$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example Let $S_{\bullet }$ be a simplicial set and let $v$ be a vertex of $S_{\bullet }$. Then $v$ can be identified with a map of simplicial sets $\Delta ^0 \rightarrow S_{\bullet }$. This map is automatically a monomorphism (note that $\Delta ^0$ has only a single $n$-simplex for every $n \geq 0$), whose image is a simplicial subset of $S_{\bullet }$. It will often be convenient to denote this simplicial subset by $\{ v \} $. For example, we can identify vertices of the standard $n$-simplex $\Delta ^ n$ with integers $i$ satisfying $0 \leq i \leq n$; every such integer $i$ determines a simplicial subset $\{ i \} \subseteq \Delta ^ n$ (whose $k$-simplices are the constant maps $[k] \rightarrow [n]$ taking the value $i$).