John Baez's "This Week's Finds in Mathematical Physics (Week 209)" is about one of John's favorite topics, namely category theory.
Incidentally, the previous week 208 was discussed here.
You know, category theory is the most abstract and "universal" part of rigorous mathematics, and it is also known as the (generalized) abstract nonsense. Actually the supporters of category theory don't view this name as an insult but rather as a cool compliment. Many of us have had a period in which the mathematical rigor was almost irresistable.
Category theory involves various theorems about sets, categories, functors, homomorphisms, and other abstract objects. Instead of saying "we perform an analogous procedure with a different kind of theory", category theorists give you a framework that translates vague words such as "analogous" into a mathematically rigorous structure.
This approach is meant to unify all of mathematics and all of rational thinking. You would think that the result must be very deep and hard to understand. But the actual results and tools of category theory are often closer to ordinary diagrams (combinatorial graphs) with arrows in between the objects - similar diagrams like those used by the people from the commercial sector in their PowerPoint presentations.
The mathematically oriented part of the string theory community is mostly excited by category theory. The so-called "derived categories (of coherent sheaves)" have been proposed as a generalization of K-theory to describe D-branes of various dimensions. Unlike K-theory that only knows about the allowed conserved charges of D-branes in various backgrounds, category theory should also know - and perhaps knows - something about their dynamics, for example the points in the moduli space at which the D-branes become (un)stable or (un)bound.
If you want to see some deep papers in this direction, see e.g.
The role of category theory in physics can therefore be described as a "progressive direction" within string theory. That's why at sci.physics.research, Aaron Bergman and Urs Schreiber replied to John Baez's week 209 that "he should be careful because he is secretly starting to work on string theory". ;-)
Nevertheless I think it is still fair to say that the dominant, physically oriented part of the string community is skeptical about the importance of the methods of category theory in physics - and, of course, for most of us the reason is simply that we don't understand where this approach came from. I've asked the same elementary questions to many people who've been trying to explain me derived categories - some of them with some success, most of them with no success whatsoever:
- Are these notions and statements of category theory something that you can prove - or at least check in many situations - to be valid for string theory as we know it, or is it just an unproven conjecture that derived categories describe D-branes?
- Or is category theory just a different language to describe the same physical insights that we know anyway, using other means?
- Or is this categorical approach meant to redefine string theory? How do you prove that such a "new" string theory makes any sense?
Probably the most serious conjecture of current high-energy theoretical physics is that string theory, once understood completely, describes the whole Universe around us exactly. Again, it's important that it is just a conjecture, albeit a very attractive one and one with a lot of circumstantial evidence. String theory itself is not just a conjecture, but rather a seemingly consistent mathematical framework.
Once we accept string theory as an objectively existing mathematical structure, a structure that we treat as a part of "generalized physics" - which is of course what all string theorists are doing every day - we can ask a lot of questions about its properties. But it's important that even in this case, the statements about string theory are just conjectures, and they need to be proved or supported by evidence, otherwise they're irrelevant and "wrong", in the physical sense.
I always feel very uneasy if the mathematically oriented people present their conjectures about physics, quantum gravity, or string theory as some sort of "obvious facts". One of the fascinating features of string theory is that its objects and investigations, even though they've been partially disconnected from the daily exchanges with the experimentalists, remained extremely physical in character. All of the objects that we deal with are analogous to some objects in well-known working physical theories, to say the least.
You may imagine that after 20 years and 15,000 papers, theoretical physicists who don't interact with the new experiments too much will end up with a completely abstract structure that does not resemble observable physics too much. It's not the case of string theory - which is a deep framework to unify all good features and facts about gauge theory, gravity, elementary particles, Higgs mechanism, confinement, and so forth. String theory is the most conservative among all the extensions of quantum field theories that go beyond the standard framework of QFT.
Well, category theory is exactly the type of science that would invalidate the previous paragraph.
Don't get me wrong: I do believe that string theory will eventually be formulated with very abstract rules - rules that will imply the features of the well-known systems and backgrounds with a geometric interpretation as an important special case. But I just don't see that this particular approach, diagrams with arrows from category theory, are getting closer to the right answer. If something is abstract, it does not have to be correct, and this particular thing just does not smell correct.
What I can imagine is that some insights of category theory will be useful to understand some things about D-branes and M5-branes - categories are possibly relevant for the "gerbes" - but it remains an open question whether category theory as we know it today will ever become a useful language for physics. No doubt, the mathematicians among us will try to impose their culture on the rest of the community. But I feel that this is the influence that would move the string theorists away from physics.
It's great if some people study string theory because of purely mathematical motivations, and I don't want to question the meaning of mathematics. What I want to question is the self-confidence of many mathematicians with which they want to use their mathematical ideas directly in physics. String theory is first of all a physical theory, and it should be studied because of physical motivations - the primary physical motivation is to locate the right ideas and equations that describe the real world.
Category theory has been used by many to achieve completely wrong physical conclusions - for example, by considering the "pompously foolish" quantization functor, many people have claimed that everything that happens in a classical theory has a counterpart in the "corresponding" quantum theory. This is just a lousy idea that we know to be wrong because of hundreds of different reasons - yet, the people who have a religious respect to the diagrams in category theory view this incorrect map as a "self-obvious dogma". Incidentally, this dogma is one of the reasons that makes the category theory community and the loop quantum gravity community correlated. It may be nice to be rigorous, but it's always more important to be correct: if the specific kind of rigor leads us to stupid conclusions in physics, we should avoid it.
Categories and "continuous vs. discrete"
Urs Schreiber on sci.physics.research describes "categorization" as "stringification" - which is the step from "quantum mechanics" to "conformal field theory". In other words, "categorization" is nothing else that adding dimensions to your auxiliary "worldvolume". Well, adding dimensions is not such a mysterious thing. Moreover, we know that sometimes it can be done and sometimes it cannot. Even if it can be done, some things are preserved, but others are completely changed.
Category theory often resembles linguistics (or even postmodern literary criticism): it is a science about arrows between different objects and about creating new objects from these arrows, but it does not really care too much whether the objects exist and what are their real properties. Although "categorization" is making things "more continuous" (e.g. by adding dimensions; making quantization), the actual diagrams that category theorists eventually talk about are extremely discrete objects.
If a complete discrete diagram tries to encode an extremely continuous, infinite-dimensional object, we should always carefully ask whether we know that the non-trivial, infinite-dimensional, highly continuous object really exists. Its existence is never guaranteed and drawing simple-minded diagrams from otherwise "rigorous and deep" category theory can't help us to decide about the "hard questions" associated with quantum field theory, string theory, or any other objects that require some sort of "infinite-dimensional continuous" structure. If we create physical theories as complex as quantum field theory or more, the difficult part is always the task to make the highly continuous, infinite-dimensional objects and expressions meaningful. On the other hand, one can invent zillions of different discrete rules - such as the rules for n-categories - and it is simply not guaranteed that a set of combinatorial, discrete rules will be relevant for anything in physics (which includes string theory).
In my opinion - and not only mine - it is important for all of us to realize that we have not yet solved the fundamental physical goal of string theory - to predict and confirm new phenomena and facts about the real world. And category theory does not seem to help us in this goal, I think. Do you disagree?