John Baez' idea: GR described as a 3-group gauge theory.

It's the beginnings of an idea. You link to the n-cat café where they discuss the draft Baez Huerta paper and where he points to page 37. To give a taste, I'll quote what he says in that passage:

==quote draft Baez Huerta==
Roberts and Schreiber go on to consider an analogous sequence of 3-groups constructed starting from a 2-group. Among these, the ‘inner automorphism 3-group’ of a 2-group plays a special role, which might make it important in understanding general relativity as a higher gauge theory.

As we have already seen in Section 4.3, Palatini gravity in 4d spacetime involves an so(3, 1)-valued 1-form A and a so(3, 1)-valued 2-form B = e ∧ e.
This is precisely the data we expect for a connection on a principal G-2-bundle where G is the tangent 2-group of the Lorentz group, except that the 2-form B fails to obey the equation dt(B) = F , as required by Theorem 4.5. Is there a way around this problem?

One possibility is to follow Breen and Messing [27], who, as we note, omit the condition dt(B) = F in their work on connections on nonabelian gerbes. This denies them the advantages of computing holonomies for surfaces, but they still have a coherent theory which may offer some new insights into general relativity.

On the other hand, Schreiber [70] has argued that for any Lie 2-group G , the
3-group I N N (G ) allows us to define a version of parallel transport for particles, strings and 2-branes starting from an arbitrary g-valued 1-form A and h-valued 2-form B. The condition dt(B) = F is not required. So, to treat 4d Palatini gravity as a higher gauge theory, perhaps we can treat the basic fields as a 3-connection on an I N N (T SO(3, 1))-3-bundle. To entice the reader into pursuing this line of research, we optimistically dub this 3-group I N N (T (SO(3, 1)) the gravity 3-group.
==endquote==

draft of "Invitation to Higher Gauge Theory" is here:
http://math.ucr.edu/home/baez/invitation1.pdf

You can see that there might be some clues here as to why BF theory keeps coming up in gravity work. They point to section 4.3 where it mentions that Palatini gravity in 4d spacetime involves an so(3, 1)-valued 1-form A and a so(3, 1)-valued 2-form B = e ∧ e. This is tantamount to a tie-in with BF theory, since in this context the curvature of the form A is denoted F.

So maybe we should look back to section 4.3. That's on pages 32-35

 

To be of some value in physics, there should be some uniqueness. The correct version of " parallel transport for particles, strings and 2-branes " would be expected to have some extra condition forcing D=11. It should relate to Evans paper on the link between division algebras and supersymmetry, and pass to topology via the link between division algebras, projective spaces and Hopf fibrations.

 

Thanks, I had missed this one.