Category theory & geometric measure theory?

I write this as an answer since it is a bit too long for a comment.

Category theory is being used to investigate differential equations. A first entry point is through the concept of D-module.

M.Kashiwara: D-Modules and Microlocal Calculus

Another approach is the one pioneered by Kashiwara in

M. Kashiwara, T. Kawai, T. Kimura: Foundations of Algebraic Analysis, Princeton University Press,1986

For applications to global problems I suggest looking at the memoir Ind-Sheaves by M.Kashiwara and P. Schapira. I have to warn you that the formalism is heavy and you will need to know a lot from Kashiwara and Schapira's book Sheaves on Manifolds.

The approach in the above references is very different from what you think are the traditional pde-s and I do not recommend giving up your day job to learn this stuff.

I say this from experience. I am trained in pde. I spent a year learning about derived categories (see these notes). While this helped me understand better various topological problems, they did not enhance my understanding of pde-s. In particular, I don't see how category theory will help you understand the concept of entropy solution. Probably only physics could.

You might want to look at the notion of magnitude:

The magnitude of a metric space: from category theory to geometric measure theory by Tom Leinster and Mark W. Meckes