Higher commutator conditions for extensions in Mal'tsev categories

Abstract

We define a Galois structure on the category of pairs of equivalence relations in an exact Mal'tsev category, and characterize central and double central extensions in terms of higher commutator conditions. These results generalize both the ones related to the abelianization functor in exact Mal'tsev categories, and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids. Some examples and applications are given in the categories of groups, precrossed modules, precrossed Lie algebras, and compact groups.

Galois theory and the categorical Peiffer commutator

“…in PXMod B (Lie K ), we have ∂(k) = ∂ ′ f (k) = 0 for all k ∈ K[f ], so that the Peiffer commutator K[f ], X is generated by the terms [k, x] and ξ(∂(x))(k). It is thus the same ideal as in Example 5 of [17], and thus we find the characterization of central extensions given there as a special case of Theorem 2.6. Moreover, given a short exact sequence (14) in the category of Lie algebra precrossed modules, we obtain an exact sequence of Lie algebra precrossed modules…”

“…in RG B (C) that are Grpd B (C)-central, namely central with respect to the Birkhoff reflection (3). As shown in [17] (by extending a result in [20]), it turns out that this is the case if and only if the following Smith centrality condition holds…”

“…Let C be a semi-abelian category satisfying (UA), and let Proof. By Corollary 3 of [17], the double extension…”

“…In [17], a characterization of double central extensions for the adjunction (3) using Smith-Pedicchio commutators was given. This allows to state the corresponding result for the adjunction (5):…”

“…A characterization of the extensions in RG B (C) that are central with respect to the adjunction (3) was established in [17], in the general context of exact Mal'tsev categories, i.e. in exact categories where any reflexive relation is an equivalence relation [13].…”