Calculus of functors and model categories

Details Calculus of functors and model categories Calculus of functors and model categories

2006-01-10

arxiv.org/abs/math/0601221v2

Mathematics

Algebraic Topology

22 pages. Exposition is substantially improved. Few minor mistakes are corrected

Scientific paper

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. 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 $n$-th derivative is a Quillen functor to the category of spectra with $\Sigma_n$-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.

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