Linear logic and monoidal categories

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。从计算机科学的并发计算理论到物理学的拓扑量子场论，一元范畴的例子在许多场合都出现过。单轴结构对于理解这些应用是至关重要的，而且线性逻辑的思想已经成功地阐明了这些单轴结构。在接下来的四年里，我的工作将是继续进行这项分析，并进一步开发这些应用。我对继续研究离散量子因果动力学特别感兴趣，正如我与合著者Prakash Panangaden和Ivan Ivanov共同定义的那样。我们构建的结构类似于拓扑量子场论中常见的协共范畴，但使用了线性逻辑中更精细的连接词。因此，我们能够捕捉到在TQFT的通常表述中所缺乏的纠缠的概念。我希望我们发现的新例子，使用像吉拉德的概率相干空间这样的结构，也会给TQFT带来新的启示。我最近的工作已经导致了一些意想不到的联系，代数拓扑，我也打算探索。检查线性逻辑的某些模型导致了形状理论的一元版本，一个由Borsuk提出的同调不变理论。新实例的性质是它们的单轴结构通常提升到相关的形状类别。我打算探究这种现象有多普遍。

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 quantum field theory in physics. The monoidal structure is crucial in understanding these applications, and furthermore ideas from linear logic have successfully shed light on these monoidal structures. My work over the next four years will be to continue with this analysis and to develop these applications further. I am especially interested in continuing work on discrete quantum causal dynamics, as defined in joint work with my coauthors Prakash Panangaden and Ivan Ivanov. The structure we constructed is similar to the usual cobordism category of topological quantum field theory, but uses the more refined connectives of linear logic. We are thus able to capture a notion of entanglement which is lacking in the usual formulation of TQFT. I am hopeful that the new examples we have discovered, using such structures as Girard's probabilistic coherence spaces, will shed new light on TQFT as well. My more recent work has led to some unexpected connections to algebraic topology, which I also intend to explore. Examining certain models of linear logic led to a monoidal version of shape theory, a homology-invariant theory due to Borsuk. The new examples have the property that their monoidal structure lifts canonically to the associated shape category. I intend to explore how common this phenomenon is.