Class-combinatorial model categories

We develop an extension of the framework of combinatorialmodel categories. The category of small presheaves over largeindexing categories and ind-categories are the basic examplesof non-combinatorial model categories embraced by the newmachinery called class-combinatorial model categories.The definition of the new class of model categories is based onthe corresponding extension of the theory of locally presentableand accessible categories developed in the companion paper,where we introduced the concepts of class-locally presentableand class-accessible categories.In this work we prove that the category of weak equivalencesof a nice class-combinatorial model category is class-accessible.Our extension of J. Smith's localization theorem depends on theverification of a cosolution-set condition. The deepest result isthat the (left Bousfield) localization of a class-combinatorialmodel category with respect to a strongly class-accessible localizationfunctor is class-combinatorial again.