Model Comparisons and Foundational Developments in Higher Category Theory

数学的主要作用之一是将一个循环结构公理化，抽象地研究它，然后将它应用于超出最初预期的情况。几个有趣的代数结构，如群或向量空间，由赋予一个集合一个或多个服从公理的运算组成。通常，代数拓扑学、代数几何学、数学物理学和逻辑学中的有趣现象，乍看之下似乎可以用这样的代数结构之一来形式化，尽管仔细观察，人们会意识到公理所要求的等式并不完全成立。相反，公理有效性的失败被一个“更高的同构”的存在所捕获，其性质在特定的上下文中是清楚的。为了适应这些情况，人们需要承认更高结构的存在，用更高的同构取代通常的等式关系所扮演的角色，将信息组织到某种更高的范畴中。到目前为止，许多感兴趣的对象已经被确定为具有一种特定类型的更高结构，称为（无穷大，n）-范畴。该研究项目将调查与此相关的多个问题。这种合作研究的更广泛的影响将是通过使文献更容易获得更高的类别理论的用户。PI计划组织一个由学生和早期职业研究人员组成的在线工作组，以探索各种模式的特点。由此产生的临时材料将在网上提供。（infinity，n）范畴是一种范畴结构，它的每个维中的对象和态射在n维以上是可逆的。虽然对这个概念有一个普遍的共识，但已经提出了许多（无穷大，n）-范畴定义的替代数学实现，每种方法都有自己的优点和缺点。目前，一些模型是预期的，但不知道是等效的。即使当模型被证明是抽象等价的，它往往是不容易导出的结构和结果从一个模型到另一个。该研究项目设定的目标是在开发特定模型的各个方面以及了解如何将不同模型联系起来方面推进知识。这些项目旨在提供现有（无穷大，2）-分类文献中目前缺乏的一致性结果，对（无穷大，2）-分类的模型池以及它们与严格的2-分类文献的关系进行可访问和有据可查的说明。这些措施包括发展理论的加权限制（无穷大，2）类，证明比较（无穷大，2）类神经，并研究兼容性的灰色张量积与这种神经。此外，在PI与合作者当前工作的基础上，联合研究的主要长期目标之一是建立n-复杂集与n2的所有其他主要模型（无穷大，n）-范畴的等价性。一个积极的回答这个长期存在的问题是至关重要的统一各种结果的背景下（无穷大，n）-文学。最后，PI还计划明确形式化归纳和共归纳同伦理论（无穷，无穷）-categories，了解他们如何与现有literary.This奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的智力价值和更广泛的影响审查标准。

One of the major roles of mathematics is to axiomatize a recurring structure, study it abstractly, and then apply it to situations beyond the originally intended ones. Several algebraic structures of interest, such as groups or vector spaces, consist of endowing a set with one or more operations subject to axioms. Often, interesting phenomena in algebraic topology, algebraic geometry, mathematical physics, and logic seem at first glance to be formalizable by one of such algebraic structures, although with a closer look one realizes that the equalities demanded by the axioms do not quite hold. Instead, the failure of the validity of the axioms is captured by the presence of a "higher isomorphism," whose nature is clear in the specific contexts. To accommodate those situations, one needs to acknowledge the presence of higher structures and replace the role played by the usual equality relation with higher isomorphisms, organizing the information into a higher category of some kind. Many objects of interest have by now been identified to have a specific type of higher structure called an (infinity, n)-category. The research project will investigate multiple questions related to this. Broader impacts of this collaborative research will be through making the literature more accessible to the users of higher category theory. The PI plans to organize an online working group of students and early career researchers to explore the features of the various models. Resulting expository materials will be made available online. An (infinity, n)-category is a type of categorical structure with objects and morphisms in each dimension which are furthermore invertible above dimension n. While there is a general agreement about this schematic idea, numerous alternative mathematical implementations of the definition of an (infinity, n)-category have been proposed, each approach leading to its own advantages and disadvantages. At present, some models are expected but not known to be equivalent. Even when models have been shown to be abstractly equivalent, it is often not easy to export constructions and results from a model to another. This research project sets goals to advance knowledge both in terms of developing aspects of specific models and understanding how to relate different models. The projects aim to provide consistency results that are currently lacking from the existing (infinity, 2)-categorical literature, producing an accessible and well-documented account of the pool of models of (infinity, 2)-categories as well as how they relate to the strict 2-categorical literature. These include developing a theory of weighted limits in (infinity, 2)-categories, proving a comparison of (infinity, 2)-categorical nerves, and studying the compatibility of the Gray tensor product with such nerves. Further, building on current work of the PI with collaborators, one of the main long-term aims of the joint research is to establish the equivalence of n-complicial sets with all other main models of (infinity, n)-categories for n2. A positive answer to this longstanding question is crucial to unifying various results in the context of the (infinity, n)-literature. Finally, the PI also plans to formalize explicitly the inductive and coinductive homotopy theories of (infinity, infinity)-categories, understanding how they relate with the existing literature.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.