  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. Our current approach allows for construction of
functorial factorizations and localizations in the equivariant model structures
on diagrams of spaces 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.
  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 model category generated
by classes of cofibrations and trivial cofibrations satisfying some reasonable
assumptions.
