Kerodon

Notation 3.2.2.5. Let $(X,x)$ be a pointed Kan complex and let $n$ be a nonnegative integer. Then the set of pointed morphisms $\Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow X$ can be identified with the set of $n$-simplices $\sigma : \Delta ^ n \rightarrow X$ having the property that $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$. In this case, we write $[\sigma ]$ for the image of $\sigma $ in the set $\pi _{n}(X,x)$. Note that, if $\tau $ is another $n$-simplex of $X$ for which $\tau |_{ \operatorname{\partial \Delta }^ n}$ is the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$, then the equality $[\sigma ] = [\tau ]$ holds in $\pi _{n}(X,x)$ if and only if there exists a homotopy $h: \Delta ^1 \times \Delta ^ n \rightarrow X$ such that $\sigma = h|_{ \{ 0\} \times \Delta ^ n }$, $\tau = h|_{ \{ 1\} \times \Delta ^ n }$, and $h|_{ \Delta ^1 \times \operatorname{\partial \Delta }^ n }$ is the constant map taking the value $x$.