Linear logic and monoidal categories

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量子场论。么半群结构在理解这些应用中至关重要，而且线性逻辑的思想已经成功地阐明了这些么半群结构。在接下来的四年里，我的工作将是继续进行这种分析，并进一步开发这些应用程序。我特别感兴趣的是继续研究离散量子因果动力学，正如我与我的合著者Prakash Panangelo和Ivan Ivanov共同工作所定义的那样。我们构造的结构类似于拓扑量子场论中通常的配边范畴，但使用了线性逻辑中更精细的联结词。因此，我们能够捕捉到的纠缠的概念，这是缺乏在通常的制定TQFT。我希望我们发现的新例子，使用诸如吉拉德的概率相干空间等结构，也将为TQFT带来新的启示。我最近的工作导致了一些意想不到的联系代数拓扑，我也打算探索。考察某些模型的线性逻辑导致了一个monoidal版本的形状理论，一个同源不变的理论，由于博苏克。新的例子具有这样的性质，即它们的monoidal结构正则地提升到相关的形状类别。我想知道这种现象有多普遍。

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.