On the “Three Subobjects Lemma” and its higher-order generalisations

Abstract

We solve a problem mentioned in the article [1] 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 [18,16,20]. The two are known to be different in general; for instance, in the context of loops, the definition of Bruck [6] is of the former kind, while the commutator-associator filtration of Mostovoy [27,26] and his co-authors is of the latter type. Another example, in the context of Moufang loops, is given in [1].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[31]. Our proof of this result is based on a higher-order version of the Three Subobjects Lemma of [8], which extends the classical Three Subgroups Lemma from group theory to categorical algebra. It says that any n-fold Higgins commutator [K1,…,Kn] of normal subobjects Ki◁X may be decomposed into a join of nested binary commutators.