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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be model categories, and let $F \\colon \\mathcal{C} \\rightleftarrows \\mathcal{D} \\colon G$ be a Quillen adjunction, with $F$ left adjoint to $G$.\nFor $X$ in $\\mathcal{C}$ and $Y$ in $\\mathcal{D}$, write $g^\\flat \\colon X \\to GY$ for the adjoint of a map $g \\colon FX \\to Y$.\nThe adjunction is a *Quillen equivalence* if the following holds for every cofibrant object $X$ of $\\mathcal{C}$ and every fibrant object $Y$ of $\\mathcal{D}$.\nA map $g \\colon FX \\to Y$ is a weak equivalence in $\\mathcal{D}$ if and only if $g^\\flat$ is a weak equivalence in $\\mathcal{C}$.\n\n## Remarks\n\nEquivalently, the derived adjunction $\\mathbf{L}F \\colon \\operatorname{Ho}(\\mathcal{C}) \\rightleftarrows \\operatorname{Ho}(\\mathcal{D}) \\colon \\mathbf{R}G$ is an equivalence of categories.\n\nExample: geometric realization $\\abs{\\cdot}$ from simplicial sets to topological spaces is left adjoint to the singular complex functor $\\operatorname{Sing}$.\nIt is a Quillen equivalence between the Kan–Quillen model structure and the Quillen model structure.\nEvery simplicial set is cofibrant and every space is fibrant.\nThe check reduces to two facts: the unit $K \\to \\operatorname{Sing}\\abs{K}$ is a weak equivalence, and the counit $\\abs{\\operatorname{Sing} Y} \\to Y$ is a weak homotopy equivalence.\n\nNon-example: give the category of sets the model structure whose weak equivalences are the bijections and in which every map is a cofibration and a fibration.\nThe functor $\\pi_0$ from simplicial sets to sets is left adjoint to the functor that sends a set to the discrete simplicial set on it.\nThis is a Quillen adjunction.\nIt is not a Quillen equivalence.\nTake the simplicial circle $K = \\Delta^1 / \\partial \\Delta^1$; the identity map of the one-point set $\\pi_0 K$ is a bijection.\nIts adjoint $K \\to \\Delta^0$ is not a weak equivalence, since $\\abs{K}$ is a circle.\n",
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