Algebra/Topology Seminar

Speaker: Gregory Arone

Title: The tensor triangular geometry of functor categories

Abstract: We consider the (infinity) category of excisive (aka polynomial) functors from Spectra to Spectra. Understanding this category is a basic problem in functor calculus. We will approach it from the perspective of tensor triangular geometry. Day convolution equips the category of excisive functors with the structure of a rigid monoidal triangulated category. We describe completely the Balmer spectrum of this category, i.e., its spectrum of prime tensor ideals. This leads to a Thick Subcategory Theorem for excisive functors. A key ingredient in the proof is a blueshift theorem for the generalized Tate construction associated to the family of non-transitive subgroups of products of symmetric groups. If there is time, I will say something about work in progress to extend these results to more general functor categories. Joint with Tobi Barthel, Drew Heard and Beren Sanders.