Calculus of functors and model categories

Abstract

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.

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

References in corpus (3)

• Calculus III: Taylor Series
• Goodwillie towers and chromatic homotopy: an overview
• Homotopy theory of small diagrams over large categories

Cited by in corpus (13)

• Calculus of functors and model categories II
• Locally class-presentable and class-accessible categories
• Model Categories for Orthogonal Calculus
• Goodwillie's Calculus of Functors and Higher Topos Theory
• Homotopy nilpotent groups
• Homotopy theory of G-diagrams and equivariant excision
• Equivariant calculus of functors and Z/2-analyticity of real K-theory
• Duality and small functors
• The spectrum of excisive functors
• Towers and fibered products of model structures
• A classification of small linear functors
• A variant of a Dwyer-Kan theorem for model categories
• Cofibrantly generated model structures for functor calculus