A chain rule for Goodwillie derivatives of functors from spectra to spectra

TL;DR

This paper establishes a chain rule for calculating the derivatives in Goodwillie calculus of functors from spectra to spectra, providing a formula for derivatives of composite functors.

Contribution

It introduces a chain rule for Goodwillie derivatives of composite functors from spectra to spectra, including explicit formulas for homogeneous cases.

Findings

Derived a chain rule for Goodwillie derivatives of composite functors

Provided explicit formulas for derivatives when the functor is homogeneous

Enhanced understanding of the structure of derivatives in spectral functor calculus

Abstract

We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor $FG$ at a base object $X$ are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of $F$ at $G(X)$ with the derivatives of $G$ at $X$. We also consider the question of finding $P_n(FG)$, and give an explicit formula for this when $F$ is homogeneous.