NSF Award
2204304
Higher category theory is increasingly being used as the metatheory for new results in several areas of mathematics, creating a huge barrier to entry for mathematicians whose primary technical expertise lies in another field. Past joint work of the PI reimagined the foundations of infinite-dimensional category theory with the aim of simplifying proofs by replacing "analytic" methods, that rely on the combinatorics of a particular model of infinite-dimensional categories, with "synthetic" ones that apply in any model. One part of this project seeks to develop a computer-verifiable formal language that expresses only statements about infinite-dimensional categories that are invariant under change of model. Such a language would force users to "speak no evil" by guaranteeing that every statement they express is model-independent. This project connects to the plans to recast the theory of infinite-dimensional categories in a new proposed univalent foundation system for mathematics, in which homotopical uniqueness up to a contractible space of choices becomes genuine uniqueness, permitting streamlined definitions of fundamental concepts. Both of these projects will be undertaken in part with mentees of the PI at Johns Hopkins. In parallel, the PI has concrete plans to continue her expository and outreach work which include a new book (Elements of Infinity-Category Theory, joint with Verity), lectures directed at the general public, survey articles prepared for a variety of audiences, and efforts to improve access to advanced mathematics, such as her service on the Equity, Diversity, and Inclusion Advisory Board at the Banff International Research Station.<br/><br/>The pioneers of homotopy type theory - the new proposed univalent foundation system - envisioned a computer-verifiable foundation for infinite-dimensional category theory, but some computational content is lost through the classical reasoning used in classical homotopy theory. With collaborators, the PI will develop a new model for classical homotopy theory in a particular category of cubical sets, in which cubical fibrations are required to be equivariant, respecting the symmetries of cubes defined by permuting their dimensions. A longer-term aim is to use similar methods to obtain cubical set based presentations of all infinity-topoi. A computer proof assistant based on equivariant cubical fibrations would have the correct classical semantics but would be able to restore the computational content to univalent mathematics. A final project explores homotopical macrocosms for higher category theory, aiming to prove that the collection of cartesian fibrations between (infinity,n)-categories assemble into a cartesian fibration of (infinity,n+1)-categories, which can be regarded as some sort of categorified hyperdoctrine for (infinity,n)-category theory. Results of this nature would establish a global lifting property against homotopy coherent diagrams that should aid further developments in (infinity,n)-category theory.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
