Category Theory Used in Physics.

[Sportin' Life]

I just have a quick, but odd question. Next term I will be taking a Category Theory class with a visiting professor from Moscow. The professor is known to have done most of his work in Physics. Does anyone have any experience using results from Category Theory applied to Physics?

It just seems like such a stretch to apply that kind of super-abstract Math with Physics, but there is a lot about Physics that I am sure I am unaware of.

 

[Fredrik]

I don't know much about these things, but you should click around a bit on [URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez's web site[/url]. You could start with this page.

 

[Tom Gilroy]

I'd first like to mention that the definition of a category that appears on Baez's page is incorrect. A category does not consist of "a set of objects and a set of morphisms." A category consists of a class of objects and a class of morphisms (any set is a class, but not all classes are sets). If you restrict the definition of a category to only having a set of objects and a set of morphisms, some of the most important examples of categories would not be included. Sorry to be pedantic.

I do know that there is an effort to use category theory to give a rigorous definition of a conformal field theory. The aim is to try to construct the correspondences between the algebraic, and geometric structures that are involved as functors between certain categories. I agree that constructing these correspondences is very important, however I'm not sure that category theory is the right way to do it.

I'm afraid I don't have any more information on the matter, I'm sorry I can't be more help.

 

[Landau]

I'd first like to mention that the definition of a category that appears on Baez's page is incorrect. A category does not consist of "a set of objects and a set of morphisms." A category consists of a class of objects and a class of morphisms (any set is a class, but not all classes are sets). If you restrict the definition of a category to only having a set of objects and a set of morphisms, some of the most important examples of categories would not be included. Sorry to be pedantic.

It's not only pedantic, it's not true. You are forcing your choice of foundations upon us. Apparently you like to work with sets and classes. Some people don't. They like to fix a universe in which everything takes place, all whose members are sets. They are in good company: Grothendieck. Nothing wrong with defining a category to have a set of objects and a set of morphisms. Again, then you are in good company, since Mac Lane does this in "Categories Work".

 

[Tom Gilroy]

It's not only pedantic, it's not true. You are forcing your choice of foundations upon us. Apparently you like to work with sets and classes. Some people don't. They like to fix a universe in which everything takes place, all whose members are sets. They are in good company: Grothendieck. Nothing wrong with defining a category to have a set of objects and a set of morphisms. Again, then you are in good company, since Mac Lane does this in "Categories Work".

I was aware of this, and reconsidering my post, I probably should have said that the definition of a category on Baez's page isn't accepted universally, and given the class based definition as an alternate option.

My intention was merely to point out that the set based definition does not cover all possible cases one may be interested in. If the Grothendieck approach is sufficient for your purposes, by all means use it. I really have no interest in forcing a particular choice of foundations upon anybody.

 

[HallsofIvy]

It's not only pedantic, it's not true. You are forcing your choice of foundations upon us. Apparently you like to work with sets and classes.

I had a professor who referred to such categories as "kitty-gories"!

Some people don't. They like to fix a universe in which everything takes place, all whose members are sets. They are in good company: Grothendieck. Nothing wrong with defining a category to have a set of objects and a set of morphisms. Again, then you are in good company, since Mac Lane does this in "Categories Work".

 

[Simon_Tyler]

The Baez & Lauda (draft) paper "A history of n-categorical physics" can be obtained from http://ncatlab.org/nlab/show/John+Baez

 

[Sportin' Life]

Thanks for the replies. Pedantry or not aside, sounds like some interesting stuff is out there. I'm actually most hoping for a strait course on pure abstract categories, but it is always interesting to figure out what some of these guys are actually doing with it all.

Cheers.

 

[Hurkyl]

It just seems like such a stretch to apply that kind of super-abstract Math with Physics, but there is a lot about Physics that I am sure I am unaware of.

It's only abstract if you use it abstractly. Matrix algebra, for example, is an example of an abelian category.

 

[Fredrik]

Why would those "kitty-gories" be interesting? (I like that name). Aren't the most interesting categories classes of mathematical structures, like the class of all groups (with group homomorphisms as arrows)?

 

[micromass]

Many categorical constructions, like triangulated categories and derived categories arise naturally in string theory. Other construction, such as monads, are quite useful in computer science. So, even though category theory was intended for mathematicians, it does find it's use in broader contexts...