Kerodon

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Example 5.5.3.4. Let $\operatorname{Cat}$ denote the category whose objects are (small) categories and whose morphisms are functors, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$ be a functor of ordinary categories. Composing with the nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$, we obtain a functor $\operatorname{N}_{\bullet }(\mathscr {F}): \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then there is a canonical isomorphism of simplicial sets $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\operatorname{N}_{\bullet }(\mathscr {F}) \simeq \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$, where the left hand side is the weighted nerve of Definition 5.5.3.1 and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is the category of elements introduced in Definition 5.5.2.1.