Homotopy (limits And) Colimits

These notes were written to accompany two talks given in the Algebraic Topology and Category Theory Proseminar at the University of Chicago in Winter 2009. When a category has some notion of limits and colimits associated to it, its ordinary limits and colimits are not necessarily homotopically meaningful. We describe a notion of a “homotopy colimit” for two sorts of categories with a homotopy theory: categories enriched in simplicial sets and model categories. For simplicial categories, we define an object with a “homotopical universal property” using the well-known bar construction. For model categories, we define a homotopy colimit functor to be a derived functor of the usual colimit functor. Finally, we note that in the setting of a simplicial model category, these two approaches coincide and refer the reader to appropriate sources.