Reimagining the Foundations of Infinite Dimensional Category Theory

抽象所提供的清晰的视角常常促进数学的进步。今天，范畴论最显著地延续了这一趋势，范畴论提供了一种描述一般数学现象的语言，现在已融入许多数学学科的结构中，特别是代数拓扑和代数几何。范畴论的进步往往来自于引入新的定义；这种哲学的一部分是，一旦确定了正确的观点，证明就应该很容易。这个项目将解决将范畴论扩展到无限维的极其困难的问题，通过在一个新的公理框架中重新建立这个理论，称为无限宇宙，它被用来简化和扩展这个领域以前工作的适用性范围。在同伦理论、衍生代数几何和数学物理中，许多感兴趣的数学对象自然存在于（无穷，n）-范畴中，这些范畴是无限维的范畴，其中n维以上的所有态射都是弱可逆的。为了完全精确，(∞,n)-范畴的范畴论应该参照特定的模型来发展，例如ta_n-空间、迭代的完全Segal空间或准范畴（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一个适当定义的无限宇宙的对象，这是一个满足一小部分公理的简单丰富的范畴。正在进行的工作表明，标准范畴概念可以定义，基本结果可以在任何无限宇宙中得到证明（因此可以立即应用于许多情况），并且这些定义与先前建立的准范畴定义一致。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，那就是使用无限维的类别“独立建模”。另一个目标是发展封闭无限宇宙中n- n空间或迭代Segal空间中n1的n维极限和极限理论，并将结果与其他工作进行比较。

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.