Proposition 4.8.6.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n \geq 0$ be an integer. The following conditions are equivalent:
- $(1)$
The functor $F$ is essentially $n$-categorical.
- $(2)$
For every pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]the functor $F'$ is essentially $n$-categorical.
- $(3)$
For every pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]where $\operatorname{\mathcal{D}}'$ is locally $(n-1)$-truncated, the $\infty $-category $\operatorname{\mathcal{C}}'$ is also locally $(n-1)$-truncated.
- $(4)$
For every pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^1 \ar [r] & \operatorname{\mathcal{D}}, } \]the $\infty $-category $\operatorname{\mathcal{C}}'$ is locally $(n-1)$-truncated.