Hank Chen

 

"Categorification" is a procedure by which one endows a new, higher level of structures upon existing structure in a coherent manner. This is motivated by various areas of physics, such as quantum gravity, high-energy physics and condensed matter theory; indeed, according to the categorical ladder proposal of Crane-Frenkel (or more physically Kitaev's program), categorification allows one to build physical models in higher-dimensions. In particular, excitations in an n-dimensional topological phase is characterized by a local fusion n-category, as explained in this paper.  One of the central goals of my project with Florian is to develop a categorification of classical and quantum symmetries. We in particular wish to construct a "2-quantum double" whose monoidal 2-category of 2-representations is equivalent to the Drinfel'd centre of a 2-category. This directly categorifies the defining equivalence that holds for Drinfel'd's original quantum double.

 

Update: the above goal has been achieved in the preprint here, where we defined a 2-Hopf algebra as a "categorical quantum group" based on the notion of homotopy A_∞-algebras. I have applied this framework to study 4-dimensional gapped topological phases here. Further work is now being done with David Green at OSU to connect our construction with the known theory of Hopf categories of Neuchl and Gurski.