Kerodon

Example 1.5.5.9. Let $X$ be a topological space. Then the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a contractible Kan complex if and only if the space $X$ is weakly contractible: that is, if and only if every continuous map $\sigma _0: S^{n-1} \rightarrow X$ is nullhomotopic (here $S^{n-1} \simeq | \operatorname{\partial \Delta }^ n |$ denotes the sphere of dimension $n-1$, so that $\sigma _0$ is nullhomotopic if and only if it extends to a continuous map defined on the disk $D^ n \simeq | \Delta ^ n |$). In particular, if the topological space $X$ is contractible, then the simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a contractible Kan complex.