Koszul duality and a classification of stable Weiss towers

TL;DR

This paper develops a Koszul duality framework for categories, applies it to Weiss calculus to classify towers, and resolves conjectures by Arone--Ching and Espic using categorical Fourier transforms.

Contribution

It introduces a new Koszul duality for categories and uses it to classify Weiss towers, extending previous operad-based dualities.

Findings

Derivatives in Weiss calculus form a right module over the Koszul dual category.

Resolved conjectures of Arone--Ching and Espic regarding derivatives.

Described polynomial approximations as pullbacks along a generalized norm map.

Abstract

We introduce a version of Koszul duality for categories, which extends the Koszul duality of operads and right modules. We demonstrate that the derivatives which appear in Weiss calculus (with values in spectra) form a right module over the Koszul dual of the category of vector spaces and orthogonal surjections, resolving conjectures of Arone--Ching and Espic. Using categorical Fourier transforms, we then classify Weiss towers. In particular, we describe the $n$-th polynomial approximation as a pullback of the $(n-1)$-st polynomial approximation along a ``generalized norm map''.