Kerodon

Remark 5.5.6.5 (Comparison with Pointed Spaces). Let us regard the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ as a full subcategory of the $\infty $-category $\operatorname{\mathcal{QC}}$ (Remark 5.5.4.8). The inclusion $\operatorname{\mathcal{S}}\hookrightarrow \operatorname{\mathcal{QC}}$ determines a functor of coslice $\infty $-categories $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{QC}}_{\ast }$. This functor restricts to an isomorphism from $\operatorname{\mathcal{S}}_{\ast }$ with the full subcategory of $\operatorname{\mathcal{QC}}_{\ast }$ spanned by those pairs $(\operatorname{\mathcal{C}}, C)$, where $\operatorname{\mathcal{C}}$ is a Kan complex.