Corollary 3.5.9.23. Let $f: X \rightarrow Y$ be a Kan fibration between Kan complexes and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $n$-truncated.
- $(2)$
For every nonnegative integer $m \geq n+2$, the induced map
\[ \theta : \operatorname{Fun}( \Delta ^{m}, X) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^{m}, X) \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^{m}, Y) } \operatorname{Fun}( \Delta ^{m}, Y) \]is a trivial Kan fibration.
- $(3)$
For every nonnegative integer $m \geq n+2$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & X \ar [d]^{f} \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & Y } \]has a solution.
- $(4)$
For every simplicial set $B$ and every simplicial subset $A \subseteq B$ which contains the $(n+1)$-skeleton $\operatorname{sk}_{n+1}(B)$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & X \ar [d]^{f} \\ B \ar@ {-->}[ur] \ar [r] & Y } \]admits a solution.