A Bornological Perspective on the Representability of Derived Moduli Stacks of Solutions to PDEs

Proving representability of derived moduli stacks of solutions to non-linear elliptic partial differential equations generally requires significant analytic machinery. In this paper, we instead show that representability naturally follows from an Artin-Lurie style representability theorem. This necessitates the development of a new model for derived differential geometry using an extension of $C^{\mathrm{\infty }}$-rings that we call $C^{\mathrm{\infty }}$-bornological rings. This new theory embeds into the theory of derived bornological geometry recently proposed by Ben-Bassat, Kelly, and Kremnizer.