Kerodon

Example 5.3.4.13. Let $X$ be a simplicial set, which we identify with a diagram $\mathscr {F}: [0] \rightarrow \operatorname{Set_{\Delta }}$. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ and the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}([0])$ can both be identified with $X$ (see Examples 5.3.2.2 and 5.3.3.2). Under these identifications, the taut scaffold $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}([0])$ of Construction 5.3.4.11 corresponds to the identity map $\operatorname{id}_{X}$.