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.Comment: 22 pages. Exposition is substantially improved. Few minor mistakes
are correcte
We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define homotopy n-nilpotent groups as homotopy algebras over certain simplicial algebraic theories. This notion interpolates between infinite loop spaces and loop spaces, but backwards. We study the relation to ordinary nilpotent groups. We prove that n-excisive functors of the form F factor over the category of homotopy n-nilpotent groups.55P47, 55U35; 18C10, 55P35
We prove a generalization of the classical connectivity theorem of Blakers-Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system pL , Rq in which the left class is stable by base change. We explain how to rederive the classical result, as well as the recent generalization of Chachólski, Scherer and Werndli (Ann. Inst. Fourier 66 (2016) 2641-2665). Our proof is inspired by the one given in homotopy-type theory in Favonia et al. (2016).
We achieve a classification of n-types of simplicial presheaves in terms of (n − 1)-types of presheaves of simplicial groupoids. This can be viewed as a description of the homotopy theory of higher stacks. As a special case we obtain a good homotopy theory of (weak) higher groupoids.
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