Definition 4.2.1.1. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We will say that $f$ is a left fibration if, for every pair of integers $0 \leq i < n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^{n} \ar@ {-->}[ur]^-{\sigma } \ar [r]^-{ \overline{\sigma } } & S_{} } \]
has 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 $f \circ \sigma _0$, we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_{}$ satisfying $f \circ \sigma = \overline{\sigma }$.
We say that $f$ is a right fibration if, for every pair of integers $0 < i \leq n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^{n} \ar@ {-->}[ur]^-{\sigma } \ar [r]^-{ \overline{\sigma } } & S_{} } \]
has a solution.