Weighted centrality, and a further approach to categorical commutativity

The classical notion of centrality has been generalized to categorical contexts, and in particular, to unital categories. In the present work, we extend the notion of commuting morphisms to more general contexts, and in particular, to subtractive categories with finite joins of subobjects, where commuting morphisms happen to be related to internal subtraction structures. We also investigated the notion of weighted commutators, which is known to capture classical commutators as special examples. We show how weighted commutators can be expressed in terms of another notion of commutator derived from commuting morphisms, called the Huq commutator, and in addition, show that various facts about commutators follow from this general fact.