Variant 4.8.6.25. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets.
We say that $U$ is a $0$-categorical inner fibration if, for every morphism $\Delta ^{m} \rightarrow \operatorname{\mathcal{D}}$, the fiber product $\Delta ^ m \times _{ \operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is isomorphic to the nerve of a partially ordered set.
We say that $F$ is a $(-1)$-categorical inner fibration if it induces an isomorphism from $\operatorname{\mathcal{C}}$ to a full simplicial subset of $\operatorname{\mathcal{D}}$ (Definition 4.1.2.15).
For $n \leq -2$, we say that $F$ is an $n$-categorical inner fibration if it is an isomorphsm of simplicial sets.