Theorem 3.5.6.17. Let $X$ be a Kan complex. For every $n \geq 0$, the comparison map $v: X \rightarrow \pi _{\leq n}(X)$ of Construction 3.5.6.10 exhibits $\pi _{\leq n}(X)$ as a fundamental $n$-groupoid of $X$.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

**Proof.**
It follows from Remark 3.5.6.11 that $v$ is bijective on $m$-simplices for $m < n$ and surjective on $n$-simplices. By construction, if $\sigma $ and $\sigma '$ are $n$-simplices of $X$, then $v(\sigma ) = v(\sigma ')$ if and only if $\sigma $ and $\sigma '$ are homotopic relative to $\operatorname{\partial \Delta }^{n}$. It will therefore suffice to show that $\pi _{\leq n}(X)$ is an $n$-groupoid, which follows from Corollary 3.5.6.16.
$\square$