Systematic approach to higher structures

在过去的25年里，起源于同伦理论的工具已经出现在数学的许多其他领域：在代数几何[Voe03][TTV08][Lurb]、数学物理[BD95][SS11]、辛几何[PTVV13]、逻辑和类型论[V+13]等等。每次它们要么带来长期悬而未决的问题的解决方案，要么带来全新的视角。这些新工具主要围绕着更高类别或更一般的更高结构的想法。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相等，而是在先前定义的运算之间产生显式“等价”的新运算。这些新的运算还必须满足一些同样表示为新运算的‘公理’，等等，用所有维度的运算编码一个非常丰富的结构。尽管这些想法取得了巨大的成功，但它们也非常难以使用。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。在使用它们的领域工作的许多数学家也表达了对这一困难的抱怨。我研究的主要长期目标是开发新的方法，使其能够以更简单、更高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，尽可能自动地将专家对更高结构的直观思考方式转化为精确而具体的数学陈述。这将对同伦理论之外的更高结构的未来使用产生非常重大的影响，甚至可能在数学之外。我将用我的赠款资助和培训一组HQP，他们不仅致力于这个问题，而且还致力于这些方法在更高范畴理论本身的应用以及更高结构在其他数学领域的应用。这些都是新的和活跃的数学领域，非常丰富的问题适合博士生或硕士学生，这些问题可以教给他们非常想要的技能，这些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中几个长期悬而未决的问题：-C·辛普森的严格猜想。-Hovey的问题是如何构建一个“模型结构的模型结构”。-Grothendieck高阶群胚的同伦假设。我已经在这些问题上取得了重大进展，并制定了在未来3年解决前两个问题的具体计划。在更高的范畴理论中，还有许多其他问题需要我的HQP来解决。

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.