Kerodon

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

Proposition 7.1.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. Then:

$(1)$

The object $Y$ is initial if and only if the projection map $\operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets.

$(2)$

The object $Y$ is final if and only if the projection map $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets.

Proof. We will give the proof of $(1)$; the proof of $(2)$ is similar. Proposition 4.3.6.1 guarantees that the projection map $q: \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets. Applying Proposition 4.4.2.14, we see that $q$ is a trivial Kan fibration if and only if, for each object $Z \in \operatorname{\mathcal{C}}$, the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}( Y, Z) = \operatorname{\mathcal{C}}_{Y/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $ is a contractible Kan complex. By virtue of Remark 7.1.2.7, this is equivalent to the assumption that $Y$ is an initial object of $\operatorname{\mathcal{C}}$. $\square$