Kerodon

Example 5.3.4.21 (Set-Valued Functors). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a diagram of sets, and let us abuse notation by identifying $\mathscr {F}$ with a diagram of discrete simplicial sets. Then the taut scaffold $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is an isomorphism. It follows that $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ can be identified with the nerve of the category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ (see Example 5.3.2.5).