Variant 3.5.7.16. Let $X$ be a Kan complex and let $n \geq 0$. The following conditions are equivalent:
The Kan complex $X$ is $n$-truncated.
There exists a homotopy equivalence $X \rightarrow Y$, where $Y$ is an $n$-groupoid.
The tautological map $X \rightarrow \pi _{\leq n}(X)$ is a homotopy equivalence.
Proof. The implication $(3) \Rightarrow (2)$ follows from Corollary 3.5.6.16 and the implication $(2) \Rightarrow (1)$ follows from Example 3.5.7.2. The implication $(3) \Rightarrow (1)$ follows from Proposition 3.5.7.15, since the tautological map $X \rightarrow \pi _{\leq n}(X)$ factors as a composition
where $q$ is a trivial Kan fibration (Corollary 3.5.6.15). $\square$