Corollary 4.6.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An object $Y \in \operatorname{\mathcal{C}}$ is initial if and only if, for every integer $n \geq 1$ and every morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (0) = Y$, there exists an $n$-simplex $\overline{\sigma }: \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\sigma }|_{ \operatorname{\partial \Delta }^ n} = \sigma $.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $n$ be a positive integer. Using the isomorphism
\[ \operatorname{\partial \Delta }^{n} \simeq (\emptyset \star \Delta ^{n-1} ) \coprod _{ (\emptyset \star \operatorname{\partial \Delta }^{n-1}) } ( \Delta ^{0} \star \operatorname{\partial \Delta }^{n-1} ) \]
supplied by Variant 4.3.6.18, we see that a morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (0) = Y$ can be identified with a commutative diagram
4.61
\begin{equation} \begin{gathered}\label{equation:characterize-initial-by-slice} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n-1} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{Y/} \ar [d] \\ \Delta ^{n-1} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}
and that an extension of $\sigma $ to an $n$-simplex of $\operatorname{\mathcal{C}}$ can be identified with a dotted arrow which renders the diagram commutative. By virtue of Proposition 4.6.7.10, the object $Y$ is initial if and only if every lifting problem of the form (4.61) admits a solution: that is, if and only if the projection map $\operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets. $\square$