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  "body": "\n## Definition\n\nLet $P$ be a set of [[prime numbers|01826KMG2H]].\nLet $\\Z_{(P)}$ be the [[localization|KET2GSPZFG]] of $\\Z$ at $P$: the rational numbers whose denominators are divisible by no prime in $P$.\nLet $X$ be a simply connected [[CW complex|5W95QFHXPR]].\nThe space $X$ is *$P$-local* if each [[homotopy group|P976WYEFMB]] $\\pi_n(X)$, $n \\ge 2$, is a $\\Z_{(P)}$-module, that is, multiplication by every prime $q \\notin P$ is bijective on it.\nA *$P$-localization* of $X$ is a map $f \\colon X \\to X_P$ to a simply connected $P$-local space such that $f$ induces isomorphisms\n$$\\pi_n(X) \\otimes \\Z_{(P)} \\to \\pi_n(X_P) \\quad \\text{for all } n \\ge 2.$$\nFor $P = \\emptyset$ one has $\\Z_{(P)} = \\Q$, and $f$ is called a *rationalization*, written $X \\to X_\\Q$.\n\n## Remarks\n\nEvery simply connected CW complex has a $P$-localization, unique up to homotopy equivalence under $X$.\nThe condition on homotopy groups is equivalent to: $f$ induces isomorphisms $\\tilde H_n(X; \\Z) \\otimes \\Z_{(P)} \\to \\tilde H_n(X_P; \\Z)$ in reduced [[singular homology|M0Q5CFJ7ZH]] for all $n$.\nThe localization is universal: every map from $X$ to a $P$-local space factors through $f$, uniquely up to homotopy.\nThe theory extends to nilpotent spaces.\n\nExample: for $n$ odd, the map $S^n \\to K(\\Q, n)$ that represents a generator of $H^n(S^n; \\Q)$ is a rationalization.\nIndeed, $\\pi_k(S^n) \\otimes \\Q$ is $\\Q$ for $k = n$ and $0$ otherwise, by [[Serre's finiteness theorem|Y40ZRJJNH0]], which matches the homotopy of $K(\\Q, n)$.\n\nNon-example: the identity of $S^2$ is not a rationalization, because $S^2$ is not $P$-local for $P = \\emptyset$: $\\pi_2(S^2) \\cong \\Z$ is not a $\\Q$-vector space.\n",
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