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

Kudryavtseva, Ganna (author)

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 étale Boolean range semigroups are equivalent to biample topological categories. These results yield the equivalence between étale 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.

• Language: English
• Keywords: étale groupoid, étale category, Boolean inverse semigroup, Boolean restriction semigroup, Ehresmann semigroup, non-commutative Stone duality
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 48 str.
• Numbering: Vol. 474, art. 110313
• PID: 20.500.12556/RUL-169014
• UDC: 512
• ISSN on article: 0001-8708
• DOI: 10.1016/j.aim.2025.110313
• COBISS.SI-ID: 234994947
• Publication date in RUL: 07.05.2025

Record is a part of a journal

• Title: Advances in mathematics
• Shortened title: Adv. math.
• Publisher: Elsevier
• ISSN: 0001-8708
• COBISS.SI-ID: 24891904

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: étalni grupoid, étalna kategorija, Boolova inverzna polgrupa, Boolova omejitvena polgrupa, Ehresmannova polgrupa, nekomutativna Stonova dualnost

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0288
• Name: Algebra in njena uporaba

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: J1-60025
• Name: Interakcija aritmetičnih lastnosti in algebraične strukture v nekomutativnih kolobarjih