Calculus of functors and model categories |
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Authors: | Georg Biedermann Oliver Röndigs |
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Institution: | 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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