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Construction 9.8.0.3 (The $\infty $-Category of Spaces). Let $\operatorname{Kan}$ denote the category of Kan complexes. We view $\operatorname{Kan}$ as a simplicial category, with simplicial morphism sets given by the constructoin

\[ \operatorname{Hom}_{\operatorname{Kan}}(X,Y)_{\bullet } = \operatorname{Fun}(X,Y). \]

We let $\operatorname{\mathcal{S}}$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ (Definition 2.4.3.5). We will refer to $\operatorname{\mathcal{S}}$ as the $\infty $-category of spaces.