Further Remarks on the "Smith is Huq" Condition

We compare the Smith is Huq condition (SH) with three commutator conditions in semi-abelian categories: first an apparently weaker condition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be stronger than (SH). This leads to the even stronger condition that weighted commutators in the sense of Gran, Janelidze and Ursini are independent of the chosen weight, which is known to be false for groups but turns out to be true in any twonilpotent semi-abelian category. Introduction It is well known that, when they exist, the normalisations of a pair of Smithcommuting equivalence relations [33, 29] will always Huq-commute [17, 6] and that the converse need not hold: counterexamples exist in the category DiGp of digroups [2, 4] and in the category Loop of loops [16]. A pointed Mal’tsev category satisfies the Smith is Huq condition (SH) if and only if two equivalence relations on a given object always centralise each other (= commute in the Smith sense) as soon as their normalisations commute. The condition (SH) is fundamental in the study of internal categorical structures: it is shown in [25] that, for a semi-abelian category, this condition holds if and only if every star-multiplicative graph is an internal groupoid. As explained in [18] and in [16] this is important when characterising internal crossed modules; furthermore, the condition has immediate (co)homological consequences [32]. Any pointed strongly protomodular exact category satisfies (SH) [6] (in particular, so does any Moore category [31]) as well as any action accessible category [8, 11] (in particular, any category of interest [27, 28]). Well-known concrete examples are the categories of groups, Lie algebras, associative algebras, non-unitary rings, and (pre)crossed modules of groups. In the present paper we restrict ourselves to the context of semi-abelian categories [19] and focus on three conditions which arose naturally in the study of (SH). In Section 1 we prove that the condition Date: 23rd January 2014. 2010 Mathematics Subject Classification. 18D35, 18E10, 08A30, 20J15.