Polynomial Functors from Algebras Over a Set-Operad and Nonlinear Mackey Functors

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of nonlinear 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 (finitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of nonlinear 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 explicitly by their cross-effects and maps relating them which satisfy a list of relations.