Synthetic differential geometry

\(C^\infty\)-rings are algebras of the cartesian operad of smooth functions \(f: \mathbb{R}^n \to \mathbb{R}\). 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 \(C^\infty\)-rings (doi, arxiv)
  • Short version: Joyce, 2012: An introduction to \(C^\infty\) schemes and \(C^\infty\) algebraic geometry (doi, arxiv)