Algebraic Goodwillie calculus and a cotriple model for the remainder

Abstract

Goodwillie has defined a tower of approximations for a functor from spaces to spaces that is analogous to the Taylor series of a function. His n th order approximation P n F at a space X depends on the values of F on coproducts of large suspensions of the space: F(Σ M X). We define an 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 |⊥* +1 F| → F → P alg n F, whose fiber is the simplicial space associated to a cotriple ⊥ built from the (n + 1) st cross effect of the functor F. In a range in which F commutes with realizations (for instance, when F is the identity functor of spaces), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor F in many interesting cases.