An article that we have written:
Synthetic geometry of differential equations
Part I. Jets and comonad structure
On the formulation of variational calculus, PDEs and classical field theory, in terms of differential cohesion. This is used at Prequantum covariant field theory to solve the problem of covariant pre-quantization of local field theories.
Abstract. We give an abstract (synthetic) formulation of the formal theory of 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.
A remarkable observation of Marvan 86 is that the jet bundle construction in ordinary differential geometry has the structure of a comonad, whose (Eilenberg-Moore) category of coalgebras is equivalent to Vinogradov's category of PDEs. We give a synthetic generalization of the jet bundle construction and exhibit it as the base change comonad along the unit of the “infinitesimal shape” functor, the differential geometric analog of Simpson’s “de Rham shape” operation in algebraic geometry. This comonad structure coincides with Marvan’s on ordinary manifolds. This suggests to consider PDE theory in the more general context of any topos equipped with an “infinitesimal shape” monad (a “differentially cohesive” topos).
We give a new natural definition of a category of formally integrable PDEs at this level of generality and prove that it is always equivalent to the Eilenberg-Moore category over the synthetic jet comonad. When restricted to ordinary manifolds, Marvan’s result shows that our definition of the category of PDEs coincides with Vinogradov's, meaning that it is a sensible generalization in the synthetic context.
Finally we observe that whenever the unit of the “infinitesimal shape” $\Im$ operation is epimorphic, which it is in examples of interest, the category of formally integrable PDEs with independent variables ranging in $\Sigma$ is also equivalent simply to the slice category over $\Im\Sigma$. This yields in particular a convenient site presentation of the categories of PDEs in general contexts.
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