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Proposition 4.8.9.8. Let $f: A \hookrightarrow B$ be a monomorphism of simplicial sets and let $n$ be an integer. The following conditions are equivalent:

$(1)$

The morphism $f$ is categorically $n$-connective.

$(2)$

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

\[ V: \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}) \]

is an equivalence of $\infty $-categories.

$(3)$

For every essentially $(n-1)$-categorical isofibration of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the functor $V$ is a trivial Kan fibration.

$(3)$

Every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{D}}} \]

admits a solution, provided that $U$ is an essentially $(n-1)$-categorical isofibration of $\infty $-categories.

Proof. The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ follow from Remarks 4.8.7.19 and 4.8.9.3, and the implication $(3) \Rightarrow (4)$. is immediate. We will complete the proof by showing that $(4)$ implies $(3)$. Assume that condition $(4)$ is satisfied, and let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an essentially $(n-1)$-categorical isofibration of $\infty $-categories. We wish to show that, for every simplicial set $B'$ and every simplicial subset $A' \subseteq B'$, every lifting problem

4.90
\begin{equation} \begin{gathered}\label{equation:categorically-connective-monomorphism} \xymatrix@C =50pt@R=50pt{ A' \ar [r] \ar [d] & \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \ar [d]^{V} \\ B' \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(A, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A, \operatorname{\mathcal{D}}) } \operatorname{Fun}(B, \operatorname{\mathcal{D}}) } \end{gathered} \end{equation}

admits a solution. Unwinding the definitions, we can rewrite (4.90) as a lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d]^{f} & \operatorname{Fun}(B', \operatorname{\mathcal{C}}) \ar [d]^{V'} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(A', \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A', \operatorname{\mathcal{D}}) } \operatorname{Fun}(B', \operatorname{\mathcal{D}}). } \]

The existence of a solution follows from $(4)$, since $V'$ is also an essentially $(n-1)$-categorical isofibration of $\infty $-categories (Corollary 4.8.6.20 and Proposition 4.4.5.1). $\square$