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 $n^{\text{th}}$ order approximation $P_n F$ at a space $X$ depends on the values of $F$ on coproducts of large suspensions of the space: $F(\vee \Sigma^M X)$.

We define an ``algebraic'' version of the Goodwillie tower, $P_n^{\text{alg}} F(X)$, 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 $P_n^{\text{alg}} F$ is the base of a fibration

$\vert{\bot^{*+1} F}\vert \rightarrow F \rightarrow P_n^{\text{alg}} F,$

whose fiber is the simplicial space associated to a cotriple $\bot$ built from the $(n+1)^{\text{st}}$ 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.