Kerodon

Corollary 3.5.9.17. Let $f: X \rightarrow Y$ be a Kan fibration between Kan complexes and let $n \geq -1$. Then $f$ is $n$-truncated if and only if the relative diagonal $\delta _{X/Y}: X \rightarrow X \times _{Y} X$ is $(n-1)$-truncated.