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

TL;DR

This paper extends the calculus of internal relations to enriched categories, introducing the concept of Ord-Mal'tsev categories and characterizing their properties through ideals and enriched relations.

Contribution

It develops an enriched calculus of relations for regular Ord-categories and introduces the notion of Ord-Mal'tsev categories with new characterizations.

Findings

Enriched calculus of relations using ideals in Ord-categories

Introduction of Ord-Mal'tsev categories and their properties

Examples of categories that are Ord-Mal'tsev but not Mal'tsev

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.