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 erutufon|nofuture Higher dimensional algebra

16th February 2004   #1
solve et coagula

Join Date: Jan 2003
Location: sussex
Posts: 6,095
Higher dimensional algebra

Quote:
 In article <9v5fpa\$6lo\$1@glue.ucr.edu>, Miguel Carrion writes >You don't use category theory to study groups. You use it to study >group homomorphisms. Nobody says "this is an object in the category of >groups" and stops at that. What they will say is "this map is a >morphism in the category of groups" and lots of properties of the map >will follow from that. I can see that, but, as yet, I don't know why we want to know about the properties of the map. >I don't pretend to be an expert on category theory, but there is >something that category theory definitely does for me. As you learned >group theory, and linear algebra, and rings, and modules... didn't you >get tired of seeing the same "isomorphism theorems" proven over and >over again, using essentially the same proofs? Yes, it's a long time ago, but I did get a bit cheesed with it. It was not just a matter of the repetition of the 'same' isomorphism theorems (I've moved your inverted commas, as I think that is where you meant them), I also felt that once we were studying isomorphism theorems instead of the actual structures we seemed to have taken abstraction to a point where its application to whatever it is that the structures are about had become somehow lost along the way. Category theory seems to take that process one step further, making me feel resistant. >Category theory allows you to say "the category of ... is an abelian >category" and prove a ton of analogous theorems in one line, modulo >showing that the algebraic structures and regular maps you are >studying form an abelian category, which is pretty easy. When you say >that a particular statement "looks right" by analogy with similar >theorems in other branches of mathematics, you are wanting to use >category theory. It sounds like it is saying, in a serious way, something I have long felt, and once tried to express for non-mathematicians in the following light hearted way, "The nice thing about algebra is that it shows that pretty much everything we do in mathematics boils down to the rules of addition and multiplication. In other words it is all terribly easy". But then it seems to compound the problem too, I continued:- "The horrible thing is that, because it talks in such abstract terms of things which might be something and might be something else, you mostly haven't got a clue what its talking about. This makes it terribly difficult." So I am still not sure whether category theory is making an easy problem difficult, or making a difficult one easy. Tell me, could you have understood the category theoretic demonstration of all those analogous theorems if you had not actually done the individual theorems and structures first? And if you have to do all of the specific instances anyway, what has been gained by the category theory? And does any of that help us to understand the specific structures which we would like to model in physics? Now if there were some category theoretic proof that, e.g. because QED is really 'just like' a lot of simpler structures, it can be constructed (even if we don't know how to do it) then that would be impressive. But I doubt there is any mileage in such a suggestion.

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