Calculus of functors and model categories |
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Authors: | Georg Biedermann,Oliver Rö ndigs |
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Affiliation: | a Department of Mathematics, Middlesex College, The University of Western Ontario, London, Ontario N6A 5B7, Canada b D-MATH, ETH Zentrum, 8092 Zürich, Switzerland c Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, D-33501 Bielefeld, Germany |
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Abstract: | The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)]. |
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Keywords: | primary, 55U35 secondary, 55P91, 18G55 |
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