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  "body": "\n## Summary\n\nA braided monoidal category is a [[monoidal category|M3MSQ3NKAE]] with a natural isomorphism $A \\otimes B \\to B \\otimes A$, the braiding, compatible with the associator.\nThe braiding need not square to the identity, and its behaviour is modelled by the [[braid groups|BNR4H5GCTP]] rather than the symmetric groups.\nBraided monoidal categories arise from representations of quantum groups and give invariants of knots and links.\nWhen the braiding squares to the identity the category is symmetric.\n\n## Explanations\n\n### L1 human\n\nIn a braided monoidal category two objects placed side by side can swap places.\nPicture each object as a string: swapping them makes one string cross over the other.\nCrossing over and crossing under are different moves, so swapping twice need not bring things back to the start.\nFor vector spaces the flip $v \\otimes w \\mapsto w \\otimes v$ is a braiding, and there swapping twice does give the identity.\n",
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