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

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.