Point-free geometry (XIV): Brainwave on curvature and further dynamics
We can picture these regions like a fluid! With that, we have to go back to curvature, defining a curved point-free space, in analogy to a Riemannian space.
We have a notion of density of regions, sort of given by the number of elements in each intersection set. I just imagined that if the number of elements in each intersection set was constant over each region, then we should recover a flat space. Curvature can emerge as a result of a difference in the number of elements in the intersection sets between two regions.
How does this make sense to me? We can *sort of* treat general relativity as a medium, at least to explain light deflection by matter. In any rate, the physical distance traversed by light in a given amount of time (time with respect to a local observer) will change, increase for positive curvature and decrease for negative curvature with respect to the distance traveled on flat space.
Now, we will have to slightly modify our definition of distance, right? My hope is that it is close, we'd have to hop, probably, to the nearest neighbor, or something. I don't know, I'll have to think about it. But the difference in the "sized" of the intersection sets really should correspond to curvature. And notice that there is no set number of elements in the intersection set that will give a flat space, it all depends on the neighboring regions. Everything is relative.
And then we can move onto a dynamical point-free space. We'd need to add on some additional operation here. This one is conceptually easy. This operation will take an element from the intersection set of one region, remove it from there, and place it into the intersection set of another region. Of course, the devil is in the details.
Call the region we want to move $R_m$ (m for move). There will likely be multiple regions with $R_m$ in its intersection sets; these would clearly have a non-trivial mutual intersection set, having at least $R_m$ populating it. So we don't have to remove $R_m$ from all intersection sets, but just one. Now the question is for which region do we add $R_m$ into its intersection set? It should move $R_m$ to a region inside the intersection set of the original region, but it cannot intersect this original region. In other words, it has to move it somewhere "close." I'll think on it and try to come up with a more precise definition later.
The operation $M$ acts on the intersection sets $\mathcal{I}_n$ for all regions $n$ (dare I call this the intersection bundle?). It basically shifts regions around by a small distance, as small a distance that we can get away with. So we can basically imagine these regions flowing around. I imagine this as making up a dynamical point-free space, much like general relativity describes a dynamical spacetime.
I dare not get my hopes up, though. How can noncommutativity of functions arise from this construction? Perhaps it does require the fuzzification process, after all. Or maybe it simply cannot emerge from this construct. But if I can manage to define this dynamical operator precisely, perhaps I can start working on a point-free analogue to general relativity. So there are still places to go with this. I'll continue to look into this idea. Just ... a bit more slowly now that I am teaching.
