# 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.

• Category theory hasn't really penetrated analysis, so I doubt it. – Harry Gindi Jan 29 at 21:22
• @HarryGindi, I'd disagree with a blanket claim about the irrelevance of category theory to analysis. I include this in my graduate real analysis course the point that the "correct" topology on spaces of smooth functions is demonstrably not a matter of whim, since it must be a (projective) limit of $C^k$ functions. Even more primitively, the "coarseness" of the product topology is explained by its unequivocal categorical definition. The topology on test functions must be the (strict) colimit topology. So I think the viewpoint, if not big theorems, of category theory is very relevant. – paul garrett Jan 29 at 23:13
• see eg Michal Marvan A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno ncatlab.org/nlab/files/MarvanJetComonad.pdf for something on the PDE side, it might be interesting to push this in the direction of geometric measure theory. – David Roberts Jan 29 at 23:28
• @HarryGindi may not have penetrated to the extent it has algebraic geometry, but that is not to say that it hasn't got some underappreciated connections. – David Roberts Jan 29 at 23:29
• My bet is that category theory will not help you with your hyperbolic equations. – Liviu Nicolaescu Jan 30 at 7:39

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

• Wow, looks interesting, thanks! I need of course some time to digest it. Rather than (what I understand for) Geometric Measure Theory, the paper you mention might be relevant for analysis on metric spaces and e.g. non smooth differential geometry. Do you know if something of this kind has ever been done (or tried)? Is it worth? Thanks a lot for your interesting answer. – Romeo Jan 30 at 9:05
• Sorry, I don't really know a lot about the area. Reading your question I was just reminded of an n-category post I had seen a few days earlier: golem.ph.utexas.edu/category/2019/01/… All my knowledge comes from following some of the links in that post. – Thomas Kalinowski Jan 30 at 22:50

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.

• Thanks for your (in some sense expected) answer. My feeling is that CT might be potentially useful to (re)formulate (P)DE's in some abstract and neat language, but - as far as I got - it is not suitable to fully understand and detect the structure of solutions. E.g. entropy solutions are particular distributional solutions to a hyp cons law: while distributional solutions are generally infinite, they happen to be unique and to have some fine structure - some kind of BV regularity: I do not see a proper way to cast this in the language of CT, nor I see some kind of usefulness. Thanks. – Romeo Jan 30 at 9:14
• Have you look at viscosity solutions? – Liviu Nicolaescu Jan 30 at 16:30
• Nope. Do you think there is some link between them and CT? I have heard people talking about them in conferences a few times, but mostly in talks about other kind of equations - I never saw them related to Hyp. Cons. Laws (linear or nonlinear). I'll give a look, thanks. – Romeo Jan 30 at 18:20
• Maybe you know this paper . annals.math.princeton.edu/wp-content/uploads/… – Liviu Nicolaescu Jan 30 at 20:01
• Ah yes, of course! I know that paper (albeit I have never seriously studied it). I have misunderstood your previous comment and I apologize for that. Actually I thought you were referring to something analogous to this concept of solution. I realized only now you were referring to viscosity solutions in the sense of Bressan-Bianchini. – Romeo Jan 30 at 20:49