#### Volume 6, issue 2 (2002)

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A chain rule in the calculus of homotopy functors

### John R Klein and John Rognes

Geometry & Topology 6 (2002) 853–887
 arXiv: math.AT/0301079
##### 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 $F\left(X\right)=Q\left(Map{\left(K,X\right)}_{+}\right)$.

##### Keywords
homotopy functor, chain rule, Brown representability
##### Mathematical Subject Classification 2000
Primary: 55P65
Secondary: 55P42, 55P91