Algebraic Goodwillie calculus and a cotriple model for the remainder

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 order approximation at a space depends on the values of on coproducts of large suspensions of the space: .

We define an ``algebraic' version of the Goodwillie tower, , that depends only on the behavior of on coproducts of . When is a functor to connected spaces or grouplike -spaces, the functor is the base of a fibration

whose fiber is the simplicial space associated to a cotriple built from the cross effect of the functor . In a range in which commutes with realizations (for instance, when 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 in many interesting cases.