A Dwyer-Rezk classification for polynomial functors in Weiss calculus
David Barnes, Magdalena Kędziorek, Niall Taggart
TL;DR
This work establishes a Weiss-calculus analogue of the Dwyer–Rezk classification by proving that the $\infty$-category of $d$-polynomial functors $\mathsf{Poly}^{\leq d}(\mathsf{Vect}_{\mathbb{k}}, {\sf{Sp}})$ is equivalent to the $\infty$-category of spectrum-valued functors on $\sf{OEpi}_{\leq d}$, i.e., $\mathsf{Poly}^{\leq d}(\mathsf{Vect}_{\mathbb{k}}, {\sf{Sp}}) \simeq \sf{Fun}(\sf{OEpi}_{\leq d}, {\sf{Sp}})$. The proof blends Weiss cross-effects with enriched $\infty$-category techniques (Schwede–Shipley framework), identifies a compact generator subcategory with $\sf{OEpi}_{\leq d}$ via $\Sigma^{\infty}_+\sf{OEpi}_{\leq d}$, and uses left Kan extensions to give an alternative model; this also yields a Morita-type equivalence with $\mathsf{Vect}_{\mathbb{k}, \leq d}$ and a disproof of BDGNS's conjecture about spectral Mackey functors. A parallel development yields a new, streamlined proof that $d$-homogeneous functors are equivalent to $\sf{Sp}^{B\mathsf{Aut}(d)}$, via $\mathsf{Homog}^{d}(\mathsf{Vect}_{\mathbb{k}}, {\sf{Sp}}) \simeq {\sf{Sp}}^{B\mathsf{Aut}(d)}$. The results extend the landscape of polynomial and homogeneous functors in Weiss calculus and hint at applications to other calculi and presentable stable $\infty$-categories.
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In Goodwillie calculus, unpublished work of Dwyer and Rezk provides a classification of reduced filtered colimit preserving $d$-excisive functors from pointed spaces to spectra as spectrum-valued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.