Kerodon

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Corollary 3.5.1.32. Let $n$ be a nonnegative integer. Then a simplicial set $X$ is $n$-connective if and only if it is nonempty and the diagonal map $\delta _{X}: X \rightarrow X \times X$ is $(n-1)$-connective.

Proof. Using Corollary 3.1.7.2 and Proposition 3.1.6.23, we can reduce to the situation where $X$ is a Kan complex. In this case, the follows by applying Corollary 3.5.1.31 in the special case $Y = \Delta ^0$. $\square$