Jets and differential linear logic
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We prove that the category of vector bundles over a fixed smooth manifold and its corresponding functional analytic category of convenient modules are models for intuitionistic differential linear logic in two ways. The first uses the jet comonad to model the exponential modality. In this case the Kleisli category is the category of convenient modules with linear differential operators as morphisms. The second uses the more familiar distributional comonad whose Kleisli category is the category of convenient modules and smooth morphisms. Combining the two gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, non-linear partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic and hence computational theory of non-linear partial differential equations and their solutions based on variational calculus.
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