Construction 4.6.8.29. Let $m$ and $n$ be nonnegative integers. Suppose we are given an $m$-simplex of $\Phi (\Delta ^ n)$, which we identify with a chain of subsets $\overrightarrow {S} = (S_0 \supseteq \cdots \supseteq S_ m)$ satisfying conditions $(a)$ and $(b)$ of Example 4.6.8.28. Let $\tau : [m] \rightarrow [1]$ be a nondecreasing function. Let $\overrightarrow {S}' = (S'_0 \supseteq \cdots \supseteq S'_ m)$ be the chain of subsets of $[n+1]$ given by the formula
The construction $(\overrightarrow {S}, \tau ) \mapsto \overrightarrow {S}'$ is compatible with the formation of face and degeneracy operators, and therefore determines a morphism of simplicial sets $\pi : \Phi ( \Delta ^ n) \times \Delta ^1 \rightarrow \Phi ( \Delta ^{n+1} )$.