Kerodon

Example 5.5.3.3 (Objects of $\operatorname{\mathcal{S}}_{\ast }$). By definition, an object of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ is an edge $e: \Delta ^{0} \rightarrow X$ of the simplicial set $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan})$ whose source is the Kan complex $\Delta ^{0}$. By virtue of Remark 5.5.1.3, this is the same data as a morphism $e: \Delta ^{0} \rightarrow X$ in the ordinary category of Kan complexes: that is, the data of a pointed Kan complex $(X,x)$ (Definition 3.2.1.5).