In the various comparisons we have made, we have actually overdetermined the relationships between the model categories, in that we have more Quillen equivalences than we need to show that all the models for (∞, 1)-categories are equivalent. We could ask, then, whether certain diagrams of Quillen equivalences commute even up to homotopy. For example, given a (fibrant) simplicial category, we can obtain a complete Segal space in two different ways: by applying the simplicial nerve to get a Segal category, then taking a fibrant replacement in CSS; or by applying the coherent nerve functor to obtain a quasi-category and then extending to get a complete Segal space.

In addition, we have another functor which is not given by one of the Quillen equivalences at all. If we take the simplicial nerve of a simplicial category, it is a Segal space (up to Reedy fibrant replacement), and hence we can apply the completion functor of Definition 5.5.7 to get a complete Segal space.We would like to compare the output of this functor to what we get from the chains of Quillen functors. The advantage of this procedure, while not part of a Quillen equivalence, is that it enables us to give an up-to-homotopy characterization of the resulting complete Segal space.

Lastly, if we begin with a model category, we have a number of ways to obtain a complete Segal space. We can take its simplicial localization to get a simplicial category and then apply one of the methods given above. However, we can also take the classification diagram, as described in Definition 8.3.1, to get a complete Segal space directly, at least up to Reedy fibrant replacement. Once again, we want to know that these methods result in weakly equivalent complete Segal spaces. Understanding these relationships is the subject of this chapter.

Classifying and Classification Diagrams

Recall the classifying diagram construction, which associates to any small category C a complete Segal space NC, from Definition 3.3.1. We saw that the classifying diagram of a category can be regarded as a more refined version of the nerve. Now that we have given a rigorous definition of complete Segal space, we can state the following result.