Polynomial functors from free groups to a stable infinity-category
Gregory Arone
TL;DR
This work identifies a fundamental bridge between polynomial functors from finitely generated free groups to a stable ∞-category D and excisive/Goodwillie-type functors on the category of pointed spaces, via the classifying-space functor and sifted completions.It proves that for each n, the restriction along the classifying space functor induces an equivalence Exc_n(S_*, D) ≃ Poly_n(Fr^{fg}, D), with explicit inverses provided by left/right Kan extensions, and it connects these categories to operadic right modules over Com and to divided power Lie-modules.The paper then leverages these equivalences to compute Ext groups between common polynomial functors (e.g., T^m ∘ ab, Λ^m ∘ ab, S^n ∘ ab, Pa_n) and various target functors, expressing results in terms of cross-effects cr_m of their ∞-categorical extensions, with rich arithmetic and torsion phenomena described (including RP^∞ and symmetric-group actions).Applications include refined stable cohomology calculations for automorphism groups of free groups, via Djament’s framework, tying Ext data to asymptotic cohomological structures and broadening Powell’s characteristic-zero results to general stable ∞-categories.
Abstract
We study the category of polynomial functors from finitely generated free groups to a stable infinity-category D. We show that this category is equivalent to the category of excisive functors from pointed animas to D, and also to truncated right comodules over the commutative operad with values in D. The latter formulation generalizes a result of Geoffrey Powell in characteristic zero. We use the equivalence of categories to calculate Ext between polynomial functors from free groups to abelian groups, extending previous results of Christine Vespa and others. Using the work of Aurelien Djament, we give applications to stable cohomology of automorphism groups of free groups with coefficients in a polynomial functor.