Synthetic geometry of differential equations: I. Jets and comonad structure

Abstract

We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A motivation for such a level of generality is the eventual goal of solving the open problem of covariant geometric pre-quantization of locally variational field theories, which may include fermions and (higher) gauge fields. (abridged)

arXiv comment72 pages, xypic and tikzcd commutative diagrams; v2: minor technical revisions, close to published version

References in corpus (3)

• Differential cohomology in a cohesive infinity-topos
• The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
• On Theories of Superalgebras of Differentiable Functions

Cited by in corpus (5)

• The stack of Yang-Mills fields on Lorentzian manifolds
• Modal Descent
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• Proper Orbifold Cohomology
• Synthetic G-jet-structures in modal homotopy type theory