Systematic approach to higher structures

更高结构的系统方法

• 批准号: RGPIN-2020-06779
• 负责人: Henry, Simon
• 金额: $ 1.68万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759208
• 关键词: Systematic; approach; higher; structures

During the last 25 years, tools originally coming from homotopy theory have appeared in many other areas of mathematics: in algebraic geometry [Voe03][TTV08][Lurb], in mathematical physics [BD95][SS11], in symplectic geometry [PTVV13], in logic and type theory [V+13], etc. Every time they brought either solutions to long standing open problems, or brand new perspectives. These new tools mostly revolve around the idea of higher categories or more generally of higher structures. They are a type of algebraic structures in which the axioms cannot be stated as mere equalities between the operations, but are new operations producing explicit "equivalence" between previously defined operations. These new operations also have to satisfy some 'axioms' again expressed as new operations and so on, encoding a very rich structure with operations of all dimensions. Despite the great success these ideas have met, they are also extremely difficult to use. Only experts in homotopy theory have been able to apply them, and this has considerably slowed down their development. Many mathematicians working in the fields where they have been used have also voiced complaints about this difficulty. The main long-term goal of my research is to develop new methods that will allow to work with higher structures in a simpler, more efficient and intuitive way. Typically, I aim to develop a framework to translate the intuitive way experts think about higher structures into precise and concrete mathematical statement, as automatically as possible. This would have a very significant impact on future uses of higher structures outside of homotopy theory, and potentially even outside of mathematics. I will use my grant to fund and train a group of HQP working on this problem, but also on the applications of these methods to the theory of higher categories itself and on applications of higher structures to other fields of mathematics. Those are new and active areas of mathematics, very rich in problems ideal for PhD or MSc students, which can teach them highly desired skill that are becoming increasingly useful in many area of mathematics. I also plan to apply these new methods to several long standing open problems in the area: - C.Simpson's strictification conjecture. - Hovey's problem of constructing a « model structures of model structures ». - The homotopy hypothesis for Grothendieck higher groupoids. I have already made significant progress on these problems, and have a concrete plan to solve the first two in the next 3 years. There are also many other problems in higher category theory that my HQP will work on.

在过去的25年里，工具最初来自同伦理论已经出现在许多其他领域的数学：在代数几何[Voe 03][TTV 08][Lurb]，在数学物理[BD 95][SS 11]，在辛几何[PTVV 13]，在逻辑和类型理论[V+13]等，每次他们带来的解决方案长期存在的开放问题，或全新的观点。这些新工具大多围绕着更高类别或更普遍的更高结构的想法。它们是一种代数结构，其中的公理不能仅仅表示为运算之间的等式，而是在先前定义的运算之间产生显式“等价”的新运算。这些新的操作也必须满足一些“公理”，再次表示为新的操作等，编码一个非常丰富的结构与所有维度的操作。尽管这些想法取得了巨大的成功，但它们也极难使用。只有同伦理论的专家才能应用它们，这大大减缓了它们的发展。许多数学家在他们已经使用的领域工作也表达了对这种困难的抱怨。我研究的主要长期目标是开发新的方法，以更简单，更有效和直观的方式处理更高的结构。通常，我的目标是开发一个框架，将专家对更高结构的直观思考方式尽可能自动地转化为精确和具体的数学陈述。这将对未来在同伦理论之外，甚至可能在数学之外使用更高结构产生非常重大的影响。我将用我的赠款，以资助和培训一组HQP工作在这个问题上，但也对应用这些方法的理论更高的类别本身和应用程序的更高的结构，以其他领域的数学。这些都是新的和活跃的数学领域，非常丰富的问题，非常适合博士或硕士学生，这可以教他们高度期望的技能，在许多数学领域变得越来越有用。我还计划将这些新方法应用于该领域的几个长期存在的开放问题：- C.辛普森的严格化猜想。- Hovey的问题是如何构造一个“模型结构的模型结构”。- Grothendieck高等群胚的同伦假设。我已经在这些问题上取得了重大进展，并有一个具体的计划，在未来3年内解决前两个问题。在高级范畴理论中，还有许多其他的问题，我的HQP将致力于解决。