Remark 3.5.1.4. Let $X$ be a Kan complex. It follows from Example 3.2.2.18 that the isomorphism class of the homotopy group $\pi _{m}(X,x)$ depends only on the connected component $[x] \in \pi _0(X)$. Consequently, if $n > 0$, then $X$ is $n$-connective if and only if it is connected and the homotopy groups $\pi _{m}(X,x)$ are trivial for $0 < m < n$ for some choice of vertex $x \in X$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$