Proposition 4.8.9.1 (05DH)

Proposition 4.8.9.1. Let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ be a functor of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The functor $F$ is categorically $n$-connective (Definition 4.8.7.1).

$(2)$

For every essentially $(n-1)$-categorical functor of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \ar [r]^-{\circ F} \ar [d]^{U \circ } & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}) \ar [d]^{ U \circ } \\ \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \ar [r]^-{\circ F} & \operatorname{Fun}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{D}}) } \]

is a categorical pullback square.

$(3)$

For every $(n-1)$-categorical isofibration $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$, precomposition with $F$ induces an equivalence of $\infty $-categories

\[ \theta _{\operatorname{\mathcal{C}}}: \operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname{\mathcal{A}}, \operatorname{\mathcal{C}}). \]

Proof. The implication $(1) \Rightarrow (2)$ is a restatement of Corollary 4.8.7.18, and the implication $(2) \Rightarrow (3)$ follows from Corollary 4.5.2.32. To show that $(3)$ implies $(1)$, we may assume without loss of generality that $F$ is an isofibration. Then the comparison map $G: \mathrm{h}_{\mathit{\leq n-1}}\mathit{(()}\operatorname{\mathcal{A}}/\operatorname{\mathcal{B}}) \rightarrow \operatorname{\mathcal{B}}$ $(n-1)$-categorical isofibration (Propositions 4.8.8.14 and 4.8.8.21). If $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ is another $(n-1)$-categorical isofibration, then we can use Proposition 4.8.8.16 to identify $\theta _{\operatorname{\mathcal{C}}}$ with the map with the functor $\operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{B}}' ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \mathrm{h}_{\mathit{\leq n-1}}\mathit{(\operatorname{\mathcal{A}}/\operatorname{\mathcal{B}})}, \operatorname{\mathcal{C}})$ given by precomposition with $G$. If condition $(3)$ is satisfied, then $G$ is an equivalence of $\infty $-categories, so that $F$ is categorically $n$-connective by virtue of Corollary 4.8.8.25. $\square$