  We extend the framework of combinatorial model categories, so that the
category of small presheaves over large indexing categories and ind-categories
would be embraced by the new machinery called class-combinatorial model
categories.
  The definition of the new class of model categories is based on the
corresponding extension of the theory of locally presentable and accessible
categories developed in the companion paper [arXiv:1110.0605], where we
introduced the concepts of locally class-presentable and class-accessible
categories.
  In this work we prove that the category of weak equivalences of a nice
class-combinatorial model category is class-accessible. Our extension of J.
Smith localization theorem depends on the verification of a cosolution-set
condition. The deepest result is that the (left Bousfield) localization of a
class-combinatorial model category with respect to a strongly class-accessible
localization functor is class-combinatorial again.
