Kerodon

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Example 6.3.1.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets and let $W$ be a collection of degenerate edges of $\operatorname{\mathcal{C}}$. Then $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if and only if it is a categorical equivalence of simplicial sets (see Proposition 4.5.3.8).