In this paper, we give a description of polynomial functors from (finitely
generated free) groups to abelian groups in terms of non-linear Mackey functors
generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili
published in 2001. This description is a consequence of our two main results: a
description of functors from (fi nitely generated free) P-algebras (for P a
set-operad) to abelian groups in terms of non-linear Mackey functors and the
isomorphism between polynomial functors on (finitely generated free) monoids
and those on (finitely generated free) groups. Polynomial functors from
(finitely generated free) P-algebras to abelian groups and from (finitely
generated free) groups to abelian groups are described explicitely by their
cross-e ffects and maps relating them which satisfy a list of relations.Comment: 58 page