Kerodon

Definition 3.2.5.1. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n \geq 0$ be a nonnegative integer. Suppose we are given a pair of maps $\sigma : \Delta ^{n} \rightarrow X_ s$ and $\tau : \Delta ^{n+1} \rightarrow S$, having the property that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ and $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ are the constant maps taking the values $x$ and $s$, respectively. We will say that $\sigma $ is incident to $\tau $ if there exists a simplex $\widetilde{\tau }: \Delta ^{n+1} \rightarrow X$ satisfying $\tau = f( \widetilde{\tau } )$, $\sigma = d^{n+1}_{0}(\widetilde{\tau })$, and $\widetilde{\tau }|_{ \Lambda ^{n+1}_{0} }: \Lambda ^{n+1}_{0} \rightarrow X$ is the constant map taking the value $x$.