A decomposition formula for the weighted commutator

We decompose the weighted subobject commutator of M. Gran, G. Janelidze and A. Ursini as a join of a binary and a ternary commutator.Comment: 7 pages. Dedicated to George Janelidze on the occasion of his sixtieth birthda

On some categorical-algebraic conditions in S-protomodular categories

In the context of protomodular categories, several additional conditions have been considered in order to obtain a closer group-like behavior. Among them are locally algebraic cartesian closedness and algebraic coherence. The recent notion of S-protomodular category, whose main examples are the category of monoids and, more generally, categories of monoids with operations and Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a description of S-protomodular categories with a strong monoid-like behavior. In this paper we consider relative versions of the conditions mentioned above, in order to exhibit the parallelism with the "absolute" protomodular context and to obtain a hierarchy among S-protomodular categories

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 two-nilpotent semi-abelian category.Comment: 13 page

Peri-abelian categories and the universal central extension condition

We study the relation between Bourn's notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular we show that a semi-abelian category is peri-abelian if and only if for each normal subobject $K\leq X$, the Higgins commutator of $K$ with itself coincides with the normalisation of the Smith commutator of the denormalisation of $K$ with itself. We show that if a category is peri-abelian, then the condition (UCE), which was introduced and studied by Casas and the second author, holds for that category. In addition we show, using amongst other things a result by Cigoli, that all categories of interest in the sense of Orzech are peri-abelian and therefore satisfy the condition (UCE).Comment: 14 pages, final version accepted for publicatio

A characterisation of Lie algebras amongst anti-commutative algebras

Let $\mathbb{K}$ be an infinite field. We prove that if a variety of anti-commutative $\mathbb{K}$-algebras - not necessarily associative, where $xx=0$ is an identity - is locally algebraically cartesian closed, then it must be a variety of Lie algebras over $\mathbb{K}$. In particular, $\mathsf{Lie}_{\mathbb{K}}$ is the largest such. Thus, for a given variety of anti-commutative $\mathbb{K}$-algebras, the Jacobi identity becomes equivalent to a categorical condition: it is an identity in~$\mathcal{V}$ if and only if $\mathcal{V}$ is a subvariety of a locally algebraically cartesian closed variety of anti-commutative $\mathbb{K}$-algebras. This is based on a result saying that an algebraically coherent variety of anti-commutative $\mathbb{K}$-algebras is either a variety of Lie algebras or a variety of anti-associative algebras over $\mathbb{K}$.Comment: Final version to appear in Journal of Pure and Applied Algebr

On the normality of Higgins commutators

In a semi-abelian context, we study the condition (NH) asking that Higgins commutators of normal subobjects are normal subobjects. We provide examples of categories that do or do not satisfy this property. We focus on the relationship with the "Smith is Huq" condition (SH) and characterise those semi-abelian categories in which both (NH) and (SH) hold in terms of reflection and preservation properties of the change of base functors of the fibration of points.Comment: 15 pages; final published versio

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

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.Comment: 20 pages; revised version, with some simplified proof