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  "body": "\n## Summary\n\nA supermanifold is a space whose functions depend on ordinary commuting coordinates and also on anticommuting coordinates.\nFormally, it is a topological space with a [[sheaf|TB6TM6R4MD]] of supercommutative [[superalgebras|6JZW4JC68Z]] that looks locally like smooth functions with values in an [[exterior algebra|0MDMZ965VE]].\nSupermanifolds are the geometric setting of supersymmetry, where bosons are described by even and fermions by odd coordinates.\nEvery smooth supermanifold is built, up to isomorphism, from an ordinary [[vector bundle|P4DZM975BV]] over an ordinary manifold.\n\n## Explanations\n\n### L1 human\n\nOn an ordinary manifold, functions are built from coordinates $x_1, \\dots, x_p$ that commute.\nA supermanifold adds coordinates $\\theta_1, \\dots, \\theta_q$ that anticommute: $\\theta_i \\theta_j = -\\theta_j \\theta_i$, so in particular $\\theta_i^2 = 0$.\nA function in one even and one odd variable is then always of the form $f(x) + g(x)\\, \\theta$, since higher powers of $\\theta$ vanish.\nThese odd coordinates have no points of their own; they live only in the algebra of functions.\n",
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