We show that the weighted normal commutator is obtained by applying the kernel functor to the Huq commutator of certain morphisms in a category of points over a fixed object. In addition, we compare the local representation (that is, an equivalence relation considered as a subobject in a category of points over a fixed object) of the Smith commutator of a pair of equivalence relations and the Huq commutator of the corresponding local representations, showing that they coincide in a normal Mal'tsev category with finite colimits.
Keywords: Weighted commutators, Huq commutator, Smith commutator, points
2020 MSC: 18E08, 18E13, 18A30, 18A20
Theory and Applications of Categories, Vol. 35, 2020, No. 39, pp 1530-1545.