An Introduction to Goodwillie Calculus

Algebraic Goodwillie Calculus and a Cotriple Model for the Remainder

We define an “algebraic” version of the Goodwillie tower, P alg n F (X), that depends only on the behavior of F on coproducts of X . When F is a functor to connected spaces or grouplike H-spaces, the functor P alg n F is the base of a fibration |⊥ F | → F → P alg n F, whose fiber is the simplicial space associated to a cotriple ⊥ built from the (n + 1) cross effect of the functor F . When the c...

The Goodwillie tower for S1 and Kuhn’s theorem

We analyze the homological behavior of the attaching maps in the 2-local Goodwillie tower of the identity evaluated at S1 . We show that they exhibit the same homological behavior as the James-Hopf maps used by N. Kuhn to prove the 2-primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie...

( ∞ , 2 ) - Categories and the Goodwillie Calculus

A Chain Rule for Goodwillie Derivatives of Functors from Spectra to Spectra

We prove a chain rule in the Goodwillie calculus of functors from spectra to spectra. We show that the derivatives of a composite FG at a base object X are given by taking the composition product of the derivatives of F at G(X) with the derivatives of G at X . Our proof also allows us to say something about the Taylor tower of FG (and not just its layers) in terms of the Taylor towers of F and G.

The Goodwillie tower and the EHP sequence

We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime 2. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the P map, and the Goodwillie differentials to the H map. F...