Algebraic Goodwillie calculus and a cotriple model for the remainder

Andrew Mauer-Oats


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.