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!
Best Answer
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
Let me know if you need any more info,
Best
Kevin