$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.5.9.29. Let $n \geq -2$ be an integer and let $f: X \rightarrow Y$ be a morphism of Kan complexes. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $n$-truncated.
- $(2)$
For every $(n+1)$-connective morphism of simplicial sets $A \rightarrow B$, the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B, X) \ar [r] \ar [d] & \operatorname{Fun}(A, X) \ar [d] \\ \operatorname{Fun}( B, Y) \ar [r] & \operatorname{Fun}(A, Y) } \]
is a homotopy pullback square.
- $(3)$
The diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, X ) \ar [d] \\ Y \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^{n+2}, Y) } \]
is a homotopy pullback square.
Proof.
The equivalence $(1) \Leftrightarrow (3)$ follows from Corollary 3.5.9.20, and the implication $(2) \Rightarrow (3)$ from Corollary 3.5.2.6. It will therefore suffice to show that $(1)$ implies $(2)$. Using Proposition 3.1.7.1, we can reduce to the case where $f$ is a Kan fibration. In this case, the map $\operatorname{Fun}(A, X) \rightarrow \operatorname{Fun}(A,Y)$ is also a Kan fibration (Corollary 3.1.3.2), so condition $(2)$ is equivalent to the requirement that the map $\theta : \operatorname{Fun}(B, X) \rightarrow \operatorname{Fun}(A, X) \times _{ \operatorname{Fun}(A,Y) } \operatorname{Fun}(B, Y)$ is a homotopy equivalence (Example 3.4.1.3). This follows from Proposition 3.5.9.24 (and Example 3.5.9.2).
$\square$