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  "body": "\n## Summary\n\nA universal bundle for a topological group $G$ is a [[principal $G$-bundle|8E37E74XVP]] $EG \\to BG$ with a classifying property.\nEvery principal $G$-bundle over a CW complex is its pullback along a map, unique up to homotopy.\nIts base $BG$ is the [[classifying space|WSZ40H5E4T]] of $G$.\nA principal bundle with contractible total space has this property under mild conditions, which gives a practical way to recognize universal bundles.\nUniversal bundles turn the classification of bundles into the computation of homotopy classes of maps.\n\n## Explanations\n\n### L1 human\n\nA universal bundle is a master bundle that contains every other bundle with the same group.\nTo get a bundle over a space $X$, one maps $X$ into the base of the master bundle and pulls the master bundle back.\nHomotopic maps give the same bundle, so bundles over $X$ correspond to maps up to deformation.\nFor the group $\\Z/2\\Z$, the double cover $S^\\infty \\to \\R P^\\infty$ is a universal bundle, and it classifies the double covers of a CW complex.\n",
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