#### 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

and

Department of Mathematics, University of Oslo

N--0316 Oslo, Norway

Email: klein@math.wayne.edu, rognes@math.uio.no

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