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 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.

💡 Research Summary

The paper develops a calculus of internal relations for regular Ord‑categories, where the appropriate notion of a relation is an ideal rather than a plain span. After recalling the standard theory of ordinary relations in regular categories, the authors adapt the concepts of monomorphism and strong epimorphism to the Ord‑enriched setting, introducing ff‑morphisms (fully faithful morphisms) and so‑morphisms (object‑surjective morphisms). A regular Ord‑category is defined by four conditions: existence of finite weighted limits, an (so‑morphism, ff‑morphism) factorisation system, stability of so‑morphisms under 2‑pullbacks, and the requirement that every so‑morphism be a bicoinserter of its comma object. The bicoinserter replaces the usual coinserter because antisymmetry is not assumed in Ord‑enrichment.

With this infrastructure, the authors turn to the 2‑permutability (Mal’tsev) property. In ordinary regular categories, a Mal’tsev category is characterised by the difunctionality of all relations, the commutativity of composition of equivalence relations, or the fact that every reflexive relation is an equivalence relation. The paper lifts these characterisations to the enriched context by replacing ordinary relations with ideals. An ideal D : X → Y satisfies an enriched difunctionality condition: for any object A and morphisms x, u : A → X and y, v : A → Y, if (x,y), (u,y) and (u,v) belong to D then (x,v) also belongs to D. This condition defines an Ord‑Mal’tsev category.

The main theoretical results are: (1) any Ord‑enrichment of a (1‑dimensional) Mal’tsev category is automatically Ord‑Mal’tsev; (2) in a regular Ord‑category, the following statements are equivalent: every ideal is difunctional, every reflexive ideal is an equivalence ideal, the composition of ideals is commutative, and each ideal is its own opposite. These are the enriched analogues of the classical Mal’tsev characterisations (Theorem 5.9). The proofs rely on a careful development of the calculus of ideals: composition is defined via the (so‑morphism, ff‑morphism) factorisation of the induced span, and the associativity of composition follows from Lemma 2.2 adapted to the enriched setting. The paper also shows that the stability of so‑morphisms under comma objects holds in regular Ord‑categories, which is essential for the bicoinserter construction.

To illustrate that Ord‑Mal’tsev categories are strictly broader than ordinary Mal’tsev categories, the authors present several examples. The category Ord itself, any regular category equipped with the discrete order, and varieties of ordered algebras are all regular Ord‑categories and hence Ord‑Mal’tsev. More interestingly, the category V‑Cat (categories enriched over a monoidal closed category V) is not Mal’tsev in the ordinary sense, yet it admits an Ord‑enrichment that makes it Ord‑Mal’tsev. The authors treat this case by an object‑wise analysis of ideals, detailed in the appendix.

Overall, the paper contributes a systematic enrichment of the calculus of relations, introduces the notions of ff‑morphisms, so‑morphisms, and bicoinserters for Ord‑categories, and shows how the classical Mal’tsev property extends to this enriched setting. This opens a new line of inquiry into how order‑enrichment interacts with algebraic properties such as 2‑permutability, and provides tools for studying categories that are not Mal’tsev but become so after an appropriate Ord‑enrichment.