Categorical Logic, Message Passing, and Relational Quantum Mechanics

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.