Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 1.1.2.1 (The Standard Simplex). Let $n \geq 0$ be an integer. We let $\Delta ^{n}$ denote the simplicial set given by the construction

\[ ([m] \in \operatorname{{\bf \Delta }}) \mapsto \operatorname{Hom}_{ \operatorname{{\bf \Delta }}}( [m], [n] ). \]

We will refer to $\Delta ^{n}$ as the standard $n$-simplex. By convention, we extend this construction to the case $n = -1$ by setting $\Delta ^{-1} = \emptyset $.