Weighted commutators in semi-abelian categories

TL;DR

This paper introduces a new framework of weighted commutators and centrality in semi-abelian categories, unifying existing commutator theories and exploring their algebraic implications.

Contribution

It defines weighted commutators and centrality, linking and generalizing Huq and Pedicchio's commutators within a unified categorical framework.

Findings

Weighted commutators generalize existing theories

Huq and Pedicchio commutators are special cases

Connections to ideal theory in universal algebra

Abstract

We introduce new notions of weighted centrality and weighted commutators corresponding to each other in the same way as centrality of congruences and commutators do in the Smith commutator theory. Both the Huq commutator of subobjects and Pedicchio's categorical generalization of Smith commutator are special cases of our weighted commutators; in fact we obtain them by taking the smallest and the largest weight respectively. At the end of the paper we briefly consider the universal-algebraic context in connection with an older work of the third author on the ideal theory version of the commutator theory.