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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]].\nA *fusion category* over $k$ is a [[monoidal category|M3MSQ3NKAE]] $(\\mathcal{C}, \\otimes, I)$ such that:\n\n1. $\\mathcal{C}$ is $k$-linear: every $\\Hom(A, B)$ is a $k$-vector space, composition is $k$-bilinear, and $\\otimes$ is $k$-bilinear on morphisms;\n2. every $\\Hom(A, B)$ is finite-dimensional;\n3. $\\mathcal{C}$ is semisimple: it is an [[abelian category|Y9CZ3MTNF1]] in which every object is a finite direct sum of simple objects;\n4. $\\mathcal{C}$ has only finitely many isomorphism classes of simple objects;\n5. $\\mathcal{C}$ is [[rigid|A10WW8GEKR]];\n6. the unit object $I$ is simple.\n\nHere an object is simple if it is nonzero and has no subobjects other than $0$ and itself.\nMany authors assume that $k$ is algebraically closed.\n\n## Remarks\n\nExample: for a finite group $G$, the category $\\mathrm{Rep}(G)$ of finite-dimensional complex [[representations|WYGRXCG2RN]] of $G$ is a fusion category over $\\C$.\nIt is semisimple by Maschke's theorem, has finitely many irreducible representations up to isomorphism, the dual of a representation gives rigidity, and the trivial representation is simple.\nNon-example: the category of finite-dimensional complex representations of the group $\\Z$ satisfies clauses 1, 2, 5 and 6.\nIt is not semisimple, and it has infinitely many simple objects, one for each $\\lambda \\in \\C^\\times$.\nNon-example: for a field $k$ of characteristic $p$ and a group $G$ of order divisible by $p$, $\\mathrm{Rep}_k(G)$ fails clause 3 by the failure of Maschke's theorem.\nIf $X_1, \\ldots, X_n$ are the simple objects, the fusion rules are the integers $N_{ij}^m$ with $X_i \\otimes X_j \\cong \\bigoplus_m N_{ij}^m X_m$.\n",
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