Category theory & geometric measure theory? My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research interests, too. 

Q. Is there any link between these areas (GMT, PDEs) and category theory? Could categories be useful to study, e.g. fine properties of BV functions? Or to understand the concept of entropy solution to a non-linear conservation law? 

I have looked for similar questions, but I have not found anything as "explicit" as I want. I am not interested into possible definitions of category theory, nor I am looking for some apologies of this area or of that area (everything is math and deserved to be studied). What I would like to know is if it is possible to frame some "fine" definitions/theorems of the areas I am working in by means of the language of CT.
 A: 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.
A: 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
