A chain rule for Goodwillie derivatives of functors from spectra to spectra
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- by Michael Ching
- Trans. Amer. Math. Soc. 362 (2010), 399-426
- DOI: https://doi.org/10.1090/S0002-9947-09-04834-X
- Published electronically: July 2, 2009
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Abstract:
We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor $FG$ at a base object $X$ are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of $F$ at $G(X)$ with the derivatives of $G$ at $X$. We also consider the question of finding $P_n(FG)$, and give an explicit formula for this when $F$ is homogeneous.References
- Thomas G. Goodwillie, Calculus. II. Analytic functors, $K$-Theory 5 (1991/92), no. 4, 295–332. MR 1162445, DOI 10.1007/BF00535644
- Thomas G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711. MR 2026544, DOI 10.2140/gt.2003.7.645
- Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR 1695653, DOI 10.1090/S0894-0347-99-00320-3
- Warren P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly 109 (2002), no. 3, 217–234. MR 1903577, DOI 10.2307/2695352
- John R. Klein and John Rognes, A chain rule in the calculus of homotopy functors, Geom. Topol. 6 (2002), 853–887. MR 1943383, DOI 10.2140/gt.2002.6.853
- Nicholas J. Kuhn, Goodwillie towers and chromatic homotopy: An overview, Proceedings of Nishida Fest (Kinosaki 2003), Geometry and Topology Monographs, vol. 10, 2007, pp. 245–279.
- Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, vol. 96, American Mathematical Society, Providence, RI, 2002. MR 1898414, DOI 10.1090/surv/096
- Randy McCarthy, Dual calculus for functors to spectra, Homotopy methods in algebraic topology (Boulder, CO, 1999) Contemp. Math., vol. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 183–215. MR 1831354, DOI 10.1090/conm/271/04357
Bibliographic Information
- Michael Ching
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 760391
- Received by editor(s): March 24, 2008
- Published electronically: July 2, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 399-426
- MSC (2000): Primary 55P42, 55P65
- DOI: https://doi.org/10.1090/S0002-9947-09-04834-X
- MathSciNet review: 2550157