Categorical Commutator Theory

Abstract

In these notes, we introduce the reader to the categorical commutator theory (of subobjects), following the formal approach given by Mantovani and Metere in 2010. Such an approach is developed along the lines provided by Higgins, based on the notion of commutator word, introduced by the author in the context of varieties of Ω-groups (groups equipped with additional algebraic operations of signature Ω). An internal interpretation of the commutator words is described, providing an intrinsic notion of Higgins commutator, which reveals to have good properties in the context of ideal determined categories. Furthermore, we will illustrate some applications of commutator theory in categorical algebra, such as a useful way to test the normality of subobjects on one side, and the construction of the abelianization functor on the other.