Example 3.5.7.5. A Kan complex $X$ is $0$-truncated if and only if it satisfies any of the following equivalent conditions:
Every connected component of $X$ is contractible.
The projection map $X \twoheadrightarrow \pi _0(X)$ is a trivial Kan fibration of simplicial sets.
The projection map $X \twoheadrightarrow \pi _0(X)$ is a homotopy equivalence.
The Kan complex $X$ is homotopy equivalent to a discrete simplicial set.