Kerodon

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Example 4.6.5.13. Let $X$ be a topological space containing a pair of points $x$ and $y$, which we regard as objects of the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$. Using Example 4.3.5.9, we obtain canonical isomorphisms of Kan complexes

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{Sing}_{\bullet }(X)}(x,y) \simeq \operatorname{Sing}_{\bullet }( P_{x,y} ) \simeq \operatorname{Hom}_{\operatorname{Sing}_{\bullet }(X)}^{\mathrm{R}}(x,y), \]

where $P_{x,y}$ denotes the topological space of continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$ (equipped with the compact-open topology). Combining this observation with Example 4.6.1.4, we can identify the pinch inclusion maps $\iota ^{\mathrm{L}}_{x,y}$ and $\iota ^{\mathrm{R}}_{x,y}$ with monomorphisms from the simplicial set $\operatorname{Sing}_{\bullet }( P_{x,y} )$ to itself. Beware that these maps are not the identity (though one can show that they are homotopic to the identity).