Kerodon

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Remark 5.3.2.9 (Comparison with the Colimit). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\{ t_{C}: \mathscr {F}(C) \rightarrow X \} _{C \in \operatorname{\mathcal{C}}}$ be a collection of morphisms which exhibit $X$ as a colimit of the diagram $\mathscr {F}$. The morphisms $t_{C}$ then determine a natural transformation $t_{\bullet }: \mathscr {F} \rightarrow \underline{X}$, where $\underline{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denotes the constant functor taking the value $X$. Using Example 5.3.2.4, we obtain a morphism of simplicial sets

$\theta : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow { \underset { \longrightarrow }{\mathrm{holim}}(t_{\bullet }) } \underset { \longrightarrow }{\mathrm{holim}}( \underline{X} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times X \rightarrow X,$

which we will refer to as the comparison map. Note that, for every vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $\theta$ to the fiber $\{ C\} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ can be identified with the morphism $t_{C}$. Since $X$ is the union of the images of the morphisms $t_{C}$, it follows that the comparison map $\theta : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \twoheadrightarrow \varinjlim (\mathscr {F})$ is an epimorphism of simplicial sets.