A chain rule in the calculus of homotopy functors

Abstract

We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum $\partial F\left(X\right)$ of such a functor $F$ at a simplicial set $X$ can be equipped with a right action by the loop group of its domain $X$, and a free left action by the loop group of its codomain $Y=F\left(X\right)$. The derivative spectrum $\partial \left(E\circ F\right)\left(X\right)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives $\partial E\left(Y\right)$ and $\partial F\left(X\right)$, with respect to the two actions of the loop group of $Y$. As an application we provide a non-manifold computation of the derivative of the functor $F\left(X\right)=Q\left(Map{\left(K,X\right)}_{+}\right)$.