Example 5.5.3.12 (0204): Pointed Sets as Pointed Spaces

Example 5.5.3.12 (Pointed Sets as Pointed Spaces). Let $\operatorname{Set}_{\ast }$ denote the category of pointed sets (see Example 4.2.3.3). Every pointed set $(X,x)$ can be regarded as a pointed Kan complex by identifying $X$ with the corresponding constant simplicial set. This construction determines a fully faithful embedding $\operatorname{Set}_{\ast } \hookrightarrow \operatorname{Kan}_{\ast }$. Composing with the equivalence of Proposition 5.5.3.8, we obtain a functor of $\infty $-categories

\[ \operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Kan}_{\ast } ) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \hookrightarrow \operatorname{\mathcal{S}}_{\ast }. \]

It follows from Remark 5.5.1.7 that this functor is fully faithful: in fact, it is an isomorphism from $\operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } )$ to the full subcategory of $\operatorname{\mathcal{S}}_{\ast }$ spanned by those pointed Kan complexes $(X,x)$ where the simplicial set $X$ is constant.