CAREER: Model-Independent Foundations for Higher Infinity-Categories

随着数学家研究的对象越来越复杂，需要更复杂的工具来组织和操纵它们之间的转换：范畴论，数学对象及其转换的形式研究，正在被（更高的）无限范畴论所取代，其中每个维度都存在态射。这一领域的一个主要挑战是，“更高的无穷范畴”的基本概念是示意性的，因此技术发展依赖于明确的模型，其中每一个都可能非常复杂。PI将进行三个相互交织的项目，推进更高的无穷大范畴理论的模型独立基础，将无穷大范畴的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，就像人们采用明智选择的抽象时经常发生的那样。PI计划共同撰写一本书，使无限范畴理论的模型独立基础对新手开放，并介绍在该计划中开发的新证明技术。PI已经写了两本书，这两本书都可以在网上免费获得，并且有创新教学法和会议组织的记录。PI将继续开展进一步的教育活动，为本科生开发一个新的数学证明的话语介绍，作为部门多样性代表扩大她在校园里的工作，并通过一个会议促进数学界内部关于专业规范的对话，然后编辑论文集。第一个提议的项目，与Verity联合，证明所有无穷范畴概念是"模型独立的"。“因此，借助任何无穷范畴模型证明的任何定理都将适用于它们。与Verity一起，PI引入了无限宇宙的概念，（更高的）无限范畴作为对象生活在其中的宇宙，并表明无限范畴理论可以从这些公理发展出来。这项工作描述了一个“综合”的方法来理论的无穷大范畴在以前的“分析”的方法。第二个项目，与舒尔曼联合，将发展一个平行的合成理论的无穷大类同伦型理论，一个新的数学基础，表达了内部逻辑的无穷大拓扑。第三个提议的项目，再次与Verity联合，是通过研究对象是复杂集合的各种无穷宇宙，从无穷范畴推广到更高的无穷范畴，PI怀疑这是一个特别经济的更高的无穷范畴模型，最终将导致一个模型独立的理论。

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.