A chain rule in the calculus of homotopy functors

Details A chain rule in the calculus of homotopy functors A chain rule in the calculus of homotopy functors

2003-01-09

arxiv.org/abs/math/0301079v1

Geom. Topol. 6(2002) 853-887

Mathematics

Algebraic Topology

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper25.abs.html

Scientific paper

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)_+).

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