The second paragraph of the Wiki entry on “Category theory” begins:

*Category theory has several faces known, not just to specialists, but to other mathematicians. “Generalized abstract nonsense” refers, not entirely affectionately, to its high level of abstraction, compared to more classical branches of mathematics.*

(Emphasis added.)

I don’t think that this explanation is that helpful.

The most abstract mathematical tools are generally the most valuable once they are widely understood and applied to appropriate problems. It’s still early days for category theory.

I also keep getting confused between functors and functions (because the symbols used are similar and I’m more used to thinking about functions) when I read about it, not to mention the different morphisms.

One thing is to keep working on it and to try to find successful ways of applying Category Theory to explaining preon or some related theory, about how the different sets of quantum numbers (including weak and strong charges, masses, etc.) of fundamental particles of physics are really related to one another by morphisms.

You don’t necessarily have to have a physical mechanism explaining how the morphism physically occurs. It can just be a mathematical representation of what happens, and what the relationships between different fundamental particles really are.

I think the key thing here is the relationship of leptons to quarks. Quark properties are only known through composites of 2 or 3 quarks, because quarks can’t be isolated. The fact of universality, e.g., similarities between lepton and quark decay processes

*muon -> electron + electron antineutrino + muon neutrino*

for leptons and

*neutron -> proton + electron + electron antineutrino*

for quarks, hints that quarks and leptons are surprisingly similar, when ignoring the strong force. (My preliminary investigations on the relationship are here,

here and here.)

The problem is how much time it takes to apply new maths to solving these physical problems.

I like the fact that although you are a mathematician, you are free to go to physics conferences and study that stuff, at least as far as your time allows. The standard mathematical tools of particle physics, like Lie and Clifford algebras, aren’t focussed at modelling the morphisms between different fundamental particles (transformations between leptons and quarks obviously haven’t been observed yet, but they probably are possible at very high energy in certain situations). That would appear to be an ideal area to try to apply Category Theory to, because you have a table of particle properties and just have to fund the correct morphisms between them. That’s very important for trying to understand how unification can occur at high energy, and could lead to quantitative, falsifiable predictions (the old unification ideas like supersymmetry are not even wrong). I hope to learn a lot more about Category Theory.

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