Reimagining the Foundations of Infinite Dimensional Category Theory

数学进步往往是由抽象提供的清晰的视角推动的。今天，范畴理论最显著地延续了这一趋势，它提供了一种描述一般数学现象的语言，这些现象现在被编织到许多数学学科的结构中，特别是代数拓扑学和代数几何。范畴理论的进步往往是通过引入新的定义来实现的；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理论扩展到无限维的极端困难的问题，通过将这一理论重新建立在一个称为无限宇宙的新公理框架中，该框架用于简化和扩展以前在这一领域的工作的适用范围。许多对同伦理论、衍生代数几何和数学物理感兴趣的数学对象自然地生活在(无穷大，n)-范畴中，这是无限维范畴，其中所有维度n以上的态射都是弱可逆的。更确切地说，(无穷大，n)-范畴的范畴理论应该参考一个特定的模型来发展，如theta_n-空间、迭代完备西格尔空间或拟范畴(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出现的，这个宇宙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，标准范畴概念可以被定义，基本结果可以在任何无限宇宙中被证明(因此在许多情况下立即适用)，并且这些定义与先前建立的准范畴的定义一致。该项目的一个目标是确定当无限宇宙发生变化时，这些范畴概念保持得有多严格，例如，从准范畴的宇宙到完备西格尔空间的宇宙。在这个方向上的结果将证明许多实践者的梦想是正确的，即独立地使用无限维类别的模型。另一个目标是在闭无限宇宙中发展关于n维极限和n维列极限的理论，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.