Corollary 3.5.1.27. Let $Y$ be a simplicial set and let $n \geq -1$ be an integer. The following conditions are equivalent:
- $(1)$
The simplicial set $Y$ is $n$-connective.
- $(2)$
The simplicial set $Y$ is nonempty and, for every vertex $y \in Y$, the inclusion map $\{ y\} \hookrightarrow Y$ is $(n-1)$-connective.
- $(3)$
There exists a vertex $y \in Y$ for which the inclusion map $\{ y\} \hookrightarrow Y$ is $(n-1)$-connective.