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.

Fixed an error pointed out by David Ayala, Aaron Mazel-Gee and Nick Rozenblyum

References in corpus (4)

• Higher Topos Theory
• Cross-effects and the classification of Taylor towers
• Chromatic fracture cubes
• Goodwillie calculus and Mackey functors

Cited by in corpus (7)

• A note on stable recollements
• A naive approach to genuine $G$-spectra and cyclotomic spectra
• Equivariant stable categories for incomplete systems of transfers
• Goodwillie calculus and Mackey functors
• On the Commutative Algebra of Categories
• On the parametrized Tate construction
• Derived Mackey functors and $C_{p^n}$-equivariant cohomology