Kerodon

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Example 5.5.4.8 ($\operatorname{Set}$-Valued Functors). Let $\operatorname{Set}$ denote the category of sets, and let us regard the nerve $\operatorname{N}_{\bullet }(\operatorname{Set})$ as a simplicial subset of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ be a morphism of simplicial sets, which we can identify with a functor of categories $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$. Using Example 5.4.3.12 and Remark 5.5.1.6, we obtain a canonical isomorphism of simplicial sets

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

where $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is the simplicial set of Definition 5.5.4.1 and $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathrm{h} \mathit{\mathscr {F}}$ is the category of elements introduced in Construction 5.5.1.1. In particular, the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left covering map.