Proposition 4.8.6.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq -1$ be an integer. The following conditions are equivalent:
- $(1)$
The functor $F$ is essentially $n$-categorical.
- $(2)$
For every integer $m \geq n+2$, every lifting problem
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]admits a solution.
- $(3)$
For every simplicial set $B$ and every simplicial subset $A \subseteq B$ which contains the $(n+1)$-skeleton of $B$, every lifting problem
\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]admits a solution.