$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 4.8.7.15. 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.
- $(2)$
The functor $F$ factors as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {U} \operatorname{\mathcal{D}}, \]
where $F'$ is a monomorphism which is bijective on $m$-simplices for $m \leq n$, and $U$ is a trivial Kan fibration.
- $(3)$
The functor $U$ factors as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {U} \operatorname{\mathcal{D}}, \]
where $F'$ is bijective on $m$-simplices for $m < n$, surjective on $n$-simplices, and $U$ is categorically $n$-connective.
Proof.
We proceed as in the proof of Corollary 3.5.2.4. The implication $(2) \Rightarrow (3)$ is clear, and the implication $(3) \Rightarrow (1)$ follows from Propositions 4.8.7.12 and 4.8.7.13. We will complete the proof by showing that $(1)$ implies $(2)$. Assume that $F$ is categorically $n$-connective. Using a variant of Exercise 3.1.7.11, we can choose a factorization of $F$ as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {U} \operatorname{\mathcal{D}}$ with the following properties;
- $(a)$
For every integer $m > n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{D}}' \ar [d]^{U} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \]
admits a solution.
- $(b)$
The morphism $F'$ can be realized as a transfinite pushout of inclusion maps $\operatorname{\partial \Delta }^ m \hookrightarrow \Delta ^ m$ for $m > n$.
It follows immediately from $(b)$ that $F$ is a monomorphism which is bijective on simplices of dimension $\leq n$. We will complete the proof by showing that $U$ is a trivial Kan fibration: that is, every lifting problem
4.85
\begin{equation} \begin{gathered}\label{equation:connective-Kan-complex-factorization-categorical} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{D}}' \ar [d]^{U} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}
admits a solution. For $m > n$, this follows from $(b)$. For $m \leq n$, we can identify (4.85) with a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}, } \]
which admits a solution by virtue of our assumption that $F$ is a categorically $n$-connective isofibration (Proposition 4.8.7.14).
$\square$