Definition 1.5.5.1. Let $q: X \rightarrow Y$ be a morphism of simplicial sets. We say that $p$ is a trivial Kan fibration if, for each $n \geq 0$, every lifting problem
\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\partial \Delta }^ n \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & Y } \]
admits a solution; here $i: \operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ denotes the inclusion map.