A Dwyer-Rezk classification for polynomial functors in Weiss calculus

TL;DR

This paper establishes a classification of polynomial functors in Weiss calculus, paralleling known results in Goodwillie calculus, by relating them to spectrum-valued functors on finite-dimensional inner product spaces.

Contribution

It provides a new proof of the classification of polynomial functors in Weiss calculus, connecting them to spectrum-valued functors on finite-dimensional inner product spaces.

Findings

Polynomial functors are equivalent to spectrum-valued functors on finite-dimensional inner product spaces.

New proof of the classification of homogeneous functors.

Analogous classification results to Goodwillie calculus are established in Weiss calculus.

Abstract

In Goodwillie calculus, unpublished work of Dwyer and Rezk provides a classification of reduced filtered colimit preserving $d$-excisive functors from pointed spaces to spectra as spectrum-valued functors on the category of finite sets of cardinality at most $d$ and epimorphisms. We prove through different methods the analogous result in Weiss calculus: $d$-polynomial functors are equivalent to spectrum-valued functors on the category of finite-dimensional inner product spaces of dimension at most $d$ and orthogonal epimorphisms. Via similar methods we obtain a new proof of the classification of homogeneous functors.