Kerodon

Construction 11.4.0.25 (The $\infty $-Category of Pointed Spaces). Let $(\operatorname{Set_{\Delta }})_{\ast }$ denote the category whose objects are pointed simplicial sets $(X,x)$ and pointed morphisms between them. We regard $(\operatorname{Set_{\Delta }})_{\ast }$ as a simplicial category, where the simplicial set of morphisms from $(X,x)$ to $(Y,y)$ is given by the fiber product $\operatorname{Fun}(X,Y) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $. Let $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{\ast }$ spanned by the pointed Kan complexes, so that $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ inherits the structure of a simplicial category. We let $\operatorname{\mathcal{S}}_{\ast }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$. Corollary 3.1.3.3 (and Remark 3.1.1.9) guarantee that the simplicial category $(\operatorname{Set}_{\Delta }^{\circ })_{\ast }$ is locally Kan, so the simplicial set $\operatorname{\mathcal{S}}_{\ast } = \operatorname{N}_{\bullet }^{\operatorname{hc}}( (\operatorname{Set}_{\Delta }^{\circ })_{\ast } )$ is an $\infty $-category (Theorem 2.4.5.1). We will refer to $\operatorname{\mathcal{S}}_{\ast }$ as the $\infty $-category of pointed spaces.