Polynomial functors from groups to spectra

Let Fr be the category of finitely generated free groups, and D a stable infinity category. We consider polynomial functors from Fr to D. We show that there are equivalences between the following infinity categories

[Polynomial functors from Fr to D] = [Excisive functors from Top${}_*$ to D] = [Truncated right modules in D over the the commutative operad] = [Right modules in D over the Lie operad, whose Koszul dual is truncated]

When D is the category of rational chain complexes, the equivalence between the first, third and fourth category was established by Geoffrey Powell. Bringing excisive functors into the picture allows one to give an “obvious” proof, that works for an arbitrary stable infinity category, and not just in characteristic zero. By way of application we can do new calculations of ext groups in the category of functors from Fr to Ab, and of stable homology of Aut(Fr) with coefficients in a polynomial functor (the last application relies crucially on work of Djament).