Kerodon

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

Remark 5.5.1.7 (Comparison with Sets). For every set $S$, let $\underline{S}$ denote the associated constant simplicial set (Construction 1.1.4.2). The construction $S \mapsto \underline{S}$ determines a fully faithful embedding from the category of sets to the category of Kan complexes. Passing to homotopy coherent nerve, we obtain a functor of $\infty $-categories $\operatorname{N}_{\bullet }( \operatorname{Set}) \rightarrow \operatorname{\mathcal{S}}$. This functor is fully faithful: in fact, it is an isomorphism from $\operatorname{N}_{\bullet }(\operatorname{Set})$ to the full subcategory of $\operatorname{\mathcal{S}}$ spanned by Kan complexes of the form $\underline{S}$. We will generally abuse notation by identifying (the nerve of) the category $\operatorname{Set}$ with its image in $\operatorname{\mathcal{S}}$: in particular, we will not distinguish between a set $S$ and the associated constant simplicial set $\underline{S}$, viewed as an object of $\operatorname{\mathcal{S}}$. We can summarize the situation informally by saying that the $\infty $-category $\operatorname{\mathcal{S}}$ is an enlargement of the ordinary category $\operatorname{Set}$.