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Jets and differential linear logic

Published online by Cambridge University Press:  24 November 2020

James Wallbridge Open the ORCID record for James Wallbridge [Opens in a new window]
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Abstract

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

Keywords

Linear logicvector bundlesdifferential operatorsdifferential categories

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Paper
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Mathematical Structures in Computer Science , Volume 30 , Issue 8 , September 2020 , pp. 865 - 891
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© The Author(s), 2020. Published by Cambridge University Press

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