Kerodon

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

Corollary 4.6.6.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories, and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The object $Y$ is initial if and only if $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$.

$(2)$

The object $Y$ is final if and only if $F(Y)$ is a final object of $\operatorname{\mathcal{D}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Note that since $F$ is an equivalence of $\infty $-categories, it is fully faithful (Theorem 4.6.2.17). If $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$, then Proposition 4.6.6.20 guarantees that the object $Y \in \operatorname{\mathcal{C}}$ is initial. To prove the converse, let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse of the functor $F$. Then $G \circ F$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$, so that $(G \circ F)(Y)$ is isomorphic to $Y$ as an object of the $\infty $-category $\operatorname{\mathcal{C}}$. If $Y$ is an initial object of $\operatorname{\mathcal{C}}$, then Corollary 4.6.6.16 guarantees that $(G \circ F)(Y)$ is also an initial object of $\operatorname{\mathcal{C}}$. Since the equivalence $G$ is fully faithful (Theorem 4.6.2.17), Proposition 4.6.6.20 guarantees that $F(Y)$ is an initial object of $\operatorname{\mathcal{D}}$. $\square$