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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\mathcal{C}$ be a [[symmetric monoidal|5S7ER95XAC]] $(\\infty, n)$-category.\nLet $\\mathrm{Bord}_n^{\\mathrm{fr}}$ be the symmetric monoidal $(\\infty, n)$-category of $n$-framed [[bordisms|T8BVGPPSZP]].\nIts objects are $n$-framed $0$-manifolds.\nFor $1 \\le k \\le n$, its $k$-morphisms are $n$-framed $k$-dimensional bordisms between $(k-1)$-dimensional bordisms.\nIts higher morphisms are diffeomorphisms and their isotopies.\nThe monoidal product is disjoint union.\nLet $\\mathrm{Fun}^{\\otimes}(\\mathrm{Bord}_n^{\\mathrm{fr}}, \\mathcal{C})$ be the $\\infty$-groupoid of symmetric monoidal functors from $\\mathrm{Bord}_n^{\\mathrm{fr}}$ to $\\mathcal{C}$.\nLet $(\\mathcal{C}^{\\mathrm{fd}})^{\\simeq}$ be the $\\infty$-groupoid of fully [[dualizable objects|8A7YXMW727]] of $\\mathcal{C}$.\nThen evaluation at the point with its standard $n$-framing is an equivalence\n$$\\mathrm{Fun}^{\\otimes}\\left(\\mathrm{Bord}_n^{\\mathrm{fr}}, \\mathcal{C}\\right) \\simeq \\left(\\mathcal{C}^{\\mathrm{fd}}\\right)^{\\simeq}.$$\n\n## Remarks\n\nA symmetric monoidal functor $Z \\colon \\mathrm{Bord}_n^{\\mathrm{fr}} \\to \\mathcal{C}$ is a fully extended framed [[topological quantum field theory|DAP92B80J2]].\nThe theorem says that such a theory is determined by its value $Z(\\mathrm{pt})$ on a point, and that every fully dualizable object occurs.\nThe point is fully dualizable in the bordism category, because bending a bordism gives the duality data; this is why $Z(\\mathrm{pt})$ must be fully dualizable.\nThe hypothesis cannot be dropped: an object of $\\mathcal{C}$ that is not fully dualizable is not the value on a point of any such theory.\nBaez and Dolan proposed the hypothesis in 1995 in the language of $n$-categories.\nLurie published a detailed outline of a proof in 2009 in the language of $(\\infty, n)$-categories.\nFor $n = 1$ and $\\mathcal{C}$ the category of vector spaces over a field, the fully dualizable objects are the finite-dimensional spaces $V$.\nThe theory with $Z(\\mathrm{pt}) = V$ assigns to a circle the number $\\dim V$, the trace of the identity of $V$.\nThere are versions for other tangential structures, such as orientations, in which framings are replaced by the structure and the object carries a homotopy fixed point structure.\n\n## Explanations\n\n### L1 human\n\nA topological field theory assigns algebraic data to manifolds and to the ways of gluing them.\nThe cobordism hypothesis says that, in the most local version, all the data are decided by what the theory assigns to a single point.\nEverything else is built by gluing points into lines, lines into surfaces, and so on.\nThe only condition on the data at the point is that it be finite in a strong algebraic sense, called full dualizability.\n\n### L2 human\n\nClassical two-dimensional field theories are classified by commutative Frobenius algebras, which uses the decomposition of surfaces into pairs of pants.\nThe cobordism hypothesis extends such classifications to all dimensions by requiring the theory to be defined down to points, using higher categories.\nIt ties [[higher category theory|3A1NMGJKJ3]] to manifold topology: the framed bordism $(\\infty, n)$-category is the free symmetric monoidal $(\\infty, n)$-category with duals on one object.\nIts proof outline relies on Morse theory to decompose bordisms into elementary handles.\n",
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