Proposition 4.8.7.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The functor $F$ is categorically $n$-connective.
- $(2)$
For every integer $0 \leq m \leq n$, 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 $B$ of dimension $\leq n$ and every simplicial subset $A \subseteq B$, 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.