Kerodon

Example 3.5.6.3. Let $X$ be a Kan complex and let $\pi _{\leq 1}(X)$ denote the fundamental groupoid of $X$ (Definition 1.4.6.12). Then the nerve $\operatorname{N}_{\bullet }( \pi _{\leq 1}(X) )$ is a $1$-groupoid (Proposition 3.5.5.8). By construction, the tautological map $u: X \rightarrow \operatorname{N}_{\bullet }( \pi _{\leq 1}(X) )$ is bijective on vertices and surjective on edges. Moreover, two edges of $X$ have the same image in $\operatorname{N}_{\bullet }( \pi _{\leq 1}(X) )$ if and only if they are homotopic relative to $\operatorname{\partial \Delta }^{1}$ (see Corollary 1.4.3.7). It follows that $u$ exhibits $\operatorname{N}_{\bullet }(\pi _{\leq 1}(X))$ as a fundamental $1$-groupoid of $X$.