Higher category theory in computer science

从根本上说，范畴论是结构和相互作用的数学理论。多年来，数学及其子领域、计算机科学和物理学之间有着紧密的联系，范畴论是一种统一的语言。高级范畴论是范畴论的扩展，使其能够研究自身和其他自然高维现象。这样的研究有可能为计算和组合性的更高范畴建模奠定一个统一框架的基础。本项目的目标分为两个主要目标：发展更高范畴理论来建模计算和组合现象;应用更高的范畴理论技术来解决问题。为了这个目标，我们开发了基于网络的证明助手'homotopy.io'，用于有限表示（无穷符号）范畴及其基础理论，特别是这个工具使我们的研究方法新颖。而不是仅仅提供笔和纸的证明，是传统的数学，我们的目标也是在适当的情况下，计算机形式化的证明。这样一个任务的需要在于更高的范畴理论的复杂性-许多定义，这是直观的日常概括的良好理解的低维结构是禁止在高维大，这本身很好地适用于计算机辅助的方法à la 'homotopy.io'。我们的第二个研究重点是（传统的）范畴理论，使用的工具，更高的范畴。在将1-范畴理论应用于理论计算机科学方面已经有了丰富的知识，通过更好地了解范畴理论本身，我们可以用更好的工具来装备自己，从而带来新的发展。例如，（乘法）线性逻辑自然地由 *-自治范畴来范畴地建模。对于更高的范畴，我们可以把 *-自治范畴看作范畴化的弗罗贝纽斯代数，并用弦图来推理它们。这是一个非常有用的工具，它使我们能够了解 *-自治范畴，这反过来又会导致对线性逻辑的新见解，最终这可以应用于计算机科学（例如，作为一种具有资源管理的编程语言（如Rust）的高级类型系统的形式化），使对应关系完整循环。图表推理，可以被视为这样一种更高类别的技术，已经被牛津大学的基础、结构和量子小组成功地利用，极大地简化了量子理论。这是一个野心，这项研究将有一天成为当代数学的一部分，需要解决长期存在的问题，在整个数学科学，如量子引力理论，或正式审查的哲学问题的平等在数学，或者更好地理解现代编程语言或复杂交互系统中高级功能的语义。这个项目属于EPSRC的福尔斯理论计算机科学研究领域。

Fundamentally, category theory is the mathematical theory of structure and interactions. For many years, there have been strong links between mathematics, its subfields, computer science, and physics, with category theory as a unifying language. Higher category theory is a extension of category theory to enable it to both study itself, and other naturally higher dimensional phenomena. Such research has the potential to lay the foundations for a consolidated framework for the higher-categorical modelling of computation and compositionality.The aims of this project fall under two main goals the development of higher category theory to model computational and compositional phenomena; the application of higher category-theoretic techniques to solve problems.Towards this aim, we develop the web-based proof assistant 'homotopy.io' for finitely-presented (infinity symbol)-categories and its underlying theory, and in particular this tool makes our research methodology novel. Rather than solely providing pen-and-paper proofs, as is traditional in mathematics, we also aim to give computer-formalised proofs where appropriate. The need for such an undertaking lies in the complexity of higher category theory - many definitions which are intuitively routine generalisations of well-understood lower-dimensional structure are prohibitively large in higher dimensions, and this lends itself well to a computer-aided approach à la 'homotopy.io'.Our second focus of study is (traditional) category theory, using the tools of higher categories. There is a wealth of knowledge already on applying 1-category theory to theoretical computer science at large, and by gaining better insight into category theory itself we equip ourselves with better tools which can lead to new developments. For example, (multiplicative) linear logic is naturally modelled categorically by *-autonomous categories. With higher categories, we can consider *-autonomous categories as a categorified Frobenius algebra and reason about them with string diagrams. This is a tremendously useful tool that allow us to learn about *-autonomous categories, which in turn leads to new insight into linear logic, and ultimately this can be applied to computer science (say, as a formalisation for an advanced type system for a programming language with resource management, like Rust), bringing the correspondence full-circle.Diagrammatic reasoning, which can be regarded as such a higher-categorical technique, has already been successfully exploited by the Foundations, Structures, and Quantum group, at Oxford, to vastly simplify quantum theory. It is an ambition that this research will one day be a piece of the contemporary mathematics required to solve long-standing problems throughout the mathematical sciences, such as a theory of quantum gravity, or a formal examination of the philosophical problem of equality in mathematics, or a greater understanding of the semantics of advanced features in modern programming languages or complex interacting systems.This project falls within the EPSRC Theoretical Computer Science research area.