Kerodon

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

Corollary 4.6.6.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

If $X$ is initial as an object of the $\infty $-category $\operatorname{\mathcal{C}}$, then it is also initial when viewed as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

$(2)$

If $\operatorname{\mathcal{C}}$ has an initial object and $X$ is initial as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, then $X$ is initial as an object of the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof. Assertion $(1)$ is immediate from the definition. To prove $(2)$, assume that $\operatorname{\mathcal{C}}$ has an initial object $Y$. Then $Y$ is also initial when viewed as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. If $X$ is an initial object of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, then $X$ and $Y$ are isomorphic when viewed as objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, hence also when viewed as objects of the $\infty $-category $\operatorname{\mathcal{C}}$. Invoking Corollary 4.6.6.16, we conclude that $X$ is also an initial object of $\operatorname{\mathcal{C}}$. $\square$