Construction (The $(\infty ,2)$-Category of $\infty $-Categories). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, endowed with the simplicial enrichment of Example We let $\operatorname{\mathbf{QCat}}$ denote the full simplicial subcategory of $\operatorname{Set_{\Delta }}$ spanned by the (small) $\infty $-categories, which we can describe concretely as follows:

  • The objects of $\operatorname{\mathbf{QCat}}$ are (small) $\infty $-categories.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then the simplicial set $\operatorname{Hom}_{\operatorname{\mathbf{QCat}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }$ is the $\infty $-category of functors $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

We let $\operatorname{ \pmb {\mathcal{QC}} }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathbf{QCat}})$. We will refer to $\operatorname{ \pmb {\mathcal{QC}} }$ as the $(\infty ,2)$-category of $\infty $-categories.