Kerodon

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

Remark 5.5.4.5. Let $\operatorname{\mathcal{C}}$ be an ordinary category equipped with a strictly unitary functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$. Then the construction $C \mapsto \operatorname{N}_{\bullet }(\mathscr {F}(C) )$ determines a functor of $\infty $-categories $\operatorname{N}_{\bullet }(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ (see Remark 5.4.4.9). In ยง5.5.5, we will construct a canonical isomorphism

\[ \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }(\mathscr {F} ) \simeq \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ), \]

where the simplicial set on the left hand side is given by Definition 5.5.4.1 and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is the category of elements introduced in Definition 5.5.2.1 (see Proposition 5.5.5.3). Stated more informally, we can regard the $\infty $-category of elements construction (Definition 5.5.4.4) as a generalization of the classical category of elements construction (Definition 5.5.2.1).