A chain rule in the calculus of homotopy functors

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 $\partial F (X)$ 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 $\partial (E \circ F) (X)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives $\partial E (Y)$ and $\partial F (X)$ , 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

References

Forward citations

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

Volume 6, issue 2 (2002)

John R Klein and John Rognes