Model Comparisons and Foundational Developments in Higher Category Theory

数学的主要作用之一是将循环结构公理化，抽象地研究它，然后将它应用于超出最初预期的情况。一些感兴趣的代数结构，如群或向量空间，由赋予集合一个或多个遵循公理的运算组成。通常，代数拓扑、代数几何、数学物理和逻辑中的有趣现象乍一看似乎可以通过这样的代数结构之一来形式化，尽管仔细观察就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“更高同构”的存在所捕捉，其性质在特定的上下文中是显而易见的。为了适应这些情况，人们需要承认存在更高的结构，并用更高的同构取代通常的平等关系所起的作用，将信息组织到某种更高的类别中。到目前为止，许多感兴趣的对象已经被识别为具有称为(无穷大，n)范畴的特定类型的较高结构。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影响将是通过使文献更容易为更高范畴理论的用户所接受。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，探索各种模式的特点。由此产生的说明性材料将在网上提供。(infinity，n)-范畴是一种范畴结构，其每个维度上的对象和态射在n维上是可逆的。虽然对这一概念有普遍的共识，但已经提出了许多可供选择的定义(infinity，n)-范畴的数学实现，每种方法都有其优缺点。目前，一些模型预计是等同的，但不知道是等同的。即使已经证明模型在抽象上是等价的，要将结构和结果从一个模型导出到另一个模型通常也不容易。这个研究项目的目标是在开发特定模型的各个方面和了解如何将不同的模型联系起来方面促进知识的发展。这些项目的目的是提供现有(无穷大，2)范畴文献目前所缺乏的一致性结果，对(无穷大，2)范畴的模型池以及它们与严格的2范畴文献的关系产生一个可访问的和有良好记录的描述。这些内容包括发展(无穷，2)-范畴的加权极限理论，证明(无穷，2)-范畴神经的比较，以及研究Gray张量积与这类神经的相容。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_2的n-互补集与(无穷，n)-范畴的所有其他主要模型的等价性。对这个长期存在的问题的肯定回答对于在(无穷大，n)文献的背景下统一各种结果至关重要。最后，PI还计划明确地将(无穷大，无穷大)范畴的归纳和余归纳同伦理论正式化，了解它们如何与现有文献相关联。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.