Linear logic and monoidal categories

线性逻辑和幺半群类别

• 批准号: 155810-2011
• 负责人: Blute, Richard
• 金额: $ 0.8万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569494
• 关键词: Linear; logic; monoidal; categories

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.

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。幺半群范畴的例子已经出现在许多环境中，从计算机科学中的并发计算理论到物理学中的拓扑和代数量子场论。幺半群结构对于理解这些应用至关重要。 与R. Seely和R. Cockett，我们发明了微分范畴的新概念。这些基本上是线性逻辑的模型，带有一个额外的微分算子，它允许人们区分态射。显然，下一个目标是考虑流形的范畴。流形之间的光滑函数集在什么范畴中可以被认为是流形？这个范畴的逻辑结构是什么？有一个基于Frolicher和Kriegl方便的向量空间的流形概念;这可能是非常相关的。在一个相关的发展中，在方便的向量空间之间也有全纯函数的概念。逻辑和范畴结构如何受到这种观点变化的影响？ 代数量子场论是一个数学上严格的框架，用于模拟量子力学和相对论的相互作用。我希望把联合收割机AQFT与最近Abramsky和Coecke的抽象量子力学结合起来。在那里，量子力学被重新表述，脱离了观测量的C*-代数理论，并以抽象的范畴术语表达。 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。该赋值将具有AQFT和Abramsky-Coecke公理所建议的属性。我们的工作已经导致了冯·诺依曼范畴的新概念。 我们目前正在进一步发展这一理论，建立冯·诺伊曼范畴的交叉积，并将多普利彻-罗伯茨定理扩展到这种环境。