Protoadditive functors, derived torsion theories and homology

TL;DR

This paper introduces protoadditive functors as a non-abelian generalization of additive functors, exploring their role in torsion theories, Galois theory, and homology, and providing new descriptions of derived functors in various categories.

Contribution

It develops the theory of protoadditive functors and shows how they induce chains of derived torsion theories, extending Galois structures and homological methods in non-abelian contexts.

Findings

Protoadditive torsion-free reflectors induce chains of derived torsion theories.

Higher central extensions can be described via protoadditive reflectors.

Explicit formulas for non-abelian derived functors are obtained in multiple categories.

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.