I went to Erlangen to learn all about Cartan Geometry from Derek Wise. I also learned about spin network quantisation, and we tried to combine the two. Unfortunately we didn't really finish this project. Still, the thesis serves as an introduction to Cartan geometry and some thoughts about quantisation.
There is an excellent question on mathoverflow asking whether there is category theory with a half-twist as graphical calculus. It turns out there is. Basically, the half-twist is a square root of the twist in a balanced braided category. The most general treatment I know of is Jeff Egger's article On Involutive Monoidal Categories.
The notion of finite dimensional spectral triples internal to an involutive monoidal dagger category is defined. It makes the definition of a noncommutative geometry with symmetry quantum group possible. Surprising relations to two-dimensional extended TQFTs with line defects are uncovered.
We develop a generalisation of the Crane-Yetter model, a topological state sum model of 4-manifolds. The new framework is used to show that the Crane-Yetter model for nonmodular ribbon fusion categories is stronger than signature and Euler characteristic.