Synthetic Differential Geometry

Synthetic differential geometry, and synthetic algebraic geometry, study the internal language of toposes similar in spirit to the classifying topos of commutative rings. Working internally, we have an axiomatically-defined ring \(R\) of "numbers", such that functions \(R \to R\) are "smooth".

Specifically, we typically have \((\hom_{R-\text{alg}}(A, R) \to R) \cong A\), as \(A\) ranges over some collection of \(R\)-algebras. Taking \(A = R[ x ]/(x^2)\), we find that functions from the "infinitesimal segment" \(\{d \in R \mid d^2 = 0\}\) to \(R\) are uniquely of the form \(f(d) = f(0) + \alpha d\), allowing us to define derivatives as \(f'(0) = \alpha\).

If we allow \(A = \mathbb{Z}[ x ]\), we find that every function \(R \to \R\) is a polynomial, giving us synthetic algebraic geometry. If we don't, we're free to have more general smooth functions; that's synthetic differential geometry.

Synthetic differential geometry is discussed in more detail in Synthetic Differential Geometry and Synthetic Geometry of Manifolds, by Anders Kock. Synthetic algebraic geometry is discussed in Ingo Blechschmidt's PhD thesis.

I've been exploring what happens if we work with toposes related to noncommutative rings instead of commutative rings. See here for details.