## 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 Σ_{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 [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].

Original language | English |
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Pages (from-to) | 92-115 |

Number of pages | 24 |

Journal | Advances in Mathematics |

Volume | 214 |

Issue number | 1 |

DOIs | |

State | Published - 10 Sep 2007 |

Externally published | Yes |

## Keywords

- Calculus of functors
- Homotopy functors
- Small functors

## ASJC Scopus subject areas

- General Mathematics