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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 F(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 (EF)(X) of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives E(Y ) and F(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
Mathematical Subject Classification 2000
Primary: 55P65
Secondary: 55P42, 55P91
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Publication
Received: 19 June 1997
Revised: 21 July 2002
Accepted: 19 December 2002
Published: 19 December 2002
Proposed: Ralph Cohen
Seconded: Gunnar Carlsson, Thomas Goodwillie
Authors
John R Klein
Department of Mathematics
Wayne State University
Detroit
Michigan 48202
USA
John Rognes
Department of Mathematics
University of Oslo
N–0316 Oslo
Norway