Example 3.5.7.25. Let $X$ be a Kan complex and let $n$ be a nonnegative integer. Then the tautological map $f: X \rightarrow \pi _{\leq n}(X)$ exhibits the fundamental $n$-groupoid $\pi _{\leq n}(X)$ as an $n$-truncation of $X$. This follows from Example 3.5.7.23 and Remark 3.5.7.22, since the quotient map $\operatorname{cosk}_{n+1}(X) \twoheadrightarrow \pi _{\leq n}(X)$ is a trivial Kan fibration (Corollary 3.5.6.15).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$