Example 1.1.0.9 (The Standard Simplex). Let $n \geq 0$ be an integer. We let $\Delta ^{n}$ denote the functor
\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad [m] \mapsto \operatorname{Hom}_{\operatorname{{\bf \Delta }}}( [m], [n] ). \]
Then $\Delta ^ n$ is a simplicial set, which we will refer to as the standard $n$-simplex. By convention, we extend this construction to the case $n = -1$ by setting $\Delta ^{-1} = \emptyset $.