One of the very basic structures of homotopical algebra is a pair \((\mathcal C,\mathcal W)\) where \(\mathcal C\) is a category and \(\mathcal W\) is a subcategory of \(\mathcal C\) containing all objects. The morphisms of \(\mathcal W\) are called weak equivalences and the pair is, in the terminology of this paper, a relative category. This paper attacks the first of two evident questions that can be asked about relative categories. This is that of the existence on the category, \(\mathcal {R}el \mathcal{C}at\), of small relative categories and weak equivalence preserving functors, of a model category structure that is a homotopy theory of homotopy theories in the sense already explored by Bergner. The answer is that there is. (The second question is that of giving a good description of the weak equivalences in this structure and will be the subject of a second paper, which follows this one in the same volume of the journal.) The proof lifts a structure due to Rezk on the category of bisimplicial sets to a Quillen equivalent one on the category of relative categories.