Notation 1.2.3.7. Let $n \geq 0$ be an integer and let $\operatorname{\mathcal{U}}$ be a collection of nonempty subsets of $[n] = \{ 0, 1, \ldots , n \} $. We will say that $\operatorname{\mathcal{U}}$ is downward closed if $\emptyset \neq I \subseteq J \in \operatorname{\mathcal{U}}$ implies that $I \in \operatorname{\mathcal{U}}$. If this condition is satisfied, we let $\Delta ^{n}_{\operatorname{\mathcal{U}}}$ denote the simplicial subset of $\Delta ^{n}$ whose $m$-simplices are nondecreasing maps $\alpha : [m] \rightarrow [n]$ for which the image of $\alpha $ is an element of $\operatorname{\mathcal{U}}$. Similarly, we set
\[ | \Delta ^{n} |_{\operatorname{\mathcal{U}}} = \{ (t_0, \ldots , t_ n) \in | \Delta ^{n} |: \{ i \in [n]: t_{i} \neq 0 \} \in \operatorname{\mathcal{U}}\} . \]