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  "body": "\n## Summary\n\nLocalizing a space at a set of primes $P$ is the topological analogue of [[localizing|4C1AE84N5N]] the integers at $P$.\nFor a simply connected space $X$ it produces a map $X \\to X_P$.\nThis map tensors every [[homotopy group|P976WYEFMB]] and every homology group with $\\Z_{(P)}$.\nThis lets one study a space one prime at a time.\nThe empty set of primes gives rationalization, the starting point of [[rational homotopy theory|8G9QPMBT7R]].\n\n## Explanations\n\n### L1 human\n\nHomotopy groups of spaces are often full of torsion at many primes, which makes them hard to compute all at once.\nLocalization keeps only the information at chosen primes and makes all other primes invertible.\nFor example, the rationalization of an odd-dimensional sphere $S^n$ is the Eilenberg–MacLane space $K(\\Q, n)$.\nRationally, an odd sphere has just one homotopy group, $\\Q$ in degree $n$, and all the torsion disappears.\n",
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