  Given any model category, or more generally any category with weak
equivalences, its simplicial localization is a simplicial category which can
rightfully be called the "homotopy theory" of the model category. There is a
model category structure on the category of simplicial categories, so taking
its simplicial localization yields a "homotopy theory of homotopy theories." In
this paper we show that there are two different categories of diagrams of
simplicial sets, each equipped with an appropriate definition of weak
equivalence, such that the resulting homotopy theories are each equivalent to
the homotopy theory arising from the model category structure on simplicial
categories. Thus, any of these three categories with their respective weak
equivalences could be considered a model for the homotopy theory of homotopy
theories. One of them in particular, Rezk's complete Segal space model category
structure on the category of simplicial spaces, is much more convenient from
the perspective of making calculations and therefore obtaining information
about a given homotopy theory.
