Kerodon

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

Corollary 4.6.6.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an initial object of $\operatorname{\mathcal{C}}$. Then an object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if it is isomorphic to $X$.

Proof. If $X$ and $Y$ are initial objects of $\operatorname{\mathcal{C}}$, then they are contained in the contractible Kan complex $\operatorname{\mathcal{C}}^{\mathrm{init}}$ of Corollary 4.6.6.15 and are therefore isomorphic when viewed as objects of $\operatorname{\mathcal{C}}$. Conversely, suppose that $X$ is initial and that there exists an isomorphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$; we wish to show that $Y$ is also initial. Fix an object $Z \in \operatorname{\mathcal{C}}$; we wish to show that the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is contractible. Let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (see Construction 4.6.8.13). Since $f$ an isomorphism in $\operatorname{\mathcal{C}}$, its homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, so composition with $[f]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in the category $\mathrm{h} \mathit{\operatorname{Kan}}$. Since the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is contractible, it follows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is also contractible. $\square$