Example 3.5.7.26. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then $f$ exhibits $Y$ as a $(-1)$-truncation of $X$ if and only if one of the following two conditions is satisfied:
Both $X$ and $Y$ are empty.
The Kan complex $X$ is nonempty and $Y$ is contractible.
See Example 3.5.7.4.