Example 3.2.2.13. Let $\mathcal{G}$ be a groupoid and let $x$ be an object of $\mathcal{G}$, which we identify with a vertex of the Kan complex $X = \operatorname{N}_{\bullet }( \mathcal{G} )$ (see Proposition 1.3.5.2). Then:
The set $\pi _{0}(X) = \pi _{0}(X,x)$ can be identified with the collection of isomorphism classes of objects of $\mathcal{G}$.
The fundamental group $\pi _{1}(X,x)$ can be identified with the automorphism group $\operatorname{Aut}_{ \mathcal{G}}(x)$ of $x$ as an object of $\mathcal{G}$.
The homotopy groups $\pi _{n}(X,x)$ are trivial for $n \geq 2$, since an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$ is determined by the restriction $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ (see Exercise 1.3.1.5).