  Given a small category C, we show that there is a universal way of expanding
C into a model category, essentially by formally adjoining homotopy colimits.
The technique of localization becomes a method for imposing `relations' into
these universal gadgets. The paper develops this formalism and discusses
various applications, for instance to the study of homotopy colimits, the
Dwyer-Kan theory of framings, and to the homotopy theory of schemes.
