Kerodon

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Definition 7.5.8.2. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$, and let $W$ denote the collection of horizontal edges of $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ (Definition 5.3.4.1). We will say that $\overline{ \mathscr {F} }$ is a categorical colimit diagram if the composite map

$\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} ) \rightarrow \overline{\mathscr {F}}( {\bf 1} )$

exhibits $\overline{\mathscr {F}}( {\bf 1} )$ as a localization of $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ with respect to $W$. (see Definition 6.3.1.9). Here ${\bf 1}$ denotes the final object of the cone $\operatorname{\mathcal{C}}^{\triangleright } \simeq \operatorname{\mathcal{C}}\star \{ {\bf 1}\}$, and the morphism on the left is the comparison map of Remark 5.3.2.9.