Algebraic Goodwillie calculus and a cotriple model for the remainder

Author:
Andrew Mauer-Oats

Journal:
Trans. Amer. Math. Soc. **358** (2006), 1869-1895

MSC (2000):
Primary 55P65

DOI:
https://doi.org/10.1090/S0002-9947-05-03936-X

Published electronically:
December 20, 2005

MathSciNet review:
2197433

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Abstract: 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 \[ |{\bot ^{*+1} F}| \rightarrow F \rightarrow P_n^{\text {alg}} F, \] 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.

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Additional Information

**Andrew Mauer-Oats**

Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Email:
amauer@math.northwestern.edu

Received by editor(s):
December 9, 2002

Published electronically:
December 20, 2005

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.