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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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

$\displaystyle \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.


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Additional Information

Andrew Mauer-Oats
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: amauer@math.northwestern.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03936-X
PII: S 0002-9947(05)03936-X
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.