Example 3.5.7.17. Let $X$ be a Kan complex. The following conditions are equivalent:
The Kan complex $X$ is $1$-truncated.
For every vertex $x \in X$, the homotopy groups $\pi _{n}(X,x)$ are trivial for $n \geq 2$.
There exists a groupoid $\mathcal{G}$ and a homotopy equivalence $X \xrightarrow {\sim } \operatorname{N}_{\bullet }( \mathcal{G} )$.
The tautological map $X \rightarrow \pi _{\leq 1}(X)$ is a homotopy equivalence.