Relating ample and biample topological categories with Boolean restriction and range semigroups

TL;DR

This paper extends the equivalence between restriction monoids and ample categories to Boolean range semigroups, establishing new categorical correspondences and characterizations in the context of topological and algebraic structures.

Contribution

It introduces the equivalence between Boolean range semigroups and ample topological categories, including étale cases, and characterizes when Boolean restriction semigroups admit compatible cosupport operations.

Findings

Boolean range semigroups are equivalent to ample topological categories.

Étale Boolean range semigroups correspond to biample topological categories.

The paper recovers known equivalences between Boolean birestriction semigroups and biample topological categories.

Abstract

We extend the equivalence by Cockett and Garner between restriction monoids and ample categories to the setting of Boolean range semigroups which are non-unital one-object versions of range categories. We show that Boolean range semigroups are equivalent to ample topological categories where the range map $r$ is open, and \'etale Boolean range semigroups are equivalent to biample topological categories. These results yield the equivalence between \'etale Boolean range semigroups and Boolean birestriction semigroups and a characterization of when a Boolean restriction semigroup admits a compatible cosupport operation. We also recover the equivalence between Boolean birestriction semigroups and biample topological categories by Kudryavtseva and Lawson. Our technique builds on the usual constructions relating inverse semigroups with ample topological groupoids via germs and slices.