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  "body": "\n## Summary\n\nA symmetric monoidal category is a [[braided monoidal category|VT8SFWT0XW]] whose braiding, applied twice, gives the identity.\nIt is the categorical form of a commutative operation: $A \\otimes B$ and $B \\otimes A$ are identified by a swap that undoes itself.\nSets with the Cartesian product, vector spaces with the [[tensor product|0BF2DE2SYK]], and modules over a commutative ring are symmetric monoidal.\nSymmetric monoidal categories are the setting of commutative monoid objects, of topological quantum field theories and of linear logic.\n\n## Explanations\n\n### L1 human\n\nIn a symmetric monoidal category, swapping two objects placed side by side and then swapping them back changes nothing.\nSo only the order of the objects matters, not how they were moved past each other.\nFor finite sets with the product, the swap $(a, b) \\mapsto (b, a)$ is the symmetry, and doing it twice gives back $(a, b)$.\nThis is the familiar commutativity, made into a structure rather than an equation.\n",
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