Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 4.5.1.1 (The Homotopy Category of $\infty $-Categories). We define a category $\mathrm{h} \mathit{\operatorname{QCat}}$ as follows:

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

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )$ is the set of isomorphism classes of objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (or, equivalently, of the homotopy category $\mathrm{h} \mathit{\operatorname{Fun}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor, we denote its isomorphism class by $[F] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Cat}}_{\infty }}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

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

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

    is characterized by the formula $[G] \circ [F] = [G \circ F]$.

We will refer to $\mathrm{h} \mathit{\operatorname{QCat}}$ as the homotopy category of $\infty $-categories.