Construction 1.1.4.10 (The Boundary of $\Delta ^ n$). Let $n \geq 0$ be an integer and let $\Delta ^{n}$ denote the standard $n$-simplex (Example 1.1.0.9). We let $\operatorname{\partial \Delta }^{n}$ denote the $(n-1)$-skeleton of $\Delta ^{n}$. We will refer to $\operatorname{\partial \Delta }^ n$ as the boundary of $\Delta ^ n$. More explicitly, the simplicial set $(\operatorname{\partial \Delta }^ n): \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is defined by the formula
\[ ( \operatorname{\partial \Delta }^{n} )( [m] ) = \{ \alpha \in \operatorname{Hom}_{\operatorname{{\bf \Delta }}}( [m], [n] ): \text{$\alpha $ is not surjective} \} . \]