Kerodon

Remark 3.2.2.21. Let $(X,x)$ be a pointed Kan complex and let $n$ be a positive integer. Suppose that $\sigma , \sigma ': \Delta ^{n} \rightarrow X$ are $n$-simplices of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ and $\sigma '|_{ \operatorname{\partial \Delta }^{n} }$ are equal to the constant map $\operatorname{\partial \Delta }^{n} \rightarrow \{ x\} \subseteq X$. It follows from Theorem 3.2.2.10 that the equality $[ \sigma ] = [\sigma ']$ holds (in the homotopy group $\pi _{n}(X,x)$) if and only if there exists an $(n+1)$-simplex $\tau $ of $X$ such that $d^{n+1}_0(\tau ) = \sigma $, $d^{n+1}_1(\tau ) = \sigma '$, and $d^{n+1}_{i}(\tau )$ is the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $2 \leq i \leq n+1$.