Universal homotopy theories
Abstract:
Begin with a small category C. In this note I explain that there is such a thing as a `universal model category built from C', which is in some sense constructed by formally adding homotopy colimits to C. There is an accompanying procedure for imposing `relations' into a model category, which is just the well-known technique of localization. The main application of these ideas is to provide a very straightforward way of setting up Morel-Voevodsky's homotopy theory of schemes, but the paper also includes applications to homotopy colimits, sheaf theory, and to the Dwyer-Kan theory of framings.