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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.10624 |
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| _version_ | 1866914312508932096 |
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| author | Clementino, Maria Manuel Rodelo, Diana |
| author_facet | Clementino, Maria Manuel Rodelo, Diana |
| contents | The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category C. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely the ideals in C. We then focus on an enriched version of the 1-dimensional algebraic 2-permutable (also called Mal'tsev) property and its well-known equivalent characterisations expressed through properties on ordinary relations. We introduce the notion of Ord-Mal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enriched aspects of calculus of relations and $2$-permutability Clementino, Maria Manuel Rodelo, Diana Category Theory The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category C. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely the ideals in C. We then focus on an enriched version of the 1-dimensional algebraic 2-permutable (also called Mal'tsev) property and its well-known equivalent characterisations expressed through properties on ordinary relations. We introduce the notion of Ord-Mal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev categories. |
| title | Enriched aspects of calculus of relations and $2$-permutability |
| topic | Category Theory |
| url | https://arxiv.org/abs/2406.10624 |