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  "body": "\n## Definition\n\nLet $\\mathcal{E}$ be a [[topos|6837RS7N82]] and let $R$ be a [[commutative ring|0310REJJY8]] object in $\\mathcal{E}$.\nLet $D = \\set{ d \\in R : d^2 = 0 }$ be the subobject of square-zero elements, and let $R^D$ be the object of maps from $D$ to $R$.\nThe pair $(\\mathcal{E}, R)$ satisfies the *Kock–Lawvere axiom* if the map\n$$\\alpha \\colon R \\times R \\to R^D, \\qquad (a, b) \\mapsto (d \\mapsto a + b d)$$\nis an isomorphism.\n*Synthetic differential geometry* is the development of differential calculus and differential geometry inside such a pair $(\\mathcal{E}, R)$.\nIts reasoning uses the internal logic of $\\mathcal{E}$, which is intuitionistic.\n\n## Remarks\n\nThe axiom says that every map $g \\colon D \\to R$ is of the form $g(d) = a + bd$ for unique $a, b \\in R$.\nFor $f \\colon R \\to R$ and $x \\in R$, applying it to $d \\mapsto f(x + d)$ defines the derivative $f'(x)$ by $f(x + d) = f(x) + d f'(x)$ for all $d \\in D$.\nTangent vectors at a point $x$ of an object $M$ are maps $t \\colon D \\to M$ with $t(0) = x$.\n\nExample: there are toposes containing the category of smooth manifolds in which the axiom holds, the well-adapted models.\nThey are built from [[C-infinity rings|4EJED0YZBB]]; for instance, the ring $C^\\infty(\\R)[\\varepsilon]/(\\varepsilon^2)$ plays the role of functions on $R \\times D$.\n\nNon-example: in the topos of sets, with $R = \\R$, the axiom fails.\nSince $\\R$ has no [[nilpotent elements|0ZB2JG1M9C]] except $0$, $D = \\set{0}$, so $R^D \\cong R$, and $\\alpha$ is not injective: $(0, 1)$ and $(0, 0)$ have the same image.\nThe axiom forces $D \\ne \\set{0}$, which is incompatible with classical logic in a field-like $R$; this is why the internal logic must be intuitionistic.\n",
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