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Definition Let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. We will say that $q$ is an interior fibration if it satisfies the following conditions:

  • Every lifting problem

    \[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{q} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]

    admits a solution, provided that $0 < i < n$ and the restriction $\overline{\sigma }|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.

  • For every vertex $X \in \operatorname{\mathcal{C}}$, the degenerate edge $\operatorname{id}_{X}$ is $q$-cartesian and $q$-cocartesian.