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  "body": "\n## Summary\n\nA graded manifold is a smooth manifold whose sheaf of functions is enlarged by anticommuting odd coordinates.\nLocally its functions are smooth functions of $n$ ordinary variables with values in a Grassmann algebra on $m$ generators.\nIt is the algebraic form, due to Kostant and to Berezin and Leites, of a [[supermanifold|Z0SB9HQBAJ]].\nGraded manifolds give a geometric home to fermionic fields and to the odd variables of the BRST and BV formalisms.\n\n## Explanations\n\n### L1 human\n\nOn an ordinary manifold a function of two variables $x, y$ is a number at each point.\nOn a graded manifold one adds symbols $\\theta_1, \\dots, \\theta_m$ that anticommute, so $\\theta_i \\theta_j = -\\theta_j \\theta_i$ and $\\theta_i^2 = 0$.\nA function is then a finite sum such as $f(x) + g(x) \\theta_1 + h(x) \\theta_1 \\theta_2$ with smooth coefficients.\nThe odd symbols carry no points of their own: setting all of them to zero gives back the ordinary manifold.\nFor example, the sections of the exterior algebra bundle of the cotangent bundle, that is the differential forms on a manifold, form a graded manifold.\n",
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