Synthetic differential geometry

Summary

Synthetic differential geometry develops differential calculus and differential geometry with genuine infinitesimals, inside a topos instead of the category of sets. Its basic object is a number line $R$ whose set $D$ of elements of square zero is not reduced to $0$. By the Kock–Lawvere axiom, every function from $D$ to $R$ is affine. Tangent vectors become maps from the infinitesimal set $D = \set{d : d^2 = 0}$ into a space, and derivatives are defined without limits. Models built from C-infinity rings show that the theory is consistent and contains ordinary smooth manifolds.