Higher Central Extensions in Mal'tsev Categories

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. Here, we further extend the scope to exact Mal’tsev categories and beyond. For this, we consider conditions on a Galois structure Γ = (C,X, I,H, η,E) which insure the existence of an induced Galois structure Γ1 = (C1,X1, I1, H1, η1,E1) such that C1 and X1 are full subcategories of the arrow category Arr(C) consisting, respectively, of all morphisms in the class E , and of all covering morphisms with respect to Γ . Moreover, we prove that Γ1 satisfies the same conditions as Γ , so that, inductively, we obtain, for each n ≥ 1, a Galois structure Γn = (Γn−1)1, whose coverings we call n+ 1-fold central extensions.