Definition 5.4.5.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$. We will say that $T$ has the inner exchange property if the following condition is satisfied:
- $(\ast )$
Let $\sigma : \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ be a $3$-simplex of $\operatorname{\mathcal{C}}$. For every triple of integers $0 \leq i < j < k \leq 3$, let $\sigma _{kji}$ be the face of $\sigma $ given by the restriction $\sigma |_{ \operatorname{N}_{\bullet }( \{ i < j < k \} )}$. Assume that the outer faces $\sigma _{210}$ and $\sigma _{321}$ belong to $T$. Then $\sigma _{310}$ belongs to $T$ if and only if $\sigma _{320}$ belongs to $T$.