Kerodon

Variant 3.2.4.12. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. Then a morphism of simplicial sets $\sigma _0: \operatorname{\partial \Delta }^ n \rightarrow X$ is nullhomotopic if and only if it can be extended to an $n$-simplex of $X$. The “if” direction follows immediately from Exercise 3.2.4.8 (and does not require the assumption that $X$ is a Kan complex). For the converse, suppose that $\sigma _0$ is homotopic to a constant map $\sigma '_{0}: \operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. Since $\sigma '_{0}$ can be extended to a map $\sigma ': \Delta ^ n \rightarrow \{ x\} \hookrightarrow X$, it follows from the homotopy extension lifting property (Remark 3.1.5.3) that $\sigma _0$ can also be extended to an $n$-simplex of $X$.