On the chain rule in Goodwillie calculus

Abstract

We prove a generalization of the Arone-Ching chain rule for Goodwillie derivatives by showing that for any pair of reduced finitary functors $F \colon \mathcal{D} \to \mathcal{E}$ and $G \colon \mathcal{C} \to \mathcal{D}$ between differentiable $\infty$-categories, there is an equivalence $\partial_*(FG) \simeq \partial_*F \circ_{\partial_*{\mathrm{id}_{\mathcal{D}}}} \partial_*G$. This confirms a conjecture of Lurie. The proof of this theorem consists of two parts, which are of independent interest. We first show that the Goodwillie derivatives can be refined to a lax functor $\partial_* \colon \mathrm{Diff} \to \mathrm{Pr}^{\mathrm{Sym}}_{\mathrm{St}}$ from the $(\infty, 2)$-category of differentiable $\infty$-categories and reduced finitary functors to a certain $(\infty, 2)$-category of generalized symmetric sequences. Such a lax structure on the Goodwillie derivatives was long believed to exist, but has not been constructed prior to this work. We then finish the proof by studying the interaction of this lax functor with Koszul duality. In order to do so, we establish a new universal property of the bar-cobar adjunction.