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

我项目的目标是继续探索一元范畴的理论，特别是使用线性逻辑的技术。线性逻辑的模型是带有单元（满足若干自然性条件的内函子）的一元范畴。在以前的工作中，我们已经定义了微分范畴的概念。这些是线性逻辑的模型，有一个额外的算子，它允许微分态射。对于任何线性逻辑模型，我们都可以联想到第二个范畴，它的Kleisli范畴。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，是由Ehrhard和Regnier引入的。这就为我们考虑抽象的微分理论提供了一个范畴和逻辑基础。

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.