Models For The Maclaurin Tower Of A Simplicial Functor Via A Derived Yoneda Embedding

TL;DR

This paper develops models for the Goodwillie tower of functors from spaces to spectra using a derived Yoneda embedding, enabling new computational approaches and classical derivative expressions.

Contribution

It introduces a novel approach to model the Goodwillie tower via a derived Yoneda embedding, linking it to pseudo-differential operators and extending computational tools.

Findings

Expressed the Goodwillie tower in terms of stable mapping spaces.

Provided a classical expression for the derivative over the basepoint.

Extended computational tools for the tower of stable mapping spaces.

Abstract

We prove that the Goodwillie tower of a weak equivalence preserving functor from spaces to spectra can be expressed in terms of the tower for stable mapping spaces. Our proof is motivated by interpreting the functors P_n and D_n as pseudo-differential operators which suggests certain `integral' presentations based on a derived Yoneda embedding. These models allow one to extend computational tools available for the tower of stable mapping spaces. As an application we give a classical expression for the derivative over the basepoint.