Kerodon

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Example 3.2.2.8. Let $X$ be a Kan complex, let $x$ be a vertex of $X$, and let $e,e': x \rightarrow x$ be edges of $X$ which begin and end at the vertex $x$. Then the equality $[e] = [e']$ holds in the fundamental group $\pi _{1}(X,x)$ if and only if $e$ is homotopic to $e'$ as a morphism in the $\infty$-category $X$ (in the sense of Definition 1.3.3.1); see Corollary 1.3.3.7.