Goodwillie calculus and Mackey functors

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.