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

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.

29 pages, LaTeX; considerably expanded section 6 to provide a correct proof of 6.1, other sections rewritten to improve exposition but no major changes; submitted for publication