I would say that Category theory as a formal system is primarily designed to formalise mathematical concepts. This is why I think that Lawvere's and Schanuel's undergraduate textbook 'Conceptual Mathematics' is a perfectly well-chosen title. I am using it too, when teaching Category Theory, as it is much more informative than 'Category Theory' when it comes down to the main relevance of Category Theory to mathematicians. (Of course, I am not suggesting to rename the field...)

I am not seeing how it would be suitable as a language of a metatheory to unify the description of the laws of Physics, though:

1. Theories of physics are fundamentally about the description of nature. The objects they are concerned with do not form categories, but based on your chosen mathematical model just live in certain categories. Hence there is a lot of "bulk" that you would not really want to concern yourselves with.

2. As a physicist you are only worried about having a suitable & convenient mathematical representation of your theory to be able to discuss it and to do calculations with to derive predictions of your theory. So using categories as a language, you are kind of modelling two steps away from what Physics is fundamentally concerned with.

3. Talking about a unifying principle for Physics, one could argue that we do have that already: the action principle and gauge theory (+ quantisation). One might argue that this has not helped us with truly addressing the issue of devising a unified framework for all of the fundamental laws of Physics, but (abstract) differential geometry (combined with functional analysis) is still a mathematical language that is much closer to what is used in Physics as of now. What would justify such a huge leap in abstraction?

Of course, I do not mean to say that category theory is not useful for physics apriori. There are interesting use cases out there. From what I am aware of Category Theory is mostly used to provide some form of semantics for mathematical formalisms physicists would like to have. I don't think it is a suitably formal framework to discuss Physics per se.

I am sending you an article on this subject.

Mathematical category theory has indeed emerged or been designed to become a unifying framework for different, potentially all mathematical theories. (When associativity is contested some theories might not fit though, but this is another question.)

Most physical theories are simply a mathematical theory where you add some Lagrangian principle for formalizing evolution. In so far it is not surprising that category theory also generalizes them.

Conceptually category theory is indeed a good choice because it allows better to recognize what is really structurally essential, and because in category theory one stays at the lowest level of complexity.

Another remark will go further : The name of category is not chosen arbitrarily for it is a reference to Aristotle's categories. I am little experienced on this subject but it seems to me that Categories in this very general sense do not only describe physical settings but more generally ontological relationships. Categories in this general sense have been a fundamental concept in logical, philosophical and theological arguments. Indeed only from the 1950s on they have taken the mathematical strict form now known.

There have been attempts to give a more formal, categorical description for language as in Joachim Lambek's calculus. What works well for grammar might show its limits at a semantic level. Poetry will always rebel against a categorization, formally and at the level of meaning.

I believe your sentiment is that Category Theory is so general that it might if applied to physics provide an insight into the overarching principles of the science. But Category Theory is not going to give you any insight that you do not already possess. It might provide a convenient notation for expressing that insight. Using Category Theory to navigate physics without a physical understanding would be like sailing a yacht without a keel.

How does physics ‘insert’ into category theory ?

I will make an attempt: physical states constitute objects of a category and transitions between physical states are the corresponding arrows. At this level we do not need to speak about real numbers, vector spaces, topology or differential geometry. If physicists do so, is that they need at a certain point to resolve equations and they need theorems to be applicable. One can make the choice to adopt the strongest mathematical theory at reach to describe the system, but then it might hide what is really essential in a physical relationship. I would rather proceed the other way round : Add only a mathematical feature when it is really needed.

It is true though that physical discoveries often are made in an over-determined system and then progressively generalized. It introduces distinctions between physical laws which, if expressed at the most general level could be seen as particular cases of a same law.

Ça dépend du modèle opératoire (le déterminisme) comme quelque physique théorique (l’unité-pression) et de la viscosité (non inductive à la la théorie pure de la mécanique nouvelle).

cf. Henri Poincaré, Science et Hypothèse. Jacques Hadamard, Psychologie de l’invention dans le domaine mathématique.

Dear @Cyprien Saito

Could you further develop your answer please and make explicit your additional hypotheses ?

Additional hypothesis ? Yes, firstly my researcher’s life started as a topology specialist in functional analysis, however, Bourbaki situation was rather close to my position in 1980s. Strangely enough, Weil’s « Sur les courbes algébriques » was beyond my vision or I found rather Grothendieck’s isolated situation in functional analysis was « sympa » , for instance, read Singer’s giant books on Bases in Banach dpace 3 vols. But I found so later problematic of Weil’s « courbes «  after the confused situation of topological field theory. Then I would like to infer Weil’s « courbes «  here gor explanatory dimplification.

page 66, logarithme replacement by zeta, obviously this situation was changed. Polylogarithm replaced the position of zeta priority in algebraic number theory by « correspondence » (Weil’s expression). I can emphasize easily on this replacement itself, it was nothing but categorical number theory if the replacement held something important for mathematical physics.

I always think of category theory as a way of making precise the structures that appear in different areas of mathematics in a visualisable way. Insofar as those structures are useful in physics (as many concepts of geometry are), then category theory and generalisations of category theory (such as higher category theory) are useful to provide structural clarity. However, I do not think category theory is unique in this regard, and I personally like graphs and trees and the oriented shapes that appear in geometric algebra.