Geometry & Topology, Vol. 6 (2002) Paper no. 25, pages 853--887.

A chain rule in the calculus of homotopy functors

John R Klein, John Rognes

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 dF(X) 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(X). The derivative spectrum d(E o F)(X)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives dE(Y) and dF(X), 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(X) = Q(Map(K, X)_+).

Keywords. Homotopy functor, chain rule, Brown representability

AMS subject classification. Primary: 55P65. Secondary: 55P42, 55P91.

DOI: 10.2140/gt.2002.6.853

E-print: arXiv:math.AT/0301079

Submitted to GT on 19 June 1997. (Revised 21 July 2002.) Paper accepted 19 December 2002. Paper published 19 December 2002.

Notes on file formats

John R Klein, John Rognes
Department of Mathematics, Wayne State University
Detroit, Michigan 48202, USA
Department of Mathematics, University of Oslo
N--0316 Oslo, Norway


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