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Construction 4.5.1.23 (The Homotopy $2$-Category of $\infty $-Categories). We define a strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ as follows:

  • The objects of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ are $\infty $-categories.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then $\underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}} }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})}$ is the homotopy category 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 on $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ is given by

    \begin{eqnarray*} \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) & = & (\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{D}},\operatorname{\mathcal{E}})}) \times (\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}) \\ & \simeq & \mathrm{h} \mathit{(\operatorname{Fun}(\operatorname{\mathcal{D}},\operatorname{\mathcal{E}}) \times \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}}))} \\ & \xrightarrow {\circ } & \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})} \\ & = & \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \end{eqnarray*}

We will refer to $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ as the homotopy $2$-category of $\infty $-categories. We let $\mathrm{h}_{2} \mathit{\operatorname{QCat}}$ denote the pith $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$, in the sense of Construction 2.2.8.9; we will refer to $\mathrm{h}_{2} \mathit{\operatorname{QCat}}$ as the homotopy $(2,1)$-category of $\infty $-categories.