Polynomial functors over free nilpotent groups

Abstract

Let $k$ be a unital commutative ring. In this paper, we study polynomial functors from the category of finitely generated free nilpotent groups to the category of $k$-modules, focusing on comparisons across different nilpotency classes and polynomial degrees. As a consequence, we obtain refinements of parts of the results of Baues and Pirashvili on polynomial functors over free nilpotent groups of class at most 2, which also recover several folklore results for free groups and free abelian groups. Furthermore, we investigate a modular analogue, formulated using dimension subgroups over a field of positive characteristic instead of lower central series. To prove the main results, we establish general criteria that guarantee equivalences between the categories of polynomial functors of different degrees or with different base categories. They are described by using a two-sided ideal of a monad associated with the base category, which encodes polynomiality of a specific degree. Inspired by the main results, we also investigate an analogous ideal for analytic functors, and show that, in most cases, no such an ideal exists.