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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.13301 |
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| _version_ | 1866917211114831872 |
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| author | Berger, Clemens Funk, Jonathon |
| author_facet | Berger, Clemens Funk, Jonathon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13301 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Locally involutive semigroups Berger, Clemens Funk, Jonathon Group Theory Category Theory Primary 20M10, 20M18, Secondary 18B40, 20M30 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. |
| title | Locally involutive semigroups |
| topic | Group Theory Category Theory Primary 20M10, 20M18, Secondary 18B40, 20M30 |
| url | https://arxiv.org/abs/2601.13301 |