# (How) is category theory actually useful in actual physics?

An answer to a recent question motivated the following question:

(how) is category theory actually useful in actual physics?

By "actual physics" I mean to refer to areas where the underlying theoretical principle has solid if not conclusive experimental justification, thus ruling out not only string theory (at least for the moment) but also everything I could notice on this nLab page (though it is possible that I missed something).

Note that I do not ask (e.g.) whether or not category theory has been used in connection with hypothetical models in physics. I've read Baez' blog from time to time over the decades and have already demonstrated knowledge of the existence of the nLab. I am dimly aware of stuff like (e.g.) the connection between between Hopf algebras and renormalization, but I have yet to encounter something that seems like it has a nontrivial category theoretic-component and cannot be expressed in some other more "traditional" language.

Note finally that I am ignorant of category theory beyond the words "morphism" and "functor" and (in my youth) "direct limit". So answers that take this into account are particularly welcome.

• Category theory serves as a formal setup with which one can organize ideas. Symplectic or Poisson manifolds are organized in a category, whose groups objects are the Lie-Poisson groups; the representations of various types of groups and of other type of symmetries organize themselves in categories, and using the language of category theory can be very helpful in expressing very complex ideas, and this is extremely helpful when dealing with complicated mathematical objects that physicists need to deal with; and so on and on. Is this "actual physics"? Well... Yes. Aug 7, 2010 at 20:49
• @Mariano: +1, and deserves to be an answer. I would like to bicker that (as you probably know) Lie-Poisson groups are not "group objects in the category of Poisson manifolds", but something related an a little weaker. Namely, the "product" of Poisson manifolds is not a categorical product, and the "inverse" map from a Lie-Poisson group to itself is not a Poisson map. A Lie group is a group object (in the classical sense) in the category of manifolds, but defining "Lie-Poisson group" categorically takes much more care. See e.g. arXiv:math/0701499v1 Aug 7, 2010 at 21:12
• Poisson-Lie groups are group objects in the category of Poisson manifolds, and anyone who tells you otherwise is using the wrong definition of group object. The inverse map is always an anti-map when anti-maps make sense (e.g. Hopf algebras are group objects in algebra^op and the inverse map is an anti-algebra map). Aug 7, 2010 at 21:47
• I realise that "actual physics" is defined in this question and hence there is a reasonable chance of answering objectively whether X is "actual physics". However I cannot but point out that in my already more than two decades as a mathematical physicist I have come across attempts to define (one could say, in fact, restrict) what Physics is more times than I care to remember. And in all this time I have come across but one satisfactory answer: "Physics is what physicists do." It may seem circular, but not completely, since the field is in constant evolution. Aug 7, 2010 at 23:56
• @Kevin: Certain fractional Hall effect systems are now believed (based on not yet completely conclusive experimental evidence) to be described effectively by Chern-Simons theory. In particular, by su(2) at some small level. Aug 8, 2010 at 14:44

Fusion categories and module categories come up in topological states of matter in solid state physics. See the research, publications, and talks at Microsoft's Station Q.

• Theories of anyons are very much related to braided fusion categories and 2-d topological quantum field theories. You could look at the survey article rmp.aps.org/abstract/RMP/v80/i3/p1083_1 (this is the only recent survey article I know of for this field ... if you have any other suggestions, I'd like to see them as well), and there are also some popular articles linked from the Microsoft Station Q website above. Aug 7, 2010 at 21:28
• Continuing my comment ... depending on who you talk to, either the $5/2$ fractional quantum Hall effect has been experimentally proven to be generated by anyons, or has not quite been proven experimentally. Topological insulators are a recent experimental discovery with related behavior. So anyons/braided fusion categories/TQFT should meet your "actual physics" criterion. Aug 7, 2010 at 21:40
• @Peter: Thanks...it's funny, I had that RMP as something that I was planning to study but hadn't yet. I note that the authors pose--and then explicitly address--the question "Why is it necessary to invoke category theory simply to specify the topological properties of non-Abelian anyons?" Aug 7, 2010 at 21:42
• @Steve: Your quote just made me realize that I should have specified non-abelian anyons in my comment. Abelian anyons are much more widely realized experimentally, but they correspond to trivial examples of braided fusion categories, and thus you don't actually need category theory to understand them. I should probably also mention that the 12/5 fractional quantum Hall effect is also believed to be generated by non-abelian anyons, but the experimental confirmation is much farther away because it is experimentally harder to work with. Aug 7, 2010 at 22:21
• @Peter: Zhenghan Wang's CBMS monograph "Topological Quantum Computation" is now available from the AMS and might fit the bill as a recent survey--particularly chapters 6 and 8 deal with this subject. Aug 8, 2010 at 0:05

Categories (and higher categories) seem to be a good way of expressing the locality of the path integral in physics. In particular, it is the idea of gluing of local structures that is important. This line of thought leads to the axiomatization of (parts of) various QFTs, with the most success in topological and conformal field theories. This idea has its origins with Atiyah, Segal, Baez-Dolan, Freed and probably a ton of other people I'm forgetting. Braided fusion categories as in the previous answer are an example of this in three dimensions. Most recently, there's Lurie's classification of TQFTs in all dimensions in terms of $(\infty,n)$ categories.

Jürgen Fuchs, Ingo Runkel and Christoph Schweigert have developed a complete treatment of Rational Conformal Field Theory based on algebra in braided tensor categories. They have applications to string theory as well as to statistical physics, most importantly to conformal defects and so-called Kramers-Wannier-dualities.

See J. Fuchs, I. Runkel, C. Schweigert: TFT construction of RCFT correlators I, II, III, IV, V for the full story or, for a summary, Schweigert's 2006 ICM talk Categorification and correlation functions in conformal field theory.

Monoidal category theory (especially dagger-compact categories) and its associated string diagram calculi are a very useful language, especially for quantum mechanics, quantum computing, and QFT. See this nice article by John Baez & Mike Stay for some of the details. It seems that quite a good deal of basic principles in quantum mechanics, such as the no-cloning theorem are really just statements about monoidal categories. And Feynman diagrams are essentially string diagrams for monoidal categories of representations.

Topos-theoretic QFT is a thing as well, though I honestly don't know anything about this approach.

It seems that background independent path integral can be rigorously defined through category theory, see https://arxiv.org/abs/2201.08963v5.

In the paper “Spin networks in gauge theory” (https://arxiv.org/abs/gr-qc/9411007), Baez gave a nice construction of spin networks states for gauge theory, which essentially gives a rigorous definition of path integral. The original construction of Baez is an analytical one. It turns out that the Baez construction has a purely categorical generalization, which is the framework of causal-net condensation illustrated in the paper "causal-net category".

• Could you indicate where in the paper this is addressed? (looks like cool stuff overall!) Aug 9 at 2:52
• @ Alec Rhea Causal-net condensation is a categorical generalization of Baez's construction of spin network states for gauge theory, which gives a rigorous definition of path integral. I need write a new paper to fully explain these ideas. Aug 9 at 8:34

First things ought to come first. And the first things in mathematics is arithmetic and Euclidean geometry. They were the first systems to be fully charactised first. Arithmetic, well before Euclid and Euclidean geometry by the eponymous Euclid. I say characterised rather than axiomatic as arithmetic was so simple that no-one bothered to axiomatise it until Peano in the early 20C. Now whilst these are thought as the iconic examples of mathematics, I would argue that they were the original physical theories that were the first ones to be axiomatised. We see examples of this being done today. For example, as one of the posts here mention, rational CFTs have been fully axiomatised.

Thus has category theory has anything to say about arithmetic? Well, traditionally, multiplication is seen as iterative addition. Category theory provides an alternative description of this as showing the coproduct (addition) is dual to the product as the naming suggests. This is new perspective on one of the oldest ideas in arithmetic is one reason why I  was attracted to category theory. But this notion doesn't stop there. The notion of a product and dually, the coproduct,  are available in any category as the universal property characterising these properties is categorical. Whether they exist is a seperate question. Thus we can ask whether they exist in categories of groups, rings, modules and algebras as well as other more esoteric categories. And they do and this gives a systematic way of proceeding compared to the adhoc, heuristic way they were first thought up. Now all these mathematical structures are important in physics: groups are important in physics as the correct way of thinking about symmetries and this is then naturally generalised to groupoids. Bundles are important in physics as a general description of field theories and their sections organise themselves into a module over the ring of functions one the base manifold. The infinitesimal description of a Lie group is a Lie algebra.

On a more sophisticated note, TQFTs are most elegantly presented via category theory. The original axioms were set out by Atiyah in the late 80s I believe. A field theory is topological when it has no local dynamics/dof and so the only dof are global.