Linear logic and monoidal categories

我的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。从计算机科学中的并行计算理论到物理学中的拓扑和代数量子场论，在许多环境中都出现了么半范畴的例子。一元体结构对于理解这些应用至关重要。 在与R.Seely和R.Cockett的共同工作中，我们发明了微分范畴的新概念。这些本质上是带有附加微分运算符的线性逻辑模型，它允许人们对态射进行微分。显而易见的下一个目标是考虑流形的类别。流形之间的光滑函数集在哪一类中可以被认为是流形？这一范畴的逻辑结构是什么？流形的概念基于Frolicher和Kriegl方便的向量空间；这很可能是非常相关的。在一个相关的发展中，还存在便利向量空间之间的全纯函数的概念。这种观点的改变对逻辑结构和范畴结构有何影响？ 代数量子场论是一个严格的数学框架，用于模拟量子力学和相对论的相互作用。我希望将AQFT与最近阿布拉姆斯基和科克的抽象量子力学结合起来。在那里，量子力学被重新表述，不再是可观测的C*-代数理论，而是用抽象的、绝对的术语表示。AQFT的这个新概念将为Minkowski空间中的每个开放区域分配一个类别。赋值将具有AQFT和阿布拉姆斯基-科克公理所建议的性质。我们的工作已经产生了冯·诺依曼范畴的新概念。我们目前正在进一步发展这一理论，建立von Neumann范畴的交叉积，并将Doplicher-Roberts定理推广到这一背景下。

The goal of my project will be to continue to explore the theory of monoidal categories, especially using the techniques of linear logic. Examples of monoidal categories have arisen in a number of settings, from the theory of concurrent computation in computer science to topological and algebraic quantum field theory in physics. The monoidal structure is crucial in understanding these applications. In joint work with R. Seely and R. Cockett, we have invented the new notion of differential category. These are essentially models of linear logic with an additional differentiation operator, which allows one to differentiate morphisms. The obvious next goal is to consider categories of manifolds. In what category can the set of smooth functions between manifolds be considered a manifold? What is the logical structure of this category? There is a notion of manifold based on the convenient vector spaces of Frolicher and Kriegl; this will likely be quite relevant. In a related development, there is also a notion of holomorphic function between convenient vector spaces. How is the logical and categorical structure affected by this change in viewpoint? Algebraic Quantum Field Theory is a mathematically rigorous framework for modelling the interaction of quantum mechanics and relativity. I wish to combine AQFT with the recent abstract quantum mechanics of Abramsky and Coecke. There, quantum mechanics is reformulated away from the theory of C*-algebras of observables and expressed in abstract, categorical terms. This new notion of AQFT will assign a category to each open region in Minkowski space. The assignment will have properties suggested by both AQFT, and the Abramsky-Coecke axiomatics. Our work has already led to the new notion of a von Neumann category. We are currently developing this theory further, establishing crossed products for von Neumann categories, and extending the Doplicher-Roberts theorem to this setting.