Kerodon

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

Remark 5.5.4.9 (Comparison with Categories). Let $\mathbf{Cat}$ denote the strict $2$-category of small categories (Example 2.2.0.4), let $\operatorname{Pith}( \mathbf{Cat} )$ denote its pith (Construction 2.3.2.10), and let us abuse notation by identifying $\operatorname{Pith}( \mathbf{Cat} )$ with the simplicial category described in Example 2.4.2.7. Concretely, this simplicial category can be described as follows:

  • The objects of $\operatorname{Pith}( \mathbf{Cat} )$ are small categories.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are objects of $\operatorname{Pith}( \mathbf{Cat} )$, then the simplicial set $\operatorname{Hom}_{\operatorname{Pith}( \mathbf{Cat} )}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }$ is the nerve of the groupoid $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ whose objects are functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and whose morphisms are natural isomorphisms.

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 $\operatorname{Pith}(\mathbf{Cat} ) \hookrightarrow \operatorname{QCat}$. Passing to homotopy coherent nerves (and invoking Example 2.4.3.11), we obtain a functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) ) \rightarrow \operatorname{\mathcal{QC}}$. Unwinding the definitions, we see that this functor induces an isomorphism from the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) )$ to the full subcategory of $\operatorname{\mathcal{QC}}$ spanned by those $\infty $-categories of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is an ordinary category.