Goodwillie calculus and Mackey functors

TL;DR

This paper establishes an equivalence between n-excisive functors from spectra to a stable ∞-category and E-valued Mackey functors, revealing new structural insights into polynomial functors and semiadditive ∞-categories.

Contribution

It introduces a novel classification of polynomial functors via Mackey functors and develops extension theorems and semiadditive ∞-category theory tools.

Findings

Equivalence between n-excisive functors and Mackey functors.

Extension theorems for polynomial functors.

Formulas for free semiadditive ∞-categories.

Abstract

We show that the category of $n$-excisive functors from the $\infty$-category of spectra to a target stable $\infty$-category $\mathbf{E}$ is equivalent to the category of $\mathbf{E}$-valued Mackey functors on an indexing category built from finite sets and surjections. This new classification of polynomial functors arises from an investigation of the structure present on cross effects. The path to this result involves a pair of surprising extension theorems for polynomial functors and a discussion of some interesting topics in semiadditive $\infty$- category theory, including a formula for the free semiadditive $\infty$-category on an $\infty$-category.