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  "body": "\n## Summary\n\nA principal bundle is a fiber bundle whose fibers are copies of a group $G$.\nThe group $G$ acts on the total space and moves points within each fiber.\nEach fiber is like the group itself, but without a chosen identity element.\nFrame bundles of manifolds and the Hopf fibration are standard examples.\nConnections on principal bundles are the language of gauge theory, and every vector bundle can be built from a principal bundle.\n\n## Explanations\n\n### L1 human\n\nOver each point of a manifold, consider all possible coordinate frames for the tangent space.\nNo frame is preferred, but any two differ by exactly one invertible matrix.\nSo the frames over one point look like the group of invertible matrices, with no special element.\nCollecting these sets over all points gives a principal bundle, the frame bundle.\n",
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