I was thinking if it may be possible to implement category theory in general relativity. I don't mean writing simply in terms of categories, but actual fundamental ideas (i.e. physics of the theory itself). For example, Prof. John Baez has written a pretty neat paper on the categorification of Yang-Mills theory. Also, is there a way or any recent development(s) to get this IR limit of general relativity from the least input (symmetries and propagating degrees of freedom) via category theory? If not, this is worth thinking about, IMO.
Here is how higher category theory (homotopy theory) arises in gravity:
First of all, the precise version of the statement that "gravity is a gauge theory" is that gravity in first order formulation is "Cartan geometry" for Minkowski spacetime regarded as the quotient of the Poincare group by the Lorentz spin-group. This statement generalizes to super-gravity, with the Poincare group replaced by the super-Poincare group ("the spacetime supersymmetry group"), see here:
The corresponding Cartan-connections encode the vielbein field and the "spin connection" that are the mathematical incarnation of the field of (super-)gravity, whose quanta are the graviton and the gravitino.
But now something special happens: higher dimensional supergravity by necessity contains not just the graviton and the gravitino, but also higher degree form fields. It is the higher degree of these form fields which is the entry point of higher category theory/homotopy theory in gravity.
Namely these tensor multiplets are no longer encoded by a Cartan-connection with values in an ordinary group (the Poincare group), but they are encoded by higher Cartan connections with values in higher categorical groups!
This is a long and fascinating story, which does not fit into this comment box here. To get started you might try these lecture notes here
or some of the links provided there.
I don't really understand your question, but since you link to my paper on higher Yang-Mills theory (which I never tried publish because it has problems, even though everything stated in it is true to the best of my knowledge), it sounds like maybe you're interested in approaches that treat gravity using ideas from higher gauge theory. For this, I urge you to read the work of Urs Schreiber. He has lots of papers on the arXiv, but a less strenous place to start is his series of articles on Physics Forums.