Kerodon

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Remark 4.6.8.14. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the enriched homotopy category of $\operatorname{\mathcal{C}}$. Then $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ has an underlying category (Example 2.1.7.5), which we will also denote by $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Concretely, this category can be described as follows:

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

• For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have

$\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( \Delta ^0, \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) ) \simeq \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ).$

In other words, $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ can be identified with the set of homotopy classes of morphisms from $X$ to $Y$ in the $\infty$-category $\operatorname{\mathcal{C}}$.

By virtue of Remark 4.6.8.10, the composition of morphisms in the category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ agrees with the composition law of Construction 1.3.5.1. In other words, we can identify $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with the homotopy category constructed in §1.3.5.