# How is Category Theory used to study differential equations?

I know that one can use Category Theory to formulate polynomial equations by modeling solutions as limits. For example, the sphere is the equalizer of the functions \begin{equation} s,t:\mathbb{R}^3\rightarrow\mathbb{R},\qquad s(x,y,z):=x^2+y^2+z^2,~t(x,y,z)=1. \label{equalizer} \end{equation} One could then find out more about the solution set by mapping the equalizer diagram into other categories. More generally, solution sets of polynomial equations (and more generally, algebraic varieties) are a central study object of algebraic geometry.

As differential equations are central to all areas of physics, I assume that there have been made a lot of attempts to generalise these ideas to solution sets of these. However, I do not yet have a lot of knowledge about algebraic geometry, topos theory or synthetic differential geometry. Thus I would be grateful if someone could explain roughly where and how Category Theory is used to study differential equations.

Can Category Theory really help to solve differential equations (for example by mapping diagrams of equations to other categories, similarly to how problems of topology are often solved by mapping topological spaces to algebraic ones in algebraic topology) or can it "only" provide schemes for generalising differential equations to other spaces/categories?

I am particularly interested in names of areas I have to look into if I want to understand this better. Also literature recommendation would be very welcome.

EDIT: I found a book by Vinogradov called Cohomological Analysis of Partial Differential Equations and Secondary Calculus where "the main result [...] is Secondary Calculus on diffieties".

However, the material is very deep and thus I am still not completely able to say whether these "new geometrical objects which are analogs of algebraic varieties" can be used to help solving PDEs or if they serve to structure the theory of PDEs or result in other applications I am not aware of. Thus further information would be very appreciated!

• On the other direction, categorical objects (Drinfeld associators) have been constructed using differential equations (the Knizhnik–Zamolodchikov equation). Feb 20 '18 at 10:19
• @NicolasHemelsoet that sounds interesting. Can you provide a reference? Feb 20 '18 at 10:32
• It helps you solve differential equations by being somewhat of a worse punishment. Namely, "if you don't solve this equation, you will have to do category theory!!!", which would make you work thrice as hard, and find a solution. Indeed, fear can be a great motivator. :P Feb 20 '18 at 13:43
• For a reference on the Knizhnik–Zamolodchikov equation, see Kassel's book on Quantum groups. It's at the end of that book. Feb 20 '18 at 14:42
• Here is an application of homological algebra to analysis : The snake lemma and extensions of functionals by Paul Garrett ! Dec 4 '18 at 13:37

There is this observation of Marvan A Note on the Category of PDEs that the jet bundle construction in ordinary differential geometry has the structure of a Comonad, whose Eilenberg-Moore category of coalgebras is equivalent to Vinogradov’s category of PDEs.

This 'synthetic' generalization of the jet bundle construction exhibits as the base change a comonad along the unit of the “infinitesimal shape” functor, which can then be shown to be the differential geometric analog of Simpson’s “de Rham shape” operation in algebraic geometry.

Edit In response to @Exchange's request for more recent developments, the work by Khavkine and Schreiber comes to mind, in a similar vein to the work undertaken by the authors above.

They were able to expand on the work of Marvan,and present a formal theory of PDEs in Synthetic Differential Geometry. Using Topos theory Khavkine and Screiber exhibit a synthetic generalization of the jet bundle construction. Under this generalisation the authors show that this is always equivalent to the Eilenberg-Moore category over the synthetic jet comonad.

They expand on this result by showing that whenever the unit of the “infinitesimal shape” $$\scr{S}$$ operation is epimorphic the category of formally integrable PDEs with independent variables ranging in some $$\Sigma$$ is also equivalent simply to the slice category over $$\scr{S} \Sigma$$. This yields in particular a convenient presentation of the categories of PDEs in general contexts.

For a more comprehensive account, see the paper from 2017

Best

Kevin

• Thank you Kevin, this is interesting. As these papers have been written in the 1980s, it would be important for me to know what the recent developments of these ideas are. Are there (more recent) books that describe how Category Theory can help to solve differential equations or how differential equations could be expressed in categorical terms/diagrams? Feb 20 '18 at 10:53
• I was able to find a book by Vinogradov: books.google.dk/books?id=XIve9AEZgZIC where "the main result [...] is Secondary Calculus on diffieties. There is a short diffiety page on the wiki: en.wikipedia.org/wiki/Diffiety and a link to a "diffiety institute": diffiety.ac.ru. However, the material is very deep and thus I am still not completely able to say whether these "new geometrical objects which are analogs of algebraic varieties" can be used to help solving PDEs or if they serve to structure the theory of PDEs or result in other applications I am not aware of. Feb 20 '18 at 11:53
• @exchange Forgive my absence, let me think about this and get back to you with an edit in around 3 hours from now. Best, Kevin.
– user284001
Feb 21 '18 at 8:54
• Physicist here; I would really appreciate some more concrete explanation. My (possibly completely mistaken) impression is that this answer says that categories are completely useless for solving what physicists call PDEs, and are solely used for defining formal analogues of PDEs. Is that right? Feb 21 '18 at 10:39
• @knzhou Hello. category theory in this context focuses on existence and uniqueness, of solutions to PDEs. One specific area where category theory plays significant role is the study of analytic solutions to systems of PDE's with analytic coefficients. Back in the 60's and 70's, Don Spencer formulated a cohomology theory (now known as Spencer cohomology) for such systems of PDE, expanded on by Sternberg, Quillen et al. So, in solving, no, studying the solutions, a yes!
– user284001
Feb 21 '18 at 10:46

I just realized that derived categories and techniques from homological algebra can help to solve differential equations. This is a very huge subject called algebraic analysis which use the tools of $D$-modules and sheaf theory. See this Mathoverflow question, this other MO question (first answer) and this book on D-modules which use heavily the langage of category. Also, Borel and Coutinho each wrote a book, you might be able to find some more informations about it.

• Wow, I would indeed not have thought that homological algebra can also help to solve PDEs. Thanks for this insight! However, as written in the MO-answers "these methods are very effective to study linear PDE but seems currently unable to deal with non-linear cases." which thus undermines the application to many theories in physics which have non-linear dynamical equations like the Navier-Stokes equation, the Einstein Field Equations or even Maxwell's equations in curved spacetime. Feb 20 '18 at 20:11

In another direction, one can use differential equations to solve a categorical problem.

If one wants to deform-quantize Poisson-Lie group, a crucial step is to deform an infinitesimally braided monoidal category into a braided monoidal category, and this boils down to find a Drinfeld associator, that is a power series in two (non-commuting) variables $X,Y$ verifying some conditions I won't write here.

It turns out that one can builds an associator from the solution of the equation $\frac{d \psi}{dz} = \psi \cdot \alpha$ where $\alpha = \sum_{a,b} d(log(z_a - z_b))t_{a,b} \in H^1(X_A, \mathbb R) \otimes \mathfrak t_A$ where $X_A$ is the configuration space of $A$ ($A$ is a finite set) points in $\mathbb C$, and $\mathfrak t_A$ is the Drinfeld-Khono Lie algebra.

This is very vast domain, a good start should be this ncatlab link defining the Drinfeld-Khono Lie algebra, and then browsing should give you a good overview. There is also good articles on the subjects, but unfortunately I don't know very complete survey, so best might be to gather different sources. A short but nice surey is given in these slides. Finally, a good reference is also the end of the book by Kassel, see the chapters "quantum groups and monodromy".

• Thanks for the detailled information! I can not accept it as the answer of the post because as you say, it answers "the other direction" but I like very much that you added it and will look at the resources that you provide to see if I could make use of it! Feb 20 '18 at 11:21
• @exchange : no need to accept, glad I could help a bit ! Feb 20 '18 at 11:23