The classical correspondance between groups and Lie algebras has been extended in the literature to various varieties of loops by associating a suitable type of algebras with each of them, e.g. Mal'cev algebras with Moufang loops and Sabinin algebras with all loops, mostly by using geometric methods.

We will present the foundations of a new, algebraic (actually functor theoretic) approach allowing to construct analogous linearizations in a much broader context, consisting of a suitable linear operad associated with any semi-abelian category. This operad recovers the types of algebras previously associated with groups and numerous varieties of loops. Extending the classical Lazard correspondance and Baker-Campbell-Hausdorff (BCH) formula to this framework using polynomial functor theory instead of the exponential function is an ongoing project.

In the two previous talks on the subject the main tool of functor calculus and a commutator theory derived from it were discussed in length. In this talk we will review the main features of these topics before focussing on the construction of the before-mentioned operad associated with a semi-abelian category or, more specifically, semi-abelian variety of universal algebras. Also the new functor theoretic approach to the BCH formula and its possible extension to the latter framework will be sketched and illustrated by a Lazard type correspondance and explicit BCH formula for the first but already remarkably complex case of 2-divisible 2-nilpotent varieties.