Calculus of functors and model categories

Georg Biedermann, Boris Chorny, Oliver Röndigs

Research output: Contribution to journalArticlepeer-review


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)].

Original languageEnglish
Pages (from-to)92-115
Number of pages24
JournalAdvances in Mathematics
Issue number1
StatePublished - 10 Sep 2007
Externally publishedYes


  • Calculus of functors
  • Homotopy functors
  • Small functors

ASJC Scopus subject areas

  • General Mathematics


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