Synthetic differential geometry

Synthetic differential geometry (SDG) is the axiomatic study of infinitesimals familiar from calculus and differential geometry.

C∞C^\infty-rings

C∞C^\infty-rings are algebras of the cartesian operad of smooth functions f:Rn→Rf: \mathbb{R}^n \to \mathbb{R}. They possess much more structure than an \(\mathbb{R\)-algebra, which is an algebra of the cartesian operad of real polynomials. The opposite of the category of \(C^\infty\)-rings serves as a model of SDG.

  • Joyce, 2019: Algebraic geometry over C∞C^\infty-rings (doi, arxiv)
    • Short version: Joyce, 2012: An introduction to C∞C^\infty schemes and C∞C^\infty algebraic geometry (doi, arxiv)