In Steve Awodey's book on category theory, he claims the latter is a branch of abstract algebra. I've never seen such a classification before. Is this really correct?
Category theory, a branch of abstract algebra?
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12There is no real definition of what being "a branch of abstract algebra" means... – 2011-11-05
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0Check Wikipedia: http://en.wikipedia.org/wiki/Category_theory – 2011-11-05
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0Why does it matter whether it's a branch of say logic instead? – 2011-11-05
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0Well, of course, it's really not important. It just struck me, while reading the book, as an unusual claim. After doing some research on the internet, I could not find a confirmation anywhere except in Awodey's own book and writings. So I guess I was just curious, that's all. – 2011-11-05
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1It seems that people who really focus on categorical type things (like Goodwillie calculus, brave new algebra, things like that) often get referred to as algebraists. – 2011-11-05
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1I don't understand why this question was closed and Arturo gave a very nice answer to it. Not all math questions have to be "how do you solve/prove X?" I think how/why mathematicians have decided to classify mathematical objects/theories is a legitimate point of inquiry – 2011-11-05
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1As I noted in my flag to moderators, I think this is a candidate for Community Wiki, rather than just a straight question; perhaps Rocco can edit it a bit and help garner reopen votes (I already cast mine). I do think this should not be a "regular question", though, but a CW. – 2011-11-05
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1+1 for Arturo's suggestion that the OP can edit the question a bit. – 2011-11-05
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0Unfortunately, I've just started using math.stackexchange, and I have no idea how to go about including this question in the Community Wiki. I can reformulate it, but that's all I could do at the moment :) Can someone give me some tips regarding the CW part? – 2011-11-06
1 Answers
One could likewise argue that General Algebra (which encompasses groups, rings, lattices, and many other structures) is the study of a special class of categories, and so argue that General Algebra is a branch of Category Theory... (For example, George Bergman's General Algebra book is heavily category-theoretically flavored).
I would say rather that Category Theory has some very large areas of intersection with General/Abstract Algebra; I remember George Bergman saying once that in the 80s (?) there was a big conference at Berkeley/MSRI that invited both Universal Algebraists and Category Theorists, and that many times over the course of the conference they discovered that there were results that each "camp" had proven independently and did not realize the other "side" knew about them, or that there were questions that had been raised on the periphery of one whose answer was well-known by the other. (I hope I'm not misremembering and/or misreporting this!).
My particular (heavily algebra-biased) experience leads me to think that Category Theory is closer to abstract algebra than to other branches of pure mathematics (e.g., topology, analysis, etc). Don't know if I would go so far as to call it a "branch" of abstract algebra, though, any more than I would call it a branch of set theory (or set theory a branch of category theory).