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Construction 5.4.4.1 (The $\infty $-Category of $\infty $-Categories). We define a simplicial category $\operatorname{QCat}$ as follows:

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

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

  • If $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty $-categories, then the composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{QCat}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})_{\bullet } \times \operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})_{\bullet } \]

    is induced by the composition map $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.

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