Category theory to me is like a shiny new car to a fellow that can’t drive: it looks impressive and I want it! Category theory, though, has always been taught as little more than an elaborate game of connect the dots (there are the dots, “objects”, and then there are the arrows, the “morphisms”). Well that’s dandy, and I’m sure if I were a grad student in math I would immediately say “I understand everything now!” However, I’m not a grad student in math, so I won’t say it.
However, undaunted by this nauseatingly complicated game of connect the dots, I have attempted (more than once!) to worm my way through two books:
Categories and Sheaves by Masaki Kashiwara and Pierre Schapira, and
Topoi: The Categorical Analysis of Logic by Robert Goldblatt
Both are excellent, once you actually know category theory. They are, as far as a first year freshman is concerned, not all that great for an introduction.
Maybe it’s me, but I can’t really learn math that well that has no application. It seems to me to be little more than Platonic sophistry to list a series of definitions and at the end say “I’m a genius, woosh!” So I am looking into the introductions to categorical logic, hoping that it would actually having applications of categories in a concrete manner.
Luckily, I managed to find a good resource: some lecture notes from Steve Awody on Categorical Logic. I though they were very good notes, I learned quite a bit about categorical logic and type theory.
Still unsatisfied, I looked further for applications of category theory in programming…since I program, and I understand a little category theory, I thought it would be the perfect opportunity to learn more about the application of categorical logic. Coincidentally, I found this technical paper: The Logic of Message Passing. (If you work on a microkernel operating system, message passing is the bread and butter of the trickier parts of the kernel.) Perhaps one could think of a quantum system as a “program”, and the interactions as “passing messages”; this is just some random thought I had when I read the paper. If you haven’t realized by my about page, I think of random things from time to time.
Perhaps it would be worth-while to examine the relational interpretation of quantum mechanics like this; it has the possible advantage of context dependency. I actually looked into using domain relational calculus for relational quantum mechanics, and I think it might be worth while (an open query being “equivalent” to an eigenvalue problem, and closed queries being “equivalent” to some quantum logic like structure). Naturally, being pressed for time as an undergrad, I didn’t look into this approach very far…but it seems feasible.
15 May 2007 at 12:04 am
For applications to physics you could try Baez and Schreiber’s papers: Higher Gauge Theory 1 and 2.
15 May 2007 at 1:50 am
If you want a gentle and luring acquaintance with categories and their relevance in physics, read everything you can find by John Baez.
A good starting point for your point of interest would be
J. Baez, Quantum Quandaries: A Category-Theoretic Perspective
That demonstrates why you want to know about categories when you are interested in quantum gravity.
If you like that, look around on John’s home page for more. He is the best expositor you will ever encounter in your life.
A similarly cool exposition of the relevance of categories in quantum physics is
Bob Koecke, Kindergarten Quantum Mechanics
As the title suggests, categories are there to make complicated-looking things become much more simple. That categories themselves have the reputation of being difficult is an urban myth. To see why, look at the above two papers!
Bob also has these great lecture notes:
Bob Coecke, Introducing categories to the practicing physicist
We have a blog where we discuss categories in mathematics, physics and philosophy, and where we do talk about applications of categories to physics a lot: The n-Category Cafe.
I once tried to collect a list of places in mathematical physics where categories appear prominently: Cats in MathPhys
15 May 2007 at 4:20 pm
Beginners in category theory might want to look into
Arbib&Manes: Arrow, Structures and Functors – The Categorical Imperative. (c)1975 Academic Press, ISBN 0-12-059060-3,
for which no prerequisite knowledge other than sets and matrix theory should be needed.
16 May 2007 at 4:20 am
This describes 2-connections, such as they appear in BF-theory in terms of their parallel transport 2-functors.
For a discussion of why expressing connections in terms of their parallel transport functors may be a good idea, already in the non-categorified setup, you could have a look at Parallel Transport and Functors.
If you are interested in LQG you might be interested in the discussion in section 5.4 there of the “generalized connections” that people use in LQG, from this point of view.
16 May 2007 at 5:53 pm
Neither can I.
27 June 2007 at 7:27 pm
Is there math that has no application? I though that was a conceit of G. H. Hardy that was by now firmly discredited.
27 June 2007 at 8:55 pm
I’m no mathematician, but it seems like a lot of category theory is not completely understood (e.g. the “tensor product” of logical propositions, etc.).
Certainly others will disagree
27 June 2007 at 10:13 pm
Well, not every category has a “tensor product”. What you’re probably looking for is the notion of a “monoidal category” (shameless self-promotion: see my next few days’ posts).
On the other hand, every category with finite products is monoidal, and the category of propositions (with proofs as morphisms) does have them. You can use conjunction as your product and True as your identity and it all works out great.
28 June 2007 at 2:17 pm
I’m relieved that someone who knows category theory can correct me