Corollary 3.5.9.20. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n \geq -2$. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $n$-truncated.
- $(2)$
The restriction map
\[ \theta : \operatorname{Fun}(\Delta ^{n+2}, X) \rightarrow \operatorname{Fun}(\Delta ^{n+2}, Y) \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, Y) } \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, X) \]is a homotopy equivalence.
- $(3)$
The diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^{n+2},X) \ar [d]^{f} \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, X) \ar [d] \\ \operatorname{Fun}(\Delta ^{n+2}, Y) \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, Y) } \]is a homotopy pullback square.
- $(4)$
The diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, X) \ar [d] \\ Y \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, Y) } \]is a homotopy pullback square.