The spectrum of excisive functors

Abstract

We prove a thick subcategory theorem for the category of -excisive functors from finite spectra to spectra. This generalizes the Hopkins–Smith thick subcategory theorem (the case) and the -equivariant thick subcategory theorem (the case). We obtain our classification theorem by completely computing the Balmer spectrum of compact -excisive functors. A key ingredient is a non-abelian blueshift theorem for the generalized Tate construction associated to the family of non-transitive subgroups of products of symmetric groups. Also important are the techniques of tensor triangular geometry and striking analogies between functor calculus and equivariant homotopy theory. In particular, we introduce a functor calculus analogue of the Burnside ring and describe its Zariski spectrum à la Dress. The analogy with equivariant homotopy theory is strengthened further through two applications: We explain the effect of changing coefficients from spectra to -modules and we establish a functor calculus analogue of transchromatic Smith–Floyd theory as developed by Kuhn–Lloyd. Our work offers a new perspective on functor calculus which builds upon the previous approaches of Arone–Ching and Glasman.