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

TL;DR

This paper introduces a new derived differential geometry framework using $C^ $-bornological rings, simplifying the proof of representability for derived moduli stacks of PDE solutions.

Contribution

It develops a novel $C^ $-bornological ring theory that integrates into derived bornological geometry, enabling a more straightforward approach to representability.

Findings

Established a new model for derived differential geometry.

Connected the new theory with existing derived bornological geometry.

Simplified the proof of representability for moduli stacks of PDE solutions.

Abstract

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 $C^\infty$-rings that we call $C^\infty$-bornological rings. This new theory embeds into the theory of derived bornological geometry recently proposed by Ben-Bassat, Kelly, and Kremnizer.