Kerodon

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

Construction 4.6.3.13 (The Enriched Homotopy Category). Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes, which we endow with the monoidal structure given by cartesian products. To every $\infty $-category $\operatorname{\mathcal{C}}$, we define an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\underline{\operatorname{Hom}}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X,Y)$ is the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 4.6.1.1.

  • For every object $X \in \operatorname{\mathcal{C}}$, the unit map $\Delta ^{0} \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,X)$ is the homotopy class of the inclusion $\{ \operatorname{id}_{X} \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition law

    \[ \circ : \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z) \]

    is given by Construction 4.6.3.9.

Note that this definition satisfies the axiomatics of Definition 2.1.7.1 by virtue of Propositions 4.6.3.11 and 4.6.3.12 We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the enriched homotopy category of the $\infty $-category $\operatorname{\mathcal{C}}$.