Commutative 2-algebra and applications.

交换2-代数及其应用。

• 批准号: 2106379
• 金额: --
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2106379
• 关键词: Commutative; algebra; applications.

Commutative algebra is a well-established area of pure mathematics, with several applications across the whole of mathematics. In particular, commutative algebra provides a foundation for the development of algebraic geometry. A useful way to understand and study commutative algebra is via category theory. Indeed, category theory provides a general approach to characterise categories of algebraic objects, establish their key properties, and describe in abstract terms the idea of commutativity and distributivity. This is achieved via monad theory, established in the '60s and '70s in the now classical works of Eilenberg, Lawvere, Beck, and others. This project seeks to develop further a relatively new subject, which may be referred to as commutative 2-algebra. This can be understood as a counterpart of commutative algebra in which instead of considering algebraic structures carried by sets, one considers algebraic structures carried by categories. As such, the subject is closely related in spirit to the idea of `categorification' that has been very prominent in representation theory in recent years. Just as commutative algebra could be studied under the lens of category theory via monads, 2-commutative algebra can be be studied using 2-dimensional category theory via 2-monads and their generalisations, which have been studied extensively by the Australian category theory school.The overall goal of the project is to develop further commutative 2-algebra by establishing counterparts of some fundamental results of classical monad theory for relative pseudo-monads, a generalisation of 2-monads that the project supervisor and some collaborators have introduced recently. Specific goals include the proof of counterparts of Beck's fundamental result on distributive laws and of Kock's characterisation of commutative monads. As an application, the project will push further the theory of operads and analytic functors, as developed by the project supervisor and his collaborators. One specific goal here is to enhance the bicategory of symmetric operads and analytic functors introduced by the supervisor and Andre' Joyal to a pseudo double category, building on work of Dwyer and Hess.The novelty of the project derives from the idea of developing the theory on the basis of the notion of a relative pseudomonad rather than that of a 2-monad, as traditionally done. This is useful because it allows us to capture important examples that are beyond the standard theory. These examples are of fundamental for our applications and are of interest also for current research in theoretical computer science.

交换代数是纯数学的一个成熟的领域，在整个数学中有几个应用。特别是，交换代数为代数几何的发展提供了基础。理解和研究交换代数的一个有用的方法是通过范畴论。事实上，范畴论提供了一个通用的方法来定义代数对象的范畴，建立它们的关键属性，并抽象地描述交换性和分配性的概念。这是通过单子理论来实现的，单子理论是在60年代和70年代建立的，现在是艾伦伯格、劳维尔、贝克和其他人的经典著作。这个项目旨在进一步发展一个相对较新的主题，它可能被称为交换2-代数。这可以被理解为交换代数的对应物，在交换代数中，不是考虑由集合承载的代数结构，而是考虑由范畴承载的代数结构。因此，这个主题在精神上与近年来在表征理论中非常突出的“范畴化”思想密切相关。正如交换代数可以通过单子在范畴论的透镜下研究一样，2-交换代数可以通过2-单子及其推广使用二维范畴论来研究，该项目的总体目标是进一步发展可交换的2-代数通过建立对应的一些基本结果的经典单子理论的相对伪单子，推广2单子的项目主管和一些合作者最近推出。具体目标包括证明同行贝克的基本结果分配法和科克的特点交换单子。作为一个应用，该项目将进一步推动由项目主管和他的合作者开发的运算和解析函子理论。这里的一个具体目标是提高双范畴的对称运算和分析函子介绍了主管和安德烈' Joyal的伪双范畴，建设工作的德怀尔和Hess.The新奇的项目源于发展的想法理论的基础上的概念相对pseudomonad，而不是一个2-monad，传统上做的。这是有用的，因为它使我们能够捕捉到超越标准理论的重要例子。这些例子是我们的应用程序的基础，也是当前理论计算机科学研究的兴趣。