Variant 3.1.7.12. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, and let $n$ be a nonnegative integer. Arguing as in the proof of Proposition 3.1.7.1, we see that $f$ admits a factorization $X \xrightarrow {f'} Q(f) \xrightarrow {f''} Y$ with the following properties:
- $(a)$
The morphism $f'$ can be realized as a transfinite pushout of horn inclusions $\Lambda ^{m}_{i} \hookrightarrow \Delta ^ m$ for $0 \leq i \leq m$ and $m > n$.
- $(b)$
For $0 \leq i \leq m$ and $m > n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r] \ar [d] & Q(f) \ar [d]^{f''} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & Y } \]admits a solution.
It follows from $(a)$ that morphism $f'$ is a monomorphism which is bijective on $k$-simplices for $k < n$.
Now suppose that the morphism $f$ satisfies the following additional condition:
- $(\ast )$
For $0 \leq i \leq m$ and $0 < m \leq n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r] \ar [d] & X \ar [d]^{f} \\ \Delta ^{m} \ar [r] \ar@ {-->}[ur] & Y } \]admits a solution.
Since $f'$ is bijective on $k$-simplices for $k < n$, it follows that the morphism $f''$ also satisfies condition $(\ast )$. Combining this with assumption $(b)$, we conclude that $f''$ is a Kan fibration.