Capturing Goodwillie's Derivative

TL;DR

This paper develops a new model category framework to better understand Goodwillie's derivative, providing a clearer classification of n-homogeneous functors and setting the stage for future comparisons with orthogonal calculus.

Contribution

It introduces a novel model category for the derivative of functors, enhancing the understanding and classification of n-homogeneous functors in Goodwillie's calculus.

Findings

Derived a new model category for the derivative as a right Quillen functor.

Provided a streamlined proof that n-homogeneous functors are classified by spectra with symmetric group actions.

Set the groundwork for comparing Goodwillie's and Weiss's calculus in future work.

Abstract

Recent work of Biedermann and R\"ondigs has translated Goodwillie's calculus of functors into the language of model categories. Their work focuses on symmetric multilinear functors and the derivative appears only briefly. In this paper we focus on understanding the derivative as a right Quillen functor to a new model category. This is directly analogous to the behaviour of Weiss's derivative in orthogonal calculus. The immediate advantage of this new category is that we obtain a streamlined and more informative proof that the n-homogeneous functors are classified by spectra with an action of the symmetric group on n objects. In a later paper we will use this new model category to give a formal comparison between the orthogonal calculus and Goodwillie's calculus of functors.