Proposition 3.5.7.7. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. Then $X$ is $n$-truncated if and only if it satisfies the following condition for every integer $m > n$:
- $(\ast _ m)$
For every vertex $x \in X$, the homotopy group $\pi _{m}(X,x)$ is trivial.