Kerodon

Example 1.4.5.5. Let $X$ be a topological space, and regard the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ as an $\infty $-category. Then the homotopy category $\mathrm{h} \mathit{\operatorname{Sing}}_{\bullet }(X)$ can be identified with the fundamental groupoid $\pi _{\leq 1}(X)$. More precisely, we can regard the contents of §1.4, when specialized to $\infty $-categories of the form $\operatorname{Sing}_{\bullet }(X)$, as providing a construction of the fundamental groupoid of $X$. By virtue of Exercise 1.4.3.4 and Example 1.4.4.5, the resulting category $\mathrm{h} \mathit{\operatorname{Sing}}_{\bullet }(X)$ matches the informal description of $\pi _{\leq 1}(X)$ given in the introduction to §1.4.5.