Kerodon

Proof of Proposition 9.1.7.1 from Proposition 9.1.7.9. Let $\operatorname{\mathcal{C}}$ be a small filtered category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a colimit diagram in the category of simplicial sets; we wish to show that $\overline{\mathscr {F}}$ is a categorical colimit diagram. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$. Using Proposition 9.1.7.9, we can choose a levelwise categorical equivalence $\alpha : \mathscr {E} \rightarrow \mathscr {F}$, where $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant. Let $\overline{\mathscr {E}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a colimit diagram extending $\mathscr {E}$, so that $\alpha $ extends uniquely to a natural transformation $\overline{\alpha }: \overline{\mathscr {E}} \rightarrow \overline{\mathscr {F}}$. Applying Corollary 4.5.7.2, we deduce that $\overline{\alpha }$ is also a levelwise categorical equivalence. Consequently, to show that $\overline{\mathscr {F}}$ is a categorical colimit diagram, it will suffice to show that $\overline{\mathscr {E}}$ is a categorical colimit diagram (Corollary 7.5.8.6). This follows from Corollary 7.5.8.7, since $\mathscr {E}$ is projectively cofibrant. $\square$