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  "body": "\n## Statement\n\nLet $G$ be a compact Hausdorff topological [[group|G5S1M9]].\nLet $\\mathrm{Rep}(G)$ be the category of finite-dimensional continuous complex [[representations|WYGRXCG2RN]] of $G$, and let $\\omega \\colon \\mathrm{Rep}(G) \\to \\mathbf{Vect}_\\C$ be the forgetful functor.\nLet $T(G)$ be the group of families $u = (u_V)_{V \\in \\mathrm{Rep}(G)}$ of linear automorphisms $u_V \\colon \\omega(V) \\to \\omega(V)$ such that:\n\n1. $u$ is a [[natural transformation|KYCPCC57FT]] $\\omega \\to \\omega$;\n2. $u_{V \\otimes W} = u_V \\otimes u_W$ for all $V, W$, and $u$ is the identity on the trivial representation $\\C$;\n3. $u_{\\overline{V}} = \\overline{u_V}$ for every $V$, where $\\overline{V}$ is the complex conjugate representation.\n\nThen the map $G \\to T(G)$ that sends $g$ to the family of operators $(\\pi_V(g))_V$ is an isomorphism of groups.\n\n## Remarks\n\nThis is Tannaka's theorem: a compact group is determined by its category of representations together with the forgetful functor.\nProof idea: by the Peter–Weyl theorem, the matrix coefficients of the representations span a dense subalgebra $\\mathcal{R}(G)$ of the algebra $C(G)$ of continuous functions.\nThese functions also separate the points of $G$.\nAn element of $T(G)$ defines an algebra homomorphism $\\mathcal{R}(G) \\to \\C$ that commutes with complex conjugation.\nOne shows that every such homomorphism is evaluation at a unique point of $G$.\n\nWithout clause 3 the group is larger.\nFor $G = U(1)$, the representations are the $\\Z$-graded spaces, and families satisfying clauses 1 and 2 are given by $t \\in \\C^\\times$ acting by $t^n$ in degree $n$; clause 3 forces $\\abs{t} = 1$.\nFor a compact Lie group, clauses 1 and 2 alone give its complexification.\nKrein's part of the duality describes which categories with such a functor arise from compact groups.\nThe algebraic version concerns a neutral Tannakian category over a field $k$.\nIt says that such a category is equivalent, as a rigid symmetric monoidal category, to the category of finite-dimensional representations of an affine group scheme over $k$.\n\n## Explanations\n\n### L1 human\n\nA compact group can be rebuilt from the way it acts on vector spaces.\nOne records all its finite-dimensional representations, all the maps between them, and how tensor products of representations behave.\nThe group is then recovered as the set of consistent ways to act on all these spaces at once.\nThis generalizes Pontryagin duality, which rebuilds a compact abelian group from its characters.\n\n### L2 human\n\nFor a compact abelian group, every irreducible representation is one-dimensional, and the theorem reduces to recovering $G$ from its character group, as in Pontryagin duality.\nFor non-abelian groups the representations do not form a group, but they do form a [[rigid|A10WW8GEKR]] [[symmetric monoidal category|5S7ER95XAC]], and the forgetful functor is a strong [[monoidal functor|PY430TXMG0]].\nThe group appears as the automorphisms of this functor that respect the tensor product.\nThis idea is central to the theory of Tannakian categories, motivic Galois groups, and the reconstruction of quantum groups from their representation categories.\n",
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