Geometry & Topology, Vol. 7 (2003) Paper no. 19, pages 645--711.

Calculus III: Taylor Series

Thomas G Goodwillie

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 (n-1)-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

AMS subject classification. Primary: 55P99. Secondary: 55U99.

DOI: 10.2140/gt.2003.7.645

E-print: arXiv:math.AT/0310481

Submitted to GT on 8 November 2002. Paper accepted 20 October 2003. Paper published 28 October 2003.

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Thomas G Goodwillie
Department of Mathematics, Brown University
Box 1917, Providence RI 02912-0001, USA

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