Construction 11.8.0.4 (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.