The ternary commutator obstruction for internal crossed modules

Abstract

In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones. Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.Comment: Revised version; 32 page

Central reflections and nilpotency in exact Mal’tsev categories

“…Corollary 6.17. Our cubical cross-effects agree up to isomorphism with those of Hartl-Loiseau [38] and Hartl-Van der Linden [39], which are defined as kernel intersections.…”

“…, X n+1 ) is directly inspired by Goodwillie [29, pg. 676] but agrees up to isomorphism with the kernel intersection definition of Hartl-Loiseau [38] and Hartl-Van der Linden [39]. Their kernel intersection is dual to the (n + 1)-fold smash product of Carboni-Janelidze [16], cf.…”

“…The co-smash product X ⋄ Z coincides in semiabelian categories with the second cross-effect cr 2 (X, Z) of the identity functor, cf. Definition 5.1 and [54,38,39]. Since the co-smash product is in general not associative (cf.…”

Compatible actions of Lie algebras

“…A proof of this result, based on a proposition in [11], is straightforward but a bit involved, and can be found in the paper in preparation [8]. As for the lower square, we can precompose with the regular epimorphism q♭1 M : this shows that the required commutativity is equivalent to the equation…”

Commutators for near-rings: Huq $${\neq}$$ ≠ Smith

“…However, this is not the case in general, which, in a sense, is already suggested by the commutator constructions of Higgins [9]; the first explicit counterexample ('digroups': two independent group structures on the same set with the same identity element) was constructed much later in a joint work of the first named author and Bourn (unpublished, but later mentioned, first in [3], in the form of an observation on change-of-base functors for split extensions). Another counter-example (loops) was given recently by Hartl and van der Linden [8]. The question of when these two commutators coincide, is of sufficient importance to justify a condition "Smith = Huq" in universal algebra around which several theories have been developed, see for example [14].…”