Example 3.5.9.18. Let $X$ be a Kan complex. Then the diagonal map $\delta _{X}: X \hookrightarrow X \times X$ factors as a composition
\[ X \xrightarrow {u} \operatorname{Fun}( \Delta ^1, X ) \xrightarrow {q} \operatorname{Fun}( \operatorname{\partial \Delta }^1, X) \simeq X \times X, \]
where $u$ is a homotopy equivalence and $q$ is a Kan fibration (Corollary 3.1.3.3). Combining Corollary 3.5.9.17 with Proposition 3.5.9.8, we see that the following conditions are equivalent for every integer $n \geq -1$:
The Kan complex $X$ is $n$-truncated.
The diagonal morphism $\delta _{X}: X \hookrightarrow X \times X$ is $(n-1)$-truncated.
For every pair of vertices $x,y \in X$, the path space
\[ \{ x \} \times ^{\mathrm{h}}_{X} \{ y\} = \{ (x,y) \} \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^1, X) } \operatorname{Fun}(\Delta ^1, X) \]is $(n-1)$-truncated.