On lax protomodularity of Ord-enriched categories

TL;DR

This paper explores a lax version of protomodularity in Ord-enriched categories, introducing the concept of lax protomodularity and applying it to the category of preordered abelian groups, OrdAb.

Contribution

It defines and studies lax protomodularity in Ord-enriched categories, extending the classical notion by replacing pullbacks with comma objects, and applies this to OrdAb.

Findings

OrdAb is lax preprotomodular with respect to precomma objects.

Lax protomodularity equivalences hold when replacing pullbacks with comma objects.

The concept extends classical protomodularity to enriched categorical contexts.

Abstract

Our main focus concerns a possible lax version of the algebraic property of protomodularity for Ord-enriched categories. Our motivating example is the category OrdAb of preordered abelian groups; indeed, while abelian groups form a protomodular category, OrdAb does not. Having in mind the role of comma objects in the enriched context, we consider some of the characteristic properties of protomodularity with respect to comma objects instead of pullbacks. We show that the equivalence between protomodularity and certain properties on pullbacks also holds when replacing conveniently pullbacks by comma objects in any finitely complete category enriched in Ord, and propose to call lax protomodular such Ord-enriched categories. We conclude by studying this sort of lax protomodularity for OrdAb, equipped with a suitable Ord-enrichment, and show that OrdAb fulfills the equivalent lax protomodular properties with respect to the weaker notion of precomma object; we call such categories lax preprotomodular.