Kerodon

Example 5.3.2.4 (Constant Diagrams). Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be a simplicial set, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the constant diagram taking the value $X$. Combining Remark 5.3.2.3 with Example 5.3.2.2, we obtain a canonical isomorphism of simplicial sets $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times X$. In particular, if $X = \Delta ^0$, then the projection map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an isomorphism.