Relating ample and biample topological categories with Boolean restriction and range semigroups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913824771145728 |
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| author | Kudryavtseva, Ganna |
| author_facet | Kudryavtseva, Ganna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relating ample and biample topological categories with Boolean restriction and range semigroups Kudryavtseva, Ganna Rings and Algebras Category Theory 22A22, 22A30, 20M18, 18B40, 06E15, 06E75 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. |
| title | Relating ample and biample topological categories with Boolean restriction and range semigroups |
| topic | Rings and Algebras Category Theory 22A22, 22A30, 20M18, 18B40, 06E15, 06E75 |
| url | https://arxiv.org/abs/2410.22181 |