Exercise 1.2.3.9. Show that every simplicial subset of the standard $n$-simplex $\Delta ^{n}$ has the form $\Delta ^{n}_{\operatorname{\mathcal{U}}}$, where $\operatorname{\mathcal{U}}$ is some (uniquely determined) downward closed collection of nonempty subsets of $[n]$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$