CAREER: Model-Independent Foundations for Higher Infinity-Categories

Johns Hopkins University, Baltimore MD

Investigators

Abstract

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.