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 connectivity of X is large enough (for example, when F is the identity functor and X is connected), 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.