Kerodon

Example 4.8.4.6. Let $X$ be a Kan complex, let $n$ be a nonnegative integer, and let $\pi _{\leq n}(X)$ denote the fundamental $n$-groupoid of $X$ (Notation 3.5.6.6). Then the tautological map $u: X \rightarrow \pi _{\leq n}(X)$ exhibits $\pi _{\leq n}(X)$ as a homotopy $n$-category of $X$, in the sense of Definition 4.8.4.1. Since $\pi _{\leq n}(X)$ is an $(n,1)$-category (Example 4.8.1.11), it will suffice that for every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $u$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \pi _{\leq n}(X), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{\mathcal{D}})$. By virtue of Proposition 4.4.3.17, we can replace $\operatorname{\mathcal{D}}$ by its core $\operatorname{\mathcal{D}}^{\simeq }$, and thereby reduce to the case where $\operatorname{\mathcal{D}}$ is an $n$-groupoid (Example 4.8.1.17). In this case, the desired result follows from the universal property of Proposition 3.5.6.5.