Polynomial functors from Algebras over a set-operad and non-linear Mackey functors

TL;DR

This paper characterizes polynomial functors from free groups and P-algebras to abelian groups using non-linear Mackey functors, establishing explicit descriptions and isomorphisms that generalize previous work.

Contribution

It provides a new description of polynomial functors from P-algebras and free groups to abelian groups via non-linear Mackey functors, extending prior results.

Findings

Describes polynomial functors via cross-effects and relations.

Establishes isomorphism between polynomial functors on free monoids and groups.

Generalizes Mackey functor descriptions to P-algebras.

Abstract

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.