Proof.
The implication $(2) \Rightarrow (3)$ is immediate and the implication $(3) \Rightarrow (1)$ follows from Corollary 3.5.2.2 (since the collection of $n$-connective morphisms is closed under composition; see Corollary 3.5.1.28). We will complete the proof by showing that $(1)$ implies $(2)$. Using a variant of Exercise 3.1.7.11, we can choose a factorization of $f$ as a composition $X \xrightarrow {f'} Y \xrightarrow {f''} Z$ with the following properties;
- $(a)$
For every integer $m > n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & Y \ar [d]^{f''} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & Z } \]
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 the morphism $f'$ is bijective on $k$-simplices for $0 \leq k \leq n$. We will complete the proof by showing that, if $f$ is $n$-connective, then $f''$ is a trivial Kan fibration: that is, every lifting problem
3.69
\begin{equation} \begin{gathered}\label{equation:connective-Kan-complex-factorization} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & Y \ar [d]^{f''} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & Z } \end{gathered} \end{equation}
admits a solution. For $m > n$, this follows from $(b)$. For $m \leq n$, we can identify (3.69) with a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & X \ar [d]^{f} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & Z, } \]
which admits a solution by virtue of our assumption that $f$ is an $n$-connective Kan fibration (Proposition 3.5.2.1).
$\square$