Operads and chain rules for the calculus of functors

TL;DR

This paper develops a new algebraic framework using operads and bimodules to describe the derivatives in Goodwillie's calculus of functors, leading to a chain rule for higher derivatives.

Contribution

It introduces operad and bimodule structures for derivatives, extending the chain rule in the calculus of functors to a more general setting.

Findings

Constructed new models for derivatives of functors of spectra.

Established operad and bimodule structures for derivatives.

Derived a chain rule for higher derivatives using these structures.

Abstract

We study the structure possessed by the Goodwillie derivatives of a pointed homotopy functor of based topological spaces. These derivatives naturally form a bimodule over the operad consisting of the derivatives of the identity functor. We then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending that of Klein and Rognes. This chain rule expresses the derivatives of FG as a derived composition product of the derivatives of F and G over the derivatives of the identity. There are two main ingredients in our proofs. Firstly, we construct new models for the Goodwillie derivatives of functors of spectra. These models allow for natural composition maps that yield operad and module structures. Then, we use a cosimplicial cobar construction to transfer this structure to functors of topological spaces. A form of Koszul duality for operads of spectra plays a key role in this.