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

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

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). Setting $x = y$, we obtain an isomorphism $\operatorname{End}_{\operatorname{Sing}_{\bullet }(X) } = \operatorname{Sing}_{\bullet }( \Omega (X) )$, where $\Omega (X)$ is the based loop space of $X$. See Example