Weighted commutators in semi-abelian categories

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 Pedicchioʼs 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.