Proposition 4.7.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $n \geq -2$ be an integer. Then $X$ is $n$-truncated if and only if it satisfies the following condition:
For $m \geq n+3$, every morphism $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.
Proof. Set $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{/X}$, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map. We can then restate condition $(\ast )$ as follows:
For $k \geq n+2$, every lifting problem
admits a solution.
For each $C \in \operatorname{\mathcal{C}}$, we can identify the fiber $U^{-1} \{ C\} $ with the pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( C, X )$. If condition $(\ast ')$ is satisfied, then each $U^{-1} \{ C\} $ is an $n$-truncated Kan complex, so that $X$ is an $n$-truncated object of $\operatorname{\mathcal{C}}$ (Remark 4.7.1.2).
For the converse, assume that $X$ is $n$-truncated, and suppose we are given a lifting problem of the form (4.73). Let $R: \Delta ^1 \times \Delta ^ k \rightarrow \Delta ^ k$ be the map given on vertices by the formula $R(i,j) = ij$. Then we can regard $\overline{H} = \overline{\sigma } \circ R$ as a natural transformation $\overline{\tau } \rightarrow \overline{\sigma }$, where $\overline{\tau }: \Delta ^ k \rightarrow \operatorname{\mathcal{C}}$ is the constant diagram taking the value $C = \overline{\sigma }(0)$. Since $U$ is a right fibration (Proposition 4.3.6.1), the restriction $\overline{H}|_{ \Delta ^1 \times \operatorname{\partial \Delta }^ k}$ can be lifted to a natural transformation $H_0: \tau _0 \rightarrow \sigma _0$ of natural transformations from $\operatorname{\partial \Delta }^ k$ to $\operatorname{\mathcal{E}}$ (Remark 4.2.6.3). Note that $\tau _0$ takes values in the fiber $U^{-1} \{ X\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( C, X )$. Our assumption that $X$ is $n$-truncated guarantees that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( C, X )$ is an $n$-truncated Kan complex, so that $\tau _0$ can be extended to a $k$-simplex $\tau : \Delta ^ k \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( C, X )$.
Note that $H_0$ carries the vertex $0 \in \operatorname{\partial \Delta }^ k$ to a morphism in the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( C, X )$, which is therefore an isomorphism in $\operatorname{\mathcal{E}}$. Since $U$ is an isofibration (Example 4.4.1.11), Proposition 4.4.5.8 guarantees that $H_0$ can be extended to a natural transformation $H: \Delta ^1 \times \Delta ^ k \rightarrow \operatorname{\mathcal{E}}$ satisfying $U \circ H = \overline{H}$. We can then regard $H$ as a natural transformation from $\tau $ to $\sigma $, where $\sigma : \Delta ^ k \rightarrow \operatorname{\mathcal{E}}$ is a solution to the lifting problem (4.73). $\square$