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  "body": "\n## Definition\n\nConsider simply connected [[CW complexes|5W95QFHXPR]].\nA map $f \\colon X \\to Y$ between them is a *rational homotopy equivalence* if the maps $\\pi_n(f) \\otimes \\Q \\colon \\pi_n(X) \\otimes \\Q \\to \\pi_n(Y) \\otimes \\Q$ on [[homotopy groups|P976WYEFMB]] are isomorphisms for all $n \\ge 2$.\nTwo such spaces have the same *rational homotopy type* if they are joined by a finite chain of rational homotopy equivalences, in either direction.\nEquivalently, their [[rationalizations|D0ZHVD797D]] $X_\\Q$ and $Y_\\Q$ are [[homotopy equivalent|XZG6MTSWCN]].\n*Rational homotopy theory* is the study of simply connected spaces up to rational homotopy type.\nIn other words, it studies the homotopy category of simply connected spaces with the rational homotopy equivalences inverted.\n\n## Remarks\n\nThe condition on $f$ is equivalent to: $H_n(f; \\Q)$ is an isomorphism in [[singular homology|M0Q5CFJ7ZH]] for all $n$.\n\nExample: the rationalization $S^3 \\to K(\\Q, 3)$ is a rational homotopy equivalence, since both sides have rational homotopy $\\Q$ in degree $3$ and $0$ in other degrees.\nSo $S^3$ and $K(\\Q, 3)$ have the same rational homotopy type.\n\nNon-example: the map $S^2 \\to K(\\Q, 2)$ that represents a generator of $H^2(S^2; \\Q)$ is an isomorphism on $\\pi_2 \\otimes \\Q$.\nIt is not a rational homotopy equivalence: $\\pi_3(S^2) \\otimes \\Q \\cong \\Q$, generated by the Hopf map, while $\\pi_3 K(\\Q, 2) = 0$.\n\nSullivan showed that this homotopy category, with finiteness conditions, is equivalent to a homotopy category of commutative differential graded algebras over $\\Q$.\nQuillen showed that it is also equivalent to a homotopy category of differential graded Lie algebras over $\\Q$.\n",
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