse semigroup

TL;DR

This paper characterizes inverse semigroups derived from zigzag maps on categories, showing their relation to Cuntz-Krieger $C^*$-algebras and establishing Morita equivalences among inverse semigroups.

Contribution

It provides an abstract characterization of inverse semigroups from zigzag maps and demonstrates their Morita equivalence to all inverse semigroups, linking algebraic and operator algebra structures.

Findings

Inverse semigroups from zigzag maps are characterized abstractly.

Every inverse semigroup is Morita equivalent to a zigzag map inverse semigroup.

Cuntz-Krieger $C^*$-algebras include tight $C^*$-algebras of all countable inverse semigroups.

Abstract

We study the path category of an inverse semigroup admitting unique maximal idempotents and give an abstract characterization of the inverse semigroups arising from zigzag maps on a left cancellative category. As applications we show that every inverse semigroup is Morita equivalent to an inverse semigroup of zigzag maps and hence the class of Cuntz-Krieger $C^*$-algebras of singly aligned categories include the tight $C^*$-algebras of all countable inverse semigroups, up to Morita equivalence.