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

Abstract. Let D be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from D to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger universal property than the customary homotopy localization functors do.

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory.

We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be non-cofibrantly generated [J. Adámek, H. Herrlich, J. Rosický, W. Tholen, Weak factorization systems and topological functors, Appl. Categ. Structures 10 (3) (2002) 237-249 [2]; Papers in honour of the seventieth birthday of Professor Heinrich Kleisli (Fribourg, 2000); B. Chorny, The model category of maps of spaces is not cofibrantly generated, Proc. Amer. Math. Soc. 131 (2003) 2255-2259; J.D. Christensen, M. Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2) (2002) 261-293; D.C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805-2841]. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces [E.D. Farjoun, Homotopy theories for diagrams of spaces, Proc. Amer. Math. Soc. 101 (1987) 181-189] and diagrams of chain complexes. We also formulate a non-functorial version of the argument, which applies in two different model structures on the category of pro-spaces [D.A. Edwards, H.M. Hastings,Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics, vol. 542, Springer, Berlin, 1976; D.C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805-2841].The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a class-cofibrantly generated model category, which is a 569 model category generated by classes of cofibrations and trivial cofibrations satisfying some reasonable assumptions. Crown