Model Comparisons and Foundational Developments in Higher Category Theory

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

One of the major roles of mathematics is to axiomatize a recurring structure, study it abstractly, and then apply it to situations beyond the originally intended ones. Several algebraic structures of interest, such as groups or vector spaces, consist of endowing a set with one or more operations subject to axioms. Often, interesting phenomena in algebraic topology, algebraic geometry, mathematical physics, and logic seem at first glance to be formalizable by one of such algebraic structures, although with a closer look one realizes that the equalities demanded by the axioms do not quite hold. Instead, the failure of the validity of the axioms is captured by the presence of a "higher isomorphism," whose nature is clear in the specific contexts. To accommodate those situations, one needs to acknowledge the presence of higher structures and replace the role played by the usual equality relation with higher isomorphisms, organizing the information into a higher category of some kind. Many objects of interest have by now been identified to have a specific type of higher structure called an (infinity, n)-category. The research project will investigate multiple questions related to this. Broader impacts of this collaborative research will be through making the literature more accessible to the users of higher category theory. The PI plans to organize an online working group of students and early career researchers to explore the features of the various models. Resulting expository materials will be made available online. An (infinity, n)-category is a type of categorical structure with objects and morphisms in each dimension which are furthermore invertible above dimension n. While there is a general agreement about this schematic idea, numerous alternative mathematical implementations of the definition of an (infinity, n)-category have been proposed, each approach leading to its own advantages and disadvantages. At present, some models are expected but not known to be equivalent. Even when models have been shown to be abstractly equivalent, it is often not easy to export constructions and results from a model to another. This research project sets goals to advance knowledge both in terms of developing aspects of specific models and understanding how to relate different models. The projects aim to provide consistency results that are currently lacking from the existing (infinity, 2)-categorical literature, producing an accessible and well-documented account of the pool of models of (infinity, 2)-categories as well as how they relate to the strict 2-categorical literature. These include developing a theory of weighted limits in (infinity, 2)-categories, proving a comparison of (infinity, 2)-categorical nerves, and studying the compatibility of the Gray tensor product with such nerves. Further, building on current work of the PI with collaborators, one of the main long-term aims of the joint research is to establish the equivalence of n-complicial sets with all other main models of (infinity, n)-categories for n>2. A positive answer to this longstanding question is crucial to unifying various results in the context of the (infinity, n)-literature. Finally, the PI also plans to formalize explicitly the inductive and coinductive homotopy theories of (infinity, infinity)-categories, understanding how they relate with the existing literature. 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.