Kerodon

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Exercise 1.2.5.2. Let $X$ be a topological space and let $\pi _{\leq 1}(X)$ denote its fundamental groupoid. Show that there is a unique map of simplicial sets $u: \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }( \pi _{\leq 1}(X) )$ with the following properties:

  • On $0$-simplices, $u$ carries each point $x \in X$ (regarded as a vertex of $\operatorname{Sing}_{\bullet }(X)$) to itself (regarded as an object of $\pi _{\leq 1}(X)$).

  • On $1$-simplices, $u$ carries each path $p: [0,1] \rightarrow X$ (regarded as an edge of $\operatorname{Sing}_{\bullet }(X)$) to its homotopy class $[p]$ (regarded as a morphism of the category $\pi _{\leq 1}(X)$).

Moreover, show that $u$ exhibits the fundamental groupoid $\pi _{\leq 1}(X)$ as a homotopy category of the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$. For a generalization, see Proposition 1.3.5.7.