The "applications" are speculative at best. Chemists and biologists don't seem to care. This is really just category theory for the sake of category theory.
Petri nets are a quite general model of concurrency that has also found use in many places in computer science. They are used in model checking, and also as the foundation for giving a semantics to BPMN, for example.
I think it's not unreasonable to consider Petri nets as a generalization of automata (finite or infinite) with concurrency. Just like automata, they find applications in many places depending on how you further extend or restrict them.
I honestly doubt it will have any practical applications in biology. Computation chemistry and synthetic biology run on differential equations, anything discrete is generally useless. Perhaps it will have some use in a couple of decades when synthetic biology progresses beyond brute force simulations but at the current level it is the wrong abstraction.
You can turn Petri nets into stochastic nets using some formal procedures. Stochastic nets have a "master equation" which spits out a system of PDEs representing reactions where many many things happen at the same time, and so it makes more sense to think in terms of concentrations etc.
We are working to capture this categorically as well. We are perfectly conscious of how a discrete system won't help you if you have an Avogadro number of things going around. Just give us time. :)
you have to understand that the real power behind this work is not the Petri nets but that fact that they are described so abstractly that they can take on different shapes, such as stochastic nets, or coloured nets, or...
It puts the different models on the same footing (and hence allowing you to unify tools and do more work)
Also, I'd even argue that biology/chemistry needs to embrace categorical methods and we would see some deep discoveries.
The differential equations are themselves in many cases an abstraction over statistical numbers of discrete events. Arguably, in those cases, they are the wrong abstraction - a thesis which can only be proven by demonstrating the superiority of a different abstraction. Has one been demonstrated? Not yet, for most purposes. Is this a promising avenue of study? IMO, yes.
Not at all. We have been applying category theory for the past few years to software and systems design and this paper in particular came as a result of those implementations.
Soon these methods will be usable by many people for all kinds of purposes. You can think of cryptographic contracts, business process execution, game theory, functional reactive programming, quantum protocols and digital or analogue electronics.
The only difference is what category the diagrams live in, or put another way, which semantics are you assigning to thee boxes and wires.
"Mathematical foundations" are not always useful to practitioners – or even at all. Algebraic quantum field theory is a good example of the latter, to a large degree [0].
The fact that workers in the field with the presumed "applications" don't seem to care is pretty telling. My guess is that working at a level of generality where subject-specific insights are suppressed automatically limits the usefulness of the work.
It's funny, this is exactly what people have been saying of Categorical Quantum Mechanics and ZX calculus for 14 years, until they didn't, and now ZX calculus is used to draft quantum protocols and businesses like Cambridge Quantum Computing are investing heavily in it.
The fact that mathematical foundations don't have applications now does not mean they won't have applications in the future. Gregorio Ricci-Curbastro's work was also considered pretty useless until Einstein decided to build General Relativity over it. If it were for opinions like this one, progress in many field would be incremental at best.
Your characterization of the development of coordinate-free differential geometry is incorrect. Intrinsic geometry was not a seemingly "useless" generalization; it was motivated by concrete and specific problems about surfaces and mathematical physics. For example, one of Riemann's papers was literally titled "A mathematical work that seeks to answer the question posed by the most distinguished academy of Paris." (Perhaps not literally, given that's a translation, but you get the point.)
I don't know anything about ZX calculus, but if people are using it to solve real-world problems, that sounds good to me. What I object to is pure mathematicians giving "applications" of their work that aren't useful or real. And when the authors allude to applications in chemistry and biology, and no chemists or biologists are doing anything with category-theoretical analyses of Petri nets, I think it's reasonable to point this out.
Do you expect something that's useful to be immediately acknowledged as such in the field, much less adopted?
Human cultures are more complex than that. Capitalistic pressures to produce results have had a noted impact on research and development. There will likely need to be a practitioner willing to do the field work to bring about the big "Aha!" for that field. Based on the silo'd history of so many fields, it's a safe bet to say that may be necessary in every field where CT can be applied.
Yes, I expect practitioners to care about something useful, assuming the result has been properly written up and explained in a way they can understand. Scientists want good results and will adopt new tools if you can make a case that they will help them publish impactful papers.
Again, it's pretty telling specific applications aren't mentioned, and instead there's just vague gesturing. How can we expect scientists to "do the field work" necessary to bring the applications to fruition if we can't even tell them what those applications are?
Also, regarding your first sentence, the vast majority of influential mathematics was indeed "immediately acknowledged," because it made progress on some important problem. Perfectoid spaces are a recent example.
Nice, if everyone thought like you do, number theory wouldn't have existed, elliptic curves along with it, and hence much of public key cryptography as well. Indeed, number theory has been exquisitely useless for a couple of millennia, give or take.
I feel you are one of those people that just hate category theory because they consider it too abstract and useless for any purpose. We get that a lot, from a lot of people. Still, since Grothendieck, categorical methods are absolutely central in modern algebraic geometry and topology, and this centrality is only destined to grow, because CT is the best theory we have to manage emerging complexity in describing systems. In my opinion, the approach of "if it's not immediately useful then it's useless" really will lead you farther and farther away to understand modern developments in applied mathematics.
I am a practitioner of life sciences. I use category theory to understand what I'm looking at around me in the world. I don't know if I'm even applying it properly and it's still been incredibly useful to me. I've developed basic principles for myself I use in daily life, to handle my emotions, and for raising my child.
Practitioner apathy and willingness to dismiss things as not useful can reflect a lack of intellectual humility in their cultures and lives.
The generality is useful if one practices seeking subject-specific insights from the generality. It sometimes leads to new options I haven't even thought of or leads me to reconsider decisions.
"Useful" is a judgment and most judgments like this are made on too short of a timeline. What's actually happening is someone hasn't found a use for the thing labeled as "useless" and is blaming it on the thing, themselves, or where the thing came from. Instead, a growth mindset focused on learning, accountability, and responsibility might be "I did this thing and the outcome I wanted didn't occur. I haven't yet learned how to use it for said outcome."
