Synthetic differential geometry

Remarks

The axiom says that every map $g \colon D \to R$ is of the form $g(d) = a + bd$ for unique $a, b \in R$. For $f \colon R \to R$ 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 $d \in D$. Tangent vectors at a point $x$ of an object $M$ are maps $t \colon D \to M$ with $t(0) = x$.

Example: there are toposes containing the category of smooth manifolds in which the axiom holds, the well-adapted models. They are built from C-infinity rings; for instance, the ring $C^\infty(\R)[\varepsilon]/(\varepsilon^2)$ plays the role of functions on $R \times D$.

Non-example: in the topos of sets, with $R = \R$, the axiom fails. Since $\R$ has no nilpotent elements except $0$, $D = \set{0}$, so $R^D \cong R$, and $\alpha$ is not injective: $(0, 1)$ and $(0, 0)$ have the same image. The 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.