Definition 4.5.6.1. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. We say that $\alpha $ is a levelwise categorical equivalence if, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a categorical equivalence of simplicial sets.
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