Kerodon

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

Remark 5.5.1.8 (Comparison with Groupoids). Let $\mathbf{Cat}$ denote the (strict) $2$-category of small categories, let $\mathbf{Gpd} \subseteq \mathbf{Cat}$ denote the full subcategory spanned by the groupoids, and let $\mathbf{Gpd}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.7), which we can describe concretely as follows:

  • The objects of the simplicial category $\mathbf{Gpd}_{\bullet }$ are small groupoids.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are groupoids, then the simplicial set $\operatorname{Hom}_{ \mathbf{Gpd} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }$ is the nerve of the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

Note that if $\operatorname{\mathcal{C}}$ is a groupoid, then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex (Proposition 1.2.4.2). By virtue of Proposition 1.4.3.3, the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful embedding of simplicial categories $\mathbf{Gpd}_{\bullet } \hookrightarrow \operatorname{Kan}$. Passing to homotopy coherent nerves and invoking Example 2.4.3.11, we obtain a functor of $\infty $-categories

\[ \operatorname{N}^{\operatorname{D}}_{\bullet }( \mathbf{Gpd} ) \simeq \operatorname{N}^{\operatorname{hc}}_{\bullet }( \mathbf{Gpd}_{\bullet } ) \hookrightarrow \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{Kan}) = \operatorname{\mathcal{S}}, \]

where $\operatorname{N}^{\operatorname{D}}_{\bullet }( \mathbf{Gpd} )$ is the Duskin nerve of the $2$-category $\mathbf{Gpd}$ (Construction 2.3.1.1). This functor restricts to an isomorphism of $\operatorname{N}^{\operatorname{D}}_{\bullet }( \mathbf{Gpd} )$ with the full subcategory of $\operatorname{\mathcal{S}}$ spanned by those Kan complexes of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is a small groupoid. We can informally summarize the situation informally by saying that the $\infty $-category $\operatorname{\mathcal{S}}$ is an enlargement of the $2$-category of groupoids $\mathbf{Gpd}$.