Higher Central Extensions in Mal’tsev Categories

Abstract

Abstract. Higher dimensional central extensions of groups were introduced by G. Janelidze as particular instances of the abstract notion of covering morphism from categorical Galois theory. More recently, the notion has been extended to and studied in arbitrary semi-abelian categories. In this article, we further extend the scope to exact Mal'tsev categories and beyond.

Higher commutator conditions for extensions in Mal'tsev categories

“…These conditions are a generalization of the ones given by G. Janelidze in [21] characterizing double central extensions in Grp, and later extended to Mal'tsev varieties by the second author and V. Rossi in [20], and to exact Mal'tsev categories by T. Everaert and T. Van der Linden [17]. In fact, the proofs given below are suitably adapted from the ones appearing in these two papers.…”

“…We then study the so-called double central extensions in the general context of exact Mal'tsev categories, and particularize our results to deduce a precise characterization of the double central extensions relative to the reflection RG(C)/B → Grpd(C)/B in terms of a commutator condition involving the categorical commutator of equivalence relations (see Corollary 3). Our general result also includes the characterization of double central extensions relative to the abelianization functor previously considered by T. Everaert and T. Van der Linden in [17] (see Corollary 2). Some applications in the special case of central extensions of precrossed Lie algebras are also given (Example 5).…”

“…Any extension which is Γ F -central is also Γ E -central. It is then also Γ E -normal (by Lemma 6 in [12]), and thus Γ F -normal. This proves (1) ⇒ (2); the converse holds by definition.…”

“…satisfying the axioms (E1), (E2), and (E3) above -is provided by the class E of regular epimorphisms. From now on we shall adopt an axiomatic approach to the class E that will have several advantages, including the possibility of comparing the results in our paper to the ones obtained by Everaert in [12].…”

“…In [12] T. Everaert proved that for any category C, if E is a class of extensions satisfying the properties (E1) to (E5) and such that every extension is monadic, and X is a strongly E-Birkhoff subcategory, then the Galois structure (C, X , I, E) is admissible in the sense of categorical Galois theory (i.e. every object is admissible), every central extension is also normal, CExt E (C) is a strongly E 1 -Birkhoff subcategory in Ext E (C), and every extension in E 1 is monadic.…”