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  "body": "\n## Definition\n\nA *$C^\\infty$-ring* is a set $A$ together with, for each $n \\ge 0$ and each [[smooth function|0V33BNKZ9S]] $f \\colon \\R^n \\to \\R$, a map $\\Phi_f \\colon A^n \\to A$, such that both conditions hold.\n\n1. For each $n$ and each coordinate projection $\\pi_i \\colon \\R^n \\to \\R$, $(x_1, \\dots, x_n) \\mapsto x_i$, one has $\\Phi_{\\pi_i}(a_1, \\dots, a_n) = a_i$.\n2. For smooth $f \\colon \\R^n \\to \\R$ and $g_1, \\dots, g_n \\colon \\R^m \\to \\R$, and $h = f \\circ (g_1, \\dots, g_n)$, one has\n$$\\Phi_h(a) = \\Phi_f\\big(\\Phi_{g_1}(a), \\dots, \\Phi_{g_n}(a)\\big) \\quad \\text{for all } a \\in A^m.$$\n\nA morphism of $C^\\infty$-rings is a map $\\phi \\colon A \\to B$ with $\\phi(\\Phi_f(a_1, \\dots, a_n)) = \\Phi_f(\\phi(a_1), \\dots, \\phi(a_n))$ for all $f$ and $a_i$.\n\n## Remarks\n\nEquivalently, a $C^\\infty$-ring is a model of the [[Lawvere theory|NTJ94H2CNS]] whose $n$-ary operations are the smooth maps $\\R^n \\to \\R$.\nIn other words, it is a product-preserving functor to sets from the category of the spaces $\\R^n$ with smooth maps.\nUsing $f(x, y) = x + y$, $f(x, y) = xy$ and the constant functions, every $C^\\infty$-ring is a [[commutative ring|0310REJJY8]] and an $\\R$-algebra.\n\nExample: for a [[smooth manifold|F45VQ47GPV]] $M$, let $A = C^\\infty(M)$ and $\\Phi_f(a_1, \\dots, a_n) = f \\circ (a_1, \\dots, a_n)$.\nClause 1 holds since $\\pi_i \\circ (a_1, \\dots, a_n) = a_i$, and clause 2 is associativity of composition.\nThe ring of [[germs of smooth functions|7SWSRGDT4A]] at a point is a $C^\\infty$-ring in the same way.\nThe ring $\\R[\\varepsilon]/(\\varepsilon^2)$ is a $C^\\infty$-ring with $\\Phi_f(a + b\\varepsilon) = f(a) + f'(a) b \\varepsilon$ for $n = 1$, and the analogous formula with the gradient for larger $n$; clause 2 is the chain rule.\n\nNon-example: on $A = \\R$, the rule $\\Phi_f(a_1, \\dots, a_n) = f(0, \\dots, 0)$ fails clause 1, since $\\Phi_{\\pi_1}(a) = 0 \\ne a$ for $a \\ne 0$.\n",
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