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The usual notion of equivalence for model categories is due to Quillen: given model categories $\mathcal{M}$ and $\mathcal{N}$, an adjunction $$F \dashv G : \mathcal{N} \to \mathcal{M}$$ is a Quillen equivalence if it is a Quillen adjunction (i.e. $F$ preserves cofibrations and trivial cofibrations, and $G$ preserves fibrations and trivial fibrations) such ...



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