Kerodon

Example 1.1.5.6. Let $X_{\bullet }$ be a simplicial set and let $S = X_0$ be the set of vertices of $X_{\bullet }$. It follows from Proposition 1.1.5.5 that there is a unique morphism of simplicial sets $f: \underline{S} \rightarrow X_{\bullet }$ which is the identity map on $0$-simplices. Using Proposition 1.1.4.12, we see that this map is an isomorphism from $\underline{S}$ to the $0$-skeleton $\operatorname{sk}_0(X_{\bullet })$. In particular, $f$ is a monomorphism, which is an isomorphism if and only if $X_{\bullet }$ has dimension $\leq 0$.