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. 



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> **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? 
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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.



Category theory & geometric measure theory?
You might want to look at the notion of [magnitude](https://golem.ph.utexas.edu/category/2016/08/a_survey_of_magnitude.html):


[The magnitude of a metric space: from category theory to geometric measure theory](https://arxiv.org/abs/1606.00095) by Tom Leinster and Mark W. Meckes


