CAREER: Model-Independent Foundations for Higher Infinity-Categories

随着数学家研究的对象的复杂性增加，需要更复杂的工具来组织和操纵它们之间的转换：范畴理论，即对数学对象及其转换的形式研究，正在被(更高)无限范畴理论所取代，其中态射存在于每个维度中。这一领域的一个主要挑战是，“更高无限范畴”的基本概念是示意性的，因此技术发展依赖于明确的模型，每个模型都可能非常复杂地指定。PI将进行三个相互交织的项目，以推进与模型无关的更高无穷范畴理论的基础，将无穷大范畴准范畴模型的已知结果推广到其他模型，同时简化几个技术证明，就像人们采用明智地选择抽象时经常出现的情况一样。PI计划合写一本书，使新手能够接触到无限范畴理论的独立于模型的基础，并展示在这个项目中开发的新证明技术。国际和平协会已经写了两本书，这两本书都可以在网上免费获得，并有创新的教育学和会议组织的记录。国际数学联合会将进一步推动教育活动，为本科生开发一个新的数学证明的论述导论，扩大她作为部门多样性代表在校园内的工作，并通过一次会议和随后编辑的大量论文来促进数学界关于专业规范的对话。第一个提出的项目，与Verity合作，证明所有无限范畴的概念都是与模型无关的。因此，借助任何无穷范畴模型证明的任何定理都将适用于所有这些范畴。与Verity一起，PI引入了无限宇宙的概念，这是一个(更高的)无限范畴作为对象生活的宇宙，并表明无限范畴的理论可以从这些公理发展出来。这项工作描述了一种与先前的“分析”方法不同的无限范畴理论的“综合”方法。第二个提出的项目，与舒尔曼合作，将开发同伦型理论中的无穷范畴的并行综合理论，这是一种新的单价数学基础，它猜想地表达了无穷大-拓朴的内在逻辑。第三个提议的项目，也是与Verity联合提出的，是通过研究对象是复杂集合的各种无限宇宙，将无限范畴推广到更高无限范畴，这是一种特别经济的更高无限-范畴-PI怀疑最终将导致与模型无关的理论。

As the objects that mathematicians study increase in complexity, more sophisticated tools are required to organize and manipulate the transformations between them: category theory, the formal study of mathematical objects and their transformations, is being supplanted by (higher) infinity-category theory, where morphisms exist in each dimension. A major challenge in this area is that the fundamental notion of "higher infinity-category" is schematic, and thus technical developments rely upon explicit models, each of which can be quite complicated to specify. The PI will conduct three interwoven projects that advance the model-independent foundations for higher infinity-category theory, generalizing known results from the quasi-categorical model for infinity-categories to other models while simultaneously simplifying several technical proofs, as is often the case when one employs a judiciously chosen abstraction. The PI plans to co-write a book to make the model-independent foundations of infinity-category theory accessible to novices and present the new proof techniques developed in this program. The PI has written two books already, both of which are freely available online, and has a record of innovative pedagogy and conference organization. The PI will pursue further educational initiatives, developing a new discursive introduction to mathematical proof for undergraduates, expanding her work on campus as a departmental diversity representative, and facilitating a conversation on professional norms within the mathematics community through a conference followed by an edited volume of essays.The first proposed project, joint with Verity, proves that all infinity-categorical notions are "model-independent." Hence any theorem proven with the aid of any model of infinity-categories will apply to them all. Together with Verity, the PI has introduced the notion of an infinity-cosmos, a universe in which (higher) infinity-categories live as objects, and has shown that the theory of infinity-categories can be developed from these axioms. This work describes a "synthetic" approach to the theory of infinity-categories in contrast to prior "analytic" approaches. A second proposed project, joint with Shulman, will develop a parallel synthetic theory of infinity-categories in homotopy type theory, a new univalent foundations for mathematics that conjecturally expresses the internal logic of an infinity-topos. The third proposed project, again joint with Verity, is to generalize from infinity-categories to higher infinity-categories by investigating the various infinity-cosmoi whose objects are complicial sets, a particularly economical model of higher infinity-categories that the PI suspects will ultimately lead to a model-independent theory.