Stratified categories, geometric fixed points and a generalized Arone-Ching theorem

Abstract: We develop a theory of Mackey functors on epiorbital categories which simultaneously generalizes the theory of genuine $G$-spectra for a finite group $G$ and the theory of $n$-excisive functors on the category of spectra. Using a new theory of stratifications of a stable $\infty$-category along a finite poset, we prove a simultaneous generalization of two reconstruction theorems: one by Abram and Kriz on recovering $G$-spectra from structure on their geometric fixed point spectra for abelian $G$, and one by Arone and Ching that recovers an $n$-excisive functor from structure on its derivatives. We deduce a strong tom Dieck splitting theorem for $K(n)$-local $G$-spectra and reprove a theorem of Kuhn on the $K(n)$-local splitting of Taylor towers.