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  "body": "\n## Definition\n\nA *symmetric monoidal category* is a [[braided monoidal category|VT8SFWT0XW]] $(\\mathcal{C}, \\otimes, I, \\alpha, \\lambda, \\rho, \\gamma)$ such that for all objects $A, B$ of $\\mathcal{C}$\n$$\\gamma_{B,A} \\circ \\gamma_{A,B} = \\id_{A \\otimes B}.$$\n\n## Remarks\n\nUnder this condition the two hexagon identities of the braiding are equivalent, so one of them suffices.\nExample: $\\mathbf{Vect}_k$ with $\\gamma(v \\otimes w) = w \\otimes v$; applying the swap twice returns $v \\otimes w$.\nExample: $\\Z/2$-graded vector spaces with the Koszul sign $\\gamma(v \\otimes w) = (-1)^{\\abs{v}\\abs{w}}\\, w \\otimes v$ on homogeneous elements; the signs cancel when the swap is applied twice.\nNon-example: $\\Z/3$-graded complex vector spaces with $\\gamma(v \\otimes w) = q^{\\abs{v}\\abs{w}}\\, w \\otimes v$, where $q = e^{2\\pi i/3}$.\nThis is a braiding because $(a, b) \\mapsto q^{ab}$ is a bicharacter of $\\Z/3$, but applying it twice multiplies $v \\otimes w$ by $q^{2\\abs{v}\\abs{w}}$, which is not $1$ when $\\abs{v} = \\abs{w} = 1$.\nSymmetric monoidal categories form the special case of [[monoidal categories|M3MSQ3NKAE]] that model commutative operations.\n",
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