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arXiv · 2406.10624

Enriched aspects of calculus of relations and $2$-permutability

Maria Manuel Clementino·Diana Rodelo
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Abstract

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.

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Maria Manuel Clementino, Diana Rodelo. 2024-06-15. Enriched aspects of calculus of relations and $2$-permutability. https://arxiv.org/abs/2406.10624

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