Reimagining the Foundations of Infinite Dimensional Category Theory

重新想象无限维范畴论的基础

• 批准号: 1509016
• 负责人: Emily Riehl
• 金额: $ 15.94万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1509016&HistoricalAwards=false
• 关键词: Reimagining; Foundations; Infinite; Dimensional; Category

Mathematical progress is often facilitated by the clarifying perspective provided by abstraction. Today this trend is continued most strikingly by category theory, which provides a language for describing general mathematical phenomena that is now woven into the fabric of many mathematical disciplines, particularly algebraic topology and algebraic geometry. Progress in category theory often comes through the introduction of new definitions; part of the philosophy is that the proofs should be easy once the correct perspectives are identified. This project will address the fiendishly difficult problem of extending category theory to infinite dimensions by re-grounding this theory within a new axiomatic framework, called an infinity-cosmos, which is used to both simplify and extend the range of applicability of previous work in this area.Many mathematical objects of interest in homotopy theory, derived algebraic geometry, and mathematical physics naturally live in (infinity,n)-categories, which are infinite-dimensional categories in which all of the morphisms above dimension n are weak invertible. To be fully precise, the category theory of (infinity,n)-categories should be developed in reference to a specific model, such as theta_n-spaces, iterated complete Segal spaces, or quasi-categories (in the case n=1). The PI and a collaborator have shown that each of these models arise as the objects of an appropriately defined infinity-cosmos, a simplicially-enriched category satisfying a small list of axioms. Ongoing work indicates that the standard categorical notions can be defined and the basic results can be proven in any infinity-cosmos (thus applying immediately in many contexts) and that these definitions agree with previously established ones for quasi-categories. One goal of the project is to determine how strictly these categorical notions are preserved upon change of infinity-cosmos, for instance from the cosmos for quasi-categories to the cosmos for complete Segal spaces. Results in this direction would justify the dream of many practitioners, which is to work with infinite-dimensional categories "model independently." Another goal is to develop the theory of n-dimensional limits and colimits for n1 in the closed infinity-cosmos for theta_n-spaces or iterated Segal spaces and compare the results with other work.

数学的进步往往是由抽象所提供的清晰的视角所促进的。今天，这一趋势继续最引人注目的范畴论，它提供了一种语言来描述一般的数学现象，现在编织成许多数学学科的结构，特别是代数拓扑和代数几何。范畴论的进步往往来自于新定义的引入;其哲学的一部分是，一旦确定了正确的观点，证明就应该很容易。该项目将解决将范畴论扩展到无限维度的极其困难的问题，方法是将该理论重新建立在一个称为无限宇宙的新公理框架内，该框架用于简化和扩展该领域以前工作的适用范围。同伦论、推导代数几何和数学物理中的许多感兴趣的数学对象自然存在于（infinity，n）-范畴，这是无限维范畴，其中所有大于n维的态射都是弱可逆的。为了完全精确，（infinity，n）-范畴的范畴论应该参照特定的模型来发展，例如theta_n-空间、迭代完备Segal空间或拟范畴（在n=1的情况下）。 PI和一位合作者已经证明，这些模型中的每一个都是作为一个适当定义的无限宇宙的对象而出现的，无限宇宙是一个满足一小串公理的简单丰富的范畴。正在进行的工作表明，标准范畴概念可以被定义，基本结果可以在任何无限宇宙中得到证明（从而立即适用于许多情况），这些定义与以前建立的准范畴的定义一致。该项目的一个目标是确定这些范畴概念在无限宇宙变化时的严格程度，例如从准范畴的宇宙到完全西格尔空间的宇宙。这个方向的结果将证明许多实践者的梦想是正确的，那就是独立地研究无限维范畴模型。“另一个目标是发展θ_n-空间或迭代Segal空间的n维极限和闭无限宇宙中n ~ 1的余极限理论，并将结果与其他工作进行比较。