On the universality of multiexcisive functors

TL;DR

This paper classifies polynomial endofunctors on spectra via Mackey functors, establishing a universal framework that generalizes previous results and applies to a prime-based Segal conjecture in Goodwillie calculus.

Contribution

It introduces multivariable functors called subdiagonal functors and proves their Mackey functor descriptions, providing a new universal perspective on excisive functors.

Findings

Categories of multivariable excisive functors are symmetric monoidally equivalent to spectral Mackey functors.

Specializing to univariate functors recovers and strengthens Glasman's results.

Proves a Segal conjecture in Goodwillie calculus for prime numbers.

Abstract

We provide a multiplicative classification of polynomial endofunctors on spectra in terms of their Mackey functors of cross--effects. More precisely, we prove that various categories of multivariable excisive functors from spectra to spectra are symmetric monoidally equivalent to the corresponding variants of spectral Mackey functors. The symmetric monoidal structures appearing here are the Day convolutions on both sides, and the Mackey functors we consider involve variations on the category of finite sets and surjections. The method is first to introduce certain multivariable functors we call subdiagonal functors. By considering them all at once using parametrised category theory, we prove inductively that they all admit Mackey functor descriptions as symmetric monoidal categories, endowing them with a universal property along the way. In particular, specialising this to univariate functors gives a new proof and strengthening of Glasman's result about d-excisive endofunctors on spectra. As application of our perspective, we prove a ``Segal conjecture'' in the context of Goodwillie calculus when d is a prime number.