Remark 3.5.7.20. Let $f: X \rightarrow Y$ be a morphism of Kan complexes and let $n \geq 0$. Then $f$ exhibits $Y$ as an $n$-truncation of $X$ if and only if the following conditions are satisfied:
The morphism $f$ induces a bijection from $\pi _0(X)$ to $\pi _0(Y)$.
For every vertex $x \in X$ having image $y = f(x)$, the map of homotopy groups $\pi _{m}(X,x) \rightarrow \pi _{m}(Y,y)$ is a bijection for $0 < m \leq n$.
For each vertex $y \in Y$ and every integer $m > n$, the homotopy group $\pi _{m}(Y,y)$ vanishes.
See Proposition 3.5.7.7.