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Definition (The Homotopy Category). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We define a 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 $X,Y \in \operatorname{\mathcal{C}}$, we let $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ denote the collection of homotopy classes of morphisms from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (as in Construction

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity morphism from $X$ to itself in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is given by the homotopy class $[\operatorname{id}_ X]$.

  • Composition of morphisms is defined as in Construction

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