Normalities and Commutators

TL;DR

This paper compares algebraic notions of normality categorically, introduces a new intrinsic description of Higgins' commutator, and extends a classical normal subgroup characterization to semi-abelian categories.

Contribution

It provides a new intrinsic description of Higgins' commutator and generalizes the normal subgroup characterization to semi-abelian categories.

Findings

Comparison of algebraic normality notions categorically

Introduction of an intrinsic Higgins' commutator description

Extension of normal subgroup characterization to semi-abelian categories

Abstract

We first compare several algebraic notions of normality, from a categorical viewpoint. Then we introduce an intrinsic description of Higgins' commutator for ideal-determined categories, and we define a new notion of normality in terms of this commutator. Our main result is to extend to any semi-abelian category the following well-known characterization of normal subgroups: a subobject $K$ is normal in $A$ if, and only if, $[A,K]\leq K$.