We show that quandle coverings in the sense of Eisermann form a (regular epi) reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

Abstract. We study the notion of fundamental group in the framework of descent-exact homological categories. This setting is sufficiently wide to include several categories of "algebraic" nature such as the almost abelian categories, the semi-abelian categories, and the categories of topological semiabelian algebras. For many adjunctions in this context, the fundamental groups are described by generalised Brown-Ellis-Hopf formulae for the integral homology of groups.

We prove a Seifert-van Kampen theorem in a non-additive setting, providing sufficient conditions on a functor F : C X from an algebraically coherent semi-abelian category with enough projectives C to an abelian category X for its first left derived functor L 1 F to preserve pushouts of split monomorphisms.

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For a particular class of Galois structures, we prove that the normal extensions are precisely those extensions that are "locally" split epic and trivial, and we use this to prove a "Galois theorem" for normal extensions. Furthermore, we interpret the normalisation functor as a Kan extension of the trivialisation functor.