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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer and let $k$ be a field.\nAll manifolds below are smooth, compact and [[oriented|702SNFQG0W]].\nFor closed $n$-manifolds $\\Sigma_0$ and $\\Sigma_1$, a *cobordism* from $\\Sigma_0$ to $\\Sigma_1$ is an $(n+1)$-[[manifold with boundary|7DS6B2BFK8]] $M$ with an orientation-preserving [[diffeomorphism|0Z1Y7JHGBR]] $\\partial M \\to \\overline{\\Sigma_0} \\sqcup \\Sigma_1$.\nHere $\\overline{\\Sigma}$ is $\\Sigma$ with the opposite orientation.\nAn $(n+1)$-dimensional *topological quantum field theory* $Z$ over $k$ assigns two kinds of data.\nTo each closed $n$-manifold $\\Sigma$ it assigns a finite-dimensional [[vector space|05EGHEQZAW]] $Z(\\Sigma)$ over $k$.\nTo each cobordism $M$ from $\\Sigma_0$ to $\\Sigma_1$ it assigns a [[linear map|0RJF668P1T]] $Z(M) \\colon Z(\\Sigma_0) \\to Z(\\Sigma_1)$.\nThese data satisfy the following axioms.\n\n1. Each orientation-preserving diffeomorphism $f \\colon \\Sigma \\to \\Sigma'$ gives a linear isomorphism $Z(f) \\colon Z(\\Sigma) \\to Z(\\Sigma')$, with $Z(f' \\circ f) = Z(f') \\circ Z(f)$.\n   Diffeomorphic cobordisms, by a diffeomorphism compatible with the boundaries, give equal maps after these isomorphisms.\n2. Let $M$ be a cobordism from $\\Sigma_0$ to $\\Sigma_1$, and $M'$ one from $\\Sigma_1$ to $\\Sigma_2$.\n   Let $M' \\cup_{\\Sigma_1} M$ be the cobordism obtained by gluing them along $\\Sigma_1$.\n   Then $Z(M' \\cup_{\\Sigma_1} M) = Z(M') \\circ Z(M)$.\n3. For each $\\Sigma$, the cylinder $\\Sigma \\times [0, 1]$ gives the identity map of $Z(\\Sigma)$.\n4. The empty $n$-manifold gives $Z(\\emptyset) = k$, and disjoint unions give natural isomorphisms $Z(\\Sigma \\sqcup \\Sigma') \\cong Z(\\Sigma) \\otimes Z(\\Sigma')$ under which $Z(M \\sqcup M') = Z(M) \\otimes Z(M')$, where $\\otimes$ is the [[tensor product|0G31VJ5YS5]] over $k$.\n\n## Remarks\n\nA closed $(n+1)$-manifold $M$ is a cobordism from $\\emptyset$ to $\\emptyset$, so $Z(M)$ is a linear map $k \\to k$, that is, a number $Z(M) \\in k$.\nBy clause 1 this number is a diffeomorphism invariant of $M$.\nThe axioms imply that $Z(\\overline{\\Sigma})$ is the [[dual space|0M856Q1AMZ]] of $Z(\\Sigma)$.\nIn the language of categories, a topological quantum field theory is a symmetric monoidal functor from the category of $(n+1)$-dimensional cobordisms to the category of vector spaces over $k$.\nExample: let $n = 0$ and let $V$ be a finite-dimensional vector space.\nSet $Z(+) = V$ for the positive point and $Z(-) = V^*$, and extend to finite sets of points by tensor products.\nAn arc from $\\emptyset$ to $\\set{+, -}$ gives the element of $V \\otimes V^*$ that corresponds to $\\id_V$, an arc from $\\set{+, -}$ to $\\emptyset$ gives evaluation $V \\otimes V^* \\to k$, and straight arcs give identities.\nThese choices satisfy the axioms, and the circle, which is the composite of the two arcs, gets the number $Z(S^1) = \\dim V$.\nNon-example: set $Z(\\Sigma) = k$ for all $\\Sigma$ and let every cobordism act as multiplication by $2$.\nThen the cylinder $\\Sigma \\times [0, 1]$ gives multiplication by $2$, not the identity, so clause 3 fails.\nIn dimension $1 + 1$, topological quantum field theories correspond to commutative Frobenius algebras over $k$.\n",
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