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  "body": "\n## Definition\n\nLet $\\mathcal{M}$ and $\\mathcal{N}$ be [[model categories|SDTZVZ6B4Q]], and let $F \\colon \\mathcal{M} \\rightleftarrows \\mathcal{N} \\colon G$ be an [[adjunction|TZHF6ZM92S]] with $F$ left adjoint to $G$.\nThe adjunction is a *Quillen adjunction* if the following equivalent conditions hold.\n\n1. $F$ preserves [[cofibrations|HSV1J7BJXX]] and trivial cofibrations.\n2. $G$ preserves fibrations and trivial fibrations.\n3. $F$ preserves cofibrations and $G$ preserves fibrations.\n\nThen $F$ is called a *left Quillen functor* and $G$ a *right Quillen functor*.\n\n## Remarks\n\nFor an adjunction, $F(i)$ has the left lifting property against $p$ if and only if $i$ has the left lifting property against $G(p)$.\nSince the classes of a model structure are characterized by lifting properties, the three conditions are equivalent.\n\nA Quillen adjunction induces an adjunction $\\mathbf{L}F \\dashv \\mathbf{R}G$ between the [[homotopy categories|48QQ7T98Z9]].\nHere $\\mathbf{L}F$ is computed by applying $F$ to a cofibrant replacement, and $\\mathbf{R}G$ by applying $G$ to a fibrant replacement.\nThis works because $F$ sends [[weak equivalences|E2870TEDG7]] between cofibrant objects to weak equivalences, by Ken Brown's lemma.\n\nExample: [[geometric realization|TG6WHHR7CF]] $\\abs{-}$ from simplicial sets to topological spaces is left adjoint to the [[singular simplicial set|RWBMXQW6S5]] functor $\\mathrm{Sing}$.\nWith the Kan–Quillen and Quillen model structures, $\\mathrm{Sing}$ sends Serre fibrations to Kan fibrations and trivial ones to trivial ones, so the pair is a Quillen adjunction.\n\nNon-example: let $\\mathcal{M}$ and $\\mathcal{N}$ both be the category of sets.\nGive $\\mathcal{M}$ the trivial model structure: $W$ the bijections, and all maps fibrations and cofibrations.\nGive $\\mathcal{N}$ the model structure with $W$ all maps, cofibrations the bijections, and fibrations all maps.\nThe identity adjunction fails clause 1, since a non-bijective map is a cofibration of $\\mathcal{M}$ but not of $\\mathcal{N}$.\n",
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