Linear Logic, Monoidal Categories and Abstract Models of Differentiation and Integration

我的项目的目标将是继续探索monoidal范畴理论，特别是使用线性逻辑的技术。线性逻辑的模型是一个配备有单子（满足几个自然性条件的内函子）的monoidal范畴。在以前的工作中，我们已经定义了微分范畴的概念。这些模型的线性逻辑与额外的运营商，使之能够区分态射。对于任何线性逻辑模型，我们都可以将第二个范畴，即克莱斯利范畴联系起来。对于微分范畴，这个范畴是光滑映射的范畴。相应的逻辑，微分线性逻辑，由Ehrhard和Regnier介绍。这样，我们就有了一个范畴的和逻辑的基础来考虑抽象的分化理论。

The goal of my project will be to continue to explore the theory of monoidal categories, especially using the techniques of linear logic. A model of linear logic is a monoidal category equipped with a monad (an endofunctor satisfying several naturality conditions). In previous work, we have defined the notion of differential category. These are models of linear logic with an additional operator which allows one to differentiate morphisms. To any model of linear logic, one can associate a second category, its Kleisli category. For differential categories, this category is a category of smooth maps. The corresponding logic, differential linear logic, was introduced by Ehrhard and Regnier. This then gives us a categorical and logical foundation for considering abstract theories of differentiation.