Title: 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)