Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Lemma 3.2.3.7. Let $\vec{\eta } = ( \eta _0, \eta _1, \ldots , \eta _{n+1} )$ be an $(n+2)$-tuple of elements of $\pi _{n}(X,x)$, let $1 \leq i \leq n$ be an integer, and let $\alpha $ be another element of $\pi _{n}(X,x)$. If $\vec{\eta }$ bounds, then the tuple $( \eta _0, \ldots , \eta _{i-2}, m_ i( \alpha , \eta _{i-1}), m_ i( \alpha , \eta _{i} ), \eta _{i+1}, \ldots , \eta _{n+1} )$ also bounds.

Proof. For $0 \leq i \leq n+1$, choose an element $\sigma _{i} \in \Sigma $ satisfying $[ \sigma _ i ] = \eta _ i$. Since $\vec{\eta }$ bounds, we can choose an $(n+1)$-simplex $\overline{\sigma }$ of $X$ satisfying $\sigma _ i = d_ i( \tau )$ for $0 \leq i \leq n+1$. Choose $\tau \in \Sigma $ satisfying $[\tau ] = \alpha $. Since $X$ is a Kan complex, we can choose $(n+1)$-simplices $\rho , \rho ': \Delta ^{n+1} \rightarrow X$ satisfying the identities

\[ d_{j}(\rho ) = \begin{cases} e & \text{ if } 0 \leq j < i-1 \\ \tau & \text{ if } j = i-1 \\ \sigma _{i-1} & \text{ if } j = i+1 \\ e & \text{ of } i+1 < j \leq n+1. \end{cases} \quad \quad d_{j}(\rho ') = \begin{cases} e & \text{ if } 0 \leq j < i-1 \\ \tau & \text{ if } j = i-1 \\ \sigma _{i} & \text{ if } j = i+1 \\ e & \text{ of } i+1 < j \leq n+1. \end{cases} \]

The definition of the multiplication $m_ i$ then gives $m_{i}( \alpha , \eta _{i-1} ) = [ d_ i(\rho ) ]$ and $m_ i( \alpha , \eta _ i ) = [ d_ i(\rho ')]$. The tuple $( s_{i}(\sigma _0), \ldots , s_{i}( \sigma _{i-2}), \rho , \rho ', \bullet , \overline{\sigma }, s_{i+1}(\sigma _{i+2}), \ldots , s_{i+1}( \sigma _{n+1} ) )$ then determines a map of simplicial sets $\Lambda ^{n+2}_{i+1} \rightarrow X$ (Exercise 1.1.2.14). Since $X$ is a Kan complex, this map can be extended to an $(n+2)$-simplex of $X$. Let $\overline{\sigma }'$ denote the $(i+1)$st face of this simplex. By construction, we have

\[ d_{j}( \overline{\sigma }' ) = \begin{cases} d_ i(\rho ) & \text{ if } j = i-1 \\ d_ i( \rho ' ) & \text{ if } j = i \\ \sigma _{j} & \text{otherwise,} \end{cases} \]

so that $\overline{\sigma }'$ witnesses that the tuple $( \eta _0, \ldots , \eta _{i-2}, m_ i( \alpha , \eta _{i-1}), m_ i( \alpha , \eta _{i} ), \eta _{i+1}, \ldots , \eta _{n+1} )$ bounds. $\square$