A chain rule in the calculus of homotopy functors

Klein, John R¹ ; Rognes, John²

¹ Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA
² Department of Mathematics, University of Oslo, N–0316 Oslo, Norway

Geometry & topology, Tome 6 (2002) no. 2, pp. 853-887.

Résumé

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 $F\left(X\right)=Q\left(Map{\left(K,X\right)}_{+}\right)$.

Détail

DOI : 10.2140/gt.2002.6.853

Keywords: homotopy functor, chain rule, Brown representability

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