  We define an ``algebraic'' version of the Goodwillie tower, P_n^alg 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_n^alg F is the base of
a fibration 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.
