Dimensions and Categorification

So Witten's Khovanov paper reflects this idea that categorifying in math corresponds to adding dimensions in physics. He goes from a three dimensional theory to a five dimensional theory in order to go from Jones polynomial to Khovanov homology. More brane-y, more dimensions, more categorical. See http://math.ucr.edu/home/baez/diary/fqxi_narrative.pdf

There is a lot going on here (quantum groups, 2-groups, infuriating! weird combinatorics) and I am putting this post up here mainly so I am forced to learn this soon enough to fulfill my promise to tell you about it.

Can anybody explain to me the classical-quantum correspondence in terms of categorification to me?? Is that even possible? It just seemed like very similar arguments. Danke.