Operads and chain rules for the calculus of functors

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.

💡 Research Summary

The paper investigates the algebraic structure hidden in the Goodwillie derivatives of a pointed homotopy functor F : Top*_ → Top*_ . The authors show that these derivatives are not merely spectra but carry a rich operadic and bimodule structure. The central object is the operad formed by the derivatives of the identity functor I, denoted ∂*I. For any functor F, its derivative sequence ∂*F becomes a left and right module over ∂*I, i.e., a bimodule. This observation allows the authors to formulate a chain rule for higher derivatives that generalizes the first‑order rule of Klein and Rognes.

The work proceeds in two major technical stages. First, the authors construct new models for the Goodwillie derivatives of functors of spectra. By reinterpreting the cross‑effect construction in a spectral setting, they obtain natural composition maps \