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  "body": "\n## Summary\n\nA Quillen adjunction is a pair of [[adjoint functors|TZHF6ZM92S]] between [[model categories|SDTZVZ6B4Q]] that respects the model structures.\nThe left adjoint preserves cofibrations and trivial cofibrations, or equivalently the right adjoint preserves fibrations and trivial fibrations.\nSuch a pair induces an adjunction between the homotopy categories, through the derived functors.\nWhen this derived adjunction is an equivalence, the pair is a [[Quillen equivalence|KZCXHTEYK7]], and the two model categories present the same homotopy theory.\n\n## Explanations\n\n### L1 human\n\nAn adjunction is a precise way in which two constructions between categories mirror each other.\nIn homotopy theory one also wants the constructions to respect weak equivalences, and most do not do so on the nose.\nA Quillen adjunction asks for less: the functors must respect the good maps of the model structures.\nThat is enough to give well-defined functors after one inverts the weak equivalences.\nGeometric realization of simplicial sets and the singular simplicial set functor form the basic example.\n",
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