Reimagining the Foundations of Infinite Dimensional Category Theory

Harvard University, Cambridge MA

Investigators

Abstract

Mathematical progress is often facilitated by the clarifying perspective provided by abstraction. Today this trend is continued most strikingly by category theory, which provides a language for describing general mathematical phenomena that is now woven into the fabric of many mathematical disciplines, particularly algebraic topology and algebraic geometry. Progress in category theory often comes through the introduction of new definitions; part of the philosophy is that the proofs should be easy once the correct perspectives are identified. This project will address the fiendishly difficult problem of extending category theory to infinite dimensions by re-grounding this theory within a new axiomatic framework, called an infinity-cosmos, which is used to both simplify and extend the range of applicability of previous work in this area. Many mathematical objects of interest in homotopy theory, derived algebraic geometry, and mathematical physics naturally live in (infinity,n)-categories, which are infinite-dimensional categories in which all of the morphisms above dimension n are weak invertible. To be fully precise, the category theory of (infinity,n)-categories should be developed in reference to a specific model, such as theta_n-spaces, iterated complete Segal spaces, or quasi-categories (in the case n=1). The PI and a collaborator have shown that each of these models arise as the objects of an appropriately defined infinity-cosmos, a simplicially-enriched category satisfying a small list of axioms. Ongoing work indicates that the standard categorical notions can be defined and the basic results can be proven in any infinity-cosmos (thus applying immediately in many contexts) and that these definitions agree with previously established ones for quasi-categories. One goal of the project is to determine how strictly these categorical notions are preserved upon change of infinity-cosmos, for instance from the cosmos for quasi-categories to the cosmos for complete Segal spaces. Results in this direction would justify the dream of many practitioners, which is to work with infinite-dimensional categories "model independently." Another goal is to develop the theory of n-dimensional limits and colimits for n>1 in the closed infinity-cosmos for theta_n-spaces or iterated Segal spaces and compare the results with other work.