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Definition Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$. We say that $T$ has the four-out-of-five property if it satisfies the following condition:

  • Let $\sigma : \Delta ^{4} \rightarrow \operatorname{\mathcal{C}}$ be a $4$-simplex of $\operatorname{\mathcal{C}}$. For every triple of integers $0 \leq i < j < k \leq 4$, let $\sigma _{kji}$ denote the $2$-simplex of $\operatorname{\mathcal{C}}$ given by the restriction of $\sigma $ to $\operatorname{N}_{\bullet }( \{ i < j < k \} )$. If the $2$-simplices $\sigma _{310}$, $\sigma _{420}$, $\sigma _{321}$, and $\sigma _{432}$ belong to $T$, then the $2$-simplex $\sigma _{430}$ also belongs to $T$.