  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.
