Universal Homotopy Theories

Model categories were introduced by Quillen [Q] to provide a framework through which one could apply homotopy theory in various settings. They have been astonishingly successful in this regard, and in recent years one of the first things one does when studying any homotopical situation is to try to set up a model structure. The aim of this paper is to introduce a new, but very basic, tool into the study of model categories. Our main observation is that given any small category C it is possible to expand C into a model category in a very generic way, essentially by formally adding homotopy colimits. In this way one obtains a ‘‘universal model category built from C.’’ There is an accompanying procedure which imposes relations into a model category, also in a certain universal sense. These two fundamental techniques are the subject of this paper. Although they are very formal—as any universal constructions would be—we hope to indicate that these ideas can be useful and have some relevance to quite disparate areas of model category theory. There are two general themes to single out regarding this material: