The authors 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. The authors then use these bimodule structures to give a chain rule for higher derivatives in the calculus of functors, extending the work 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 the authors' proofs. First, they 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, they 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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

• Introduction
• Basics
• Operads and modules
• Functors of spectra
• Functors of spaces
• Appendix A. Categories of operads, modules and bimodules
• Bibliography