Definition 4.1.1.1. Let $q: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. We say that $q$ is an inner fibration if, for every pair of integers $0 < i < n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & X_{} \ar [d]^{q} \\ \Delta ^{n} \ar@ {-->}[ur]^-{\sigma } \ar [r]^-{ \overline{\sigma } } & S_{} } \]
admits a solution (as indicated by the dotted arrow). That is, for every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X_{}$ and every $n$-simplex $\overline{\sigma }: \Delta ^ n \rightarrow S_{}$ extending $q \circ \sigma _0$, we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_{}$ satisfying $q \circ \sigma = \overline{\sigma }$.