# On the "three subobjects lemma" and its higher-order generalisations

## TL;DR

This paper proves that in algebraically coherent semi-abelian categories, two different definitions of the lower central series, one based on nested binary commutators and the other on higher Higgins commutators, coincide.

## Contribution

It establishes the equivalence of two approaches to defining nilpotency in semi-abelian categories, extending the Three Subobjects Lemma to higher-order commutators.

## Key findings

- The two definitions of the lower central series coincide in algebraically coherent semi-abelian categories.
- A higher-order Three Subobjects Lemma is proved, generalizing the classical lemma from group theory.
- The result applies to all Orzech categories of interest.

## Abstract

We solve a problem mentioned in an article of Berger and Bourn: we prove that in the context of an algebraically coherent semi-abelian category, two natural definitions of the lower central series coincide. In a first, "standard" approach, nilpotency is defined as in group theory via nested binary commutators of the form $[[X,X],X]$. In a second approach, higher Higgins commutators of the form $[X,X,X]$ are used to define nilpotent objects. The two are known to be different in general; for instance, in the context of loops, the definition of Bruck is of the former kind, while the commutator-associator filtration of Mostovoy and his co-authors is of the latter type. Another example, in the context of Moufang loops, is given in Berger and Bourn's paper. In this article, we show that the two streams of development agree in any algebraically coherent semi-abelian category. Such are, for instance, all Orzech categories of interest. Our proof of this result is based on a higher-order version of the Three Subobjects Lemma of Cigoli-Gray-Van der Linden, which extends the classical Three Subgroups Lemma from group theory to categorical algebra. It says that any $n$-fold Higgins commutator $[K_1, \dots,K_n]$ of normal subobjects $K_i$ of an object $X$ may be decomposed into a join of nested binary commutators.