Enriched aspects of calculus of relations and 2-permutability

The aim of this work is to further develop the calculus of (internal) relations for a regular [Formula: see text]-category [Formula: see text]. To capture the enriched features of a regular [Formula: see text]-category and obtain a good calculus, the relations we work with are precisely the ideals in [Formula: see text]. We then focus on an enriched version of the 1-dimensional algebraic [Formula: see text]-permutable (also called Mal’tsev) property and its well-known equivalent characterizations expressed through properties on ordinary relations. We introduce the notion of [Formula: see text]-Mal’tsev category and show that these may be characterized through enriched versions of the above-mentioned properties adapted to ideals. Any [Formula: see text]-enrichment of a 1-dimensional Mal’tsev category is necessarily an [Formula: see text]-Mal’tsev category. We also give some examples of categories which are not Mal’tsev categories, but are [Formula: see text]-Mal’tsev categories.