Koszul duality and a classification of stable Weiss towers

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''.