Low dimensional category theory has found application in a plethora of subjects, from pure mathematics to computer science passing through quantum theory. In particular, the higher dimensional algebra approach envisioned by Baez and Dolan provided new tools which made possible various new results. The idea is to use categorical tools to model higher dimensional versions of algebraic objects. For instance, it is well known that the fundamental group of a topological space, introduced by Poincaré in 1895, captures 1- dimensional homotopical information about the space. In a similar way one would expect a notion of n-group to be able to capture higher homotopical aspects. In dimension 2 this was studied by Whitehead and Maclane with the introduction of crossed modules and 2-groups.
This project aims to solve open problems on 3-dimensional aspects of higher dimensional algebra. We plan to further develop the theory of 3-groups, extending results which use the theory of opfibrations to 2-dimensional category theory, and define bicategories equipped with two monoidal structures interacting with each other and prove a strictification result for them. These results will have applications in homological algebra (e.g. higher cohomology groups), mathematical physics and computer science.
