Volume 7, issue 2 (2003)

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Calculus III: Taylor Series

Thomas G Goodwillie

Geometry & Topology 7 (2003) 645–711
 arXiv: math.AT/0310481
Abstract

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal $n$–excisive approximation, which may be thought of as its $n$–excisive part. Homogeneous functors, meaning $n$–excisive functors with trivial $\left(n-1\right)$–excisive part, can be classified: they correspond to symmetric functors of $n$ variables that are reduced and $1$–excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen’s algebraic $K$–theory.

Keywords
Homotopy functor, excision, Taylor tower
Secondary: 55U99
Publication
Received: 8 November 2002
Accepted: 20 October 2003
Published: 28 October 2003
Proposed: Haynes Miller
Seconded: Ralph Cohen, Gunnar Carlsson
Authors
 Thomas G Goodwillie Department of Mathematics Brown University Box 1917 Providence Rhode Island 02912–0001 USA