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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, \\otimes, I, \\alpha, \\lambda, \\rho)$ be a [[monoidal category|M3MSQ3NKAE]].\nA *braiding* on $\\mathcal{C}$ is a natural isomorphism $\\gamma$ with components $\\gamma_{A,B} \\colon A \\otimes B \\to B \\otimes A$, natural in $A$ and $B$.\nIt must satisfy the two hexagon identities for all objects $A, B, C$:\n\n1. $\\alpha_{B,C,A} \\circ \\gamma_{A, B \\otimes C} \\circ \\alpha_{A,B,C} = (\\id_B \\otimes \\gamma_{A,C}) \\circ \\alpha_{B,A,C} \\circ (\\gamma_{A,B} \\otimes \\id_C)$ as morphisms $(A \\otimes B) \\otimes C \\to B \\otimes (C \\otimes A)$;\n2. $\\alpha^{-1}_{C,A,B} \\circ \\gamma_{A \\otimes B, C} \\circ \\alpha^{-1}_{A,B,C} = (\\gamma_{A,C} \\otimes \\id_B) \\circ \\alpha^{-1}_{A,C,B} \\circ (\\id_A \\otimes \\gamma_{B,C})$ as morphisms $A \\otimes (B \\otimes C) \\to (C \\otimes A) \\otimes B$.\n\nA *braided monoidal category* is a monoidal category with a braiding.\n\n## Remarks\n\nExample: $\\mathbf{Vect}_k$ with the usual tensor product and $\\gamma_{V,W}(v \\otimes w) = w \\otimes v$.\nOn $(u \\otimes v) \\otimes w$ both sides of the first hexagon give $v \\otimes (w \\otimes u)$, and both sides of the second hexagon send $u \\otimes (v \\otimes w)$ to $(w \\otimes u) \\otimes v$.\nNon-example: on $\\mathbf{Vect}_k$ with $\\operatorname{char} k \\ne 2$, the natural isomorphism $\\gamma_{V,W}(v \\otimes w) = -\\, w \\otimes v$ is not a braiding.\nIn each hexagon the left side contains one copy of $\\gamma$ and the right side two, so for nonzero spaces the two sides differ by the sign $-1$.\nThe first hexagon says that moving $A$ past $B \\otimes C$ in one step equals moving it past $B$ and then past $C$.\nThe second says the same for moving $C$ past $A \\otimes B$.\n",
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