Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 3.1.4.12 (The $\infty $-Category of Spaces). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, which we view as a simplicial category (where the simplicial set of morphisms from $X_{}$ to $Y_{}$ is given by $\operatorname{Fun}( X_{}, Y_{} )$). Let $\operatorname{Set}_{\Delta }^{\circ }$ denote the full subcategory of $\operatorname{Set}_{\Delta }$ spanned by the Kan complexes, so that $\operatorname{Set}_{\Delta }^{\circ }$ inherits the structure of a simplicial category. We let $\operatorname{\mathcal{S}}$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set}_{\Delta }^{\circ } )$. Corollary 3.1.3.4 implies that the simplicial category $\operatorname{Set}_{\Delta }^{\circ }$ is locally Kan, so the simplicial set $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set}_{\Delta }^{\circ } )$ is an $\infty $-category (Theorem 2.4.5.1). We will refer to $\operatorname{\mathcal{S}}$ as the $\infty $-category of spaces.