Lemma 4.6.8.30 (01M2)

Lemma 4.6.8.30. Let $n \geq 0$ be an integer, and let $\pi : \Phi ( \Delta ^ n) \times \Delta ^1 \rightarrow \Phi ( \Delta ^{n+1} )$ be the morphism of simplicial sets defined in Construction 4.6.8.29. Then $\pi $ fits into a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Phi ( \Delta ^ n ) \times \{ 0\} \ar [r] \ar [d] & \Phi ( \Delta ^ n ) \times \Delta ^1 \ar [d]^{\pi } \\ \Delta ^0 \ar [r] & \Phi ( \Delta ^{n+1} ). } \]

Proof of Lemma 4.6.8.30. Fix an integer $m \geq 0$. By construction, the restriction $\pi |_{ \Phi (\Delta ^ n) \times \{ 0\} }$ is the constant map which carries each $m$-simplex of $\Phi ( \Delta ^{n} )$ to the element of $\Phi ( \Delta ^{n+1} )$ given by the constant chain $\overrightarrow {S}_0 = ( \{ 0 \} \subseteq \{ 0 \} \subseteq \cdots \subseteq \{ 0 \} )$. To complete the proof, we must show that for each $m \geq 0$, the map $\pi $ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \Phi ( \Delta ^ n)_ m \times \{ \textnormal{Nondecreasing functions $\tau : [m] \rightarrow [1]$ with $\tau (m) = 1$} \} \ar [d] \\ \Phi ( \Delta ^{n+1})_{m} \setminus \{ \overrightarrow {S}_0 \} }. \]

The inverse bijection can be described explicitly as follows: it carries an $m$-simplex $(S'_0 \supseteq \cdots \supseteq S'_ m) \neq \overrightarrow {S}_0$ of $\Phi ( \Delta ^{n+1} )$ to the pair $( \overrightarrow {S}, \tau )$, where $\overrightarrow {S} = (S_0 \supseteq \cdots \supseteq S_ m)$ is the $m$-simplex of $\Phi ( \Delta ^ n )$ given by

\[ S_{i} = \{ s-1: s \in S'_ i, s > 0 \} \quad \quad \tau (i) = \begin{cases} 0 & \textnormal{ if } 0 \in S'_ i \\ 1 & \textnormal{ if } 0 \notin S'_ i. \end{cases} \]

$\square$