Protoadditive functors, derived torsion theories and homology

Abstract

Abstract. Protoadditive functors are designed to replace additive functors in a non-abelian setting. Their properties are studied, in particular in relationship with torsion theories, Galois theory, homology and factorisation systems. It is shown how a protoadditive torsion-free reflector induces a chain of derived torsion theories in the categories of higher extensions, similar to the Galois structures of higher central extensions previously considered in semi-abelian homological algebra. Such higher central extensions are also studied, with respect to Birkhoff subcategories whose reflector is protoadditive or, more generally, factors through a protoadditive reflector. In this way we obtain simple descriptions of the non-abelian derived functors of the reflectors via higher Hopf formulae. Various examples are considered in the categories of groups, compact groups, internal groupoids in a semi-abelian category, and other ones.

Monotone-light factorisation systems and torsion theories

“…Let X be a reflective subcategory of a pointed category C with pullback-stable normal epimorphisms such that each unit η A : A → HI(A) is a normal epimorphism (=the cokernel of some morphism). Then X is torsionfree if, and only if, it is semi-left-exact (see Theorem 1.6 in [14]). Hence, in this context, any torsion-free subcategory induces a reflective factorisation system (E, M), with E the class of morphisms that are inverted by the reflector I: C → X and M the closure under pullback of the class of all morphisms in the image of the inclusion functor H. Now, replacing E by the class E ′ of morphisms stably in E, and M by the class M * of morphisms locally in M, we would like (E ′ , M * ) to be a factorisation system too.…”

“…Also in this case the factorisation system (E, M) associated with the "torsion-free" reflection induces a monotone-light factorisation system (E ′ , M * ). In view of the recent interest in torsion theories in contexts more general than the one of abelian categories (see for instance [6,10,14,17,24]), a natural question to ask is whether torsion theories induce monotone-light factorisation systems also in a non-abelian context. The aim of the present article is to show that this is indeed the case for torsion theories in a normal category [25] (as, for instance, any semi-abelian category [22]).…”

Higher commutator conditions for extensions in Mal'tsev categories

“…The category Grp(Comp) of compact (Hausdorff) groups is an exact Mal'tsev category which is monadic over the category of sets [33]. As it follows from Lemma 6.13 in [15], the Huq commutator of two normal subobjects (=closed normal subgroups) H and K of a compact group G is simply given by the closure [H, K] of the classical group-theoretic commutator [H, K] of H and K. In the category Grp(Comp) this Huq commutator of normal subobjects is the normal subobject associated with the categorical commutator [34] of the corresponding equivalence relations: this essentially follows from [3], by taking into account the fact that Grp(Comp) is a strongly protomodular category [2]. From the previous example, it then follows that an extension…”

A comonadic interpretation of Baues–Ellis homology of crossed modules

“…Theorem 5.2 may be used to obtain Hopf formulae for H n`1 pH, G, µq. Indeed, the functor A is of a type considered in [17], so that the general theory developed there applies.…”