One of the most useful formal features of the category of simplicial sets, or here its full subcategory of quasi-categories, is that it is cartesian closed and, in particular, self-enriched. It follows by Theorem 7.5.3 that its limits and colimits all satisfy simplicially enriched universal properties. Throughout much of this text, we have exploited simplicial enrichments because of their convenience, even when our interest was in mere homotopy types.

In this final chapter, following observations of Joyal and Verity, we use the self-enrichment of the category of quasi-categories established by Corollary 15.2.3 to define a 2-category of quasi-categories appropriate for its homotopy theory: equivalences in this 2-category are exactly equivalences of quasi-categories. This (strict) 2-category qCat2 is a truncation of the simplicial category of quasi-categories. The hom-spaces between quasi-categories have cells in each dimension starting from the vertices, which are ordinary maps of simplicial sets. Accordingly, in this chapter, we denote this simplicial category by qCat∞. The 2-cells in qCat2 are homotopy classes of 1-simplices in the corresponding hom-spaces; all higher-dimensional information is discarded. Our interest in these structures is predicated on their competing enrichments; for this reason, we write qCat2 and qCat∞ for the enriched categories, without the underline used elsewhere to signal enrichments. The category qCat is the common underlying category of both the 2-category qCat2 and the simplicial category qCat∞.

We illustrate through examples how qCat2 can be used to determine the appropriate quasi-categorical generalizations of categorical concepts.