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Category Theory: ?What is up with these Diagrams?

  1. Dec 7, 2007 #1
    Category Theory: ???What is up with these Diagrams???

    So I found a basic category theory book online and was trying to learn some of the basics. Many of the proofs are done in diagram form (and it seems to very greatly reduce their lengths). However, no where in the book does the author prove that the diagrams make sense, explain exactly how to interpret the diagrams, or say what things like "this diagram commutes" means. I've seen similar treatment of diagrams in category theory in a few books online as well as on websites (such as John Baez's), and I've so far just been entirely lost with them.

    Could someone point me to somewhere that explains these in at least some semi-rigorous fashion?
  2. jcsd
  3. Dec 7, 2007 #2


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    The vertices in a diagram are objects of your category. (e.g. sets in Set, groups in Grp, real vector spaces in VectR)

    The arrows in your diagram are morphisms of your category. (e.g. functions in Set, group homomorphisms in Grp, linear transformations in VectR)

    Each arrow that appears in a diagram is an assertion that the object at the starting vertex is the domain of the corresponding morphism, and the object at the ending vertex is the codomain of the corresponding morphism.

    If you have a path through the diagram, you can "evaluate" by composing the morphisms corresponding to each arrow.

    The diagram is commutative if and only if, whenever there are multiple ways to travel from a particular vertex to another particular vertex, evaluating all such paths gives you the same morphism.

    (Conventional exception: if you draw a pair of parallel arrows between two vertices, we do not require those arrows to denote the same morphism. This exception does not apply to longer paths that include those arrows)

    The standard reference for category theory is Categories for the Working Mathematician, by Saunders Mac Lane.
  4. Dec 8, 2007 #3
    Ah, looking at a few pages of the book on Amazon, it seems that he actually spends some time explaining and justifying the diagrams. That's good. There seems to be a copy at ulib.org, and I know that my school's library has a copy.

    Has anyone heard anything about the book Category Theory by Steve Awodey? He's a professor in my university's philosophy department and he teaches a course on Category Theory (which is supposedly offered in the spring but it hasn't to my knowledge been offered since 2 years ago) from printouts of his book. He claims that his book is a gentler introduction than Mac Lanes and is meant for students who haven't had as strong a background in algebra yet. However, it's not as popular, so it seems to be very difficult to find a decently priced copy.
  5. Dec 8, 2007 #4

    Chris Hillman

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    Recommend a book? But hopefully not the book you so dislike?

    Luke, for heaven's sake, what book are you talking about here?

    In elementary category theory, "commutative diagrams" are really only a very convenient shorthand for statements you could write in terms of "compositions" of "arrows" (aka "morphisms"), so there's nothing very subtle to justify! I discussed this point in an expository paper I wrote when I was a graduate student, called "A Categorical Primer", which you might be able to find somewhere.

    I'll go out on a limb and guess that you read a few pages using Google's "Search Inside the Book" feature from the undergraduate textbook by Lawvere and Schanuel, Conceptual Mathematics, Cambridge University Press, 1997. If so, that book starts gently but eventually covers quite a bit of ground and is very clear.

    A very clear and well-organized introduction is Geroch, Mathematical Physics, University of Chicago Press, 1985. (Despite the title, this is basically an introduction to category theory in the guise of an introduction to mathematical methods for physicists.)

    A more conventional textbook is McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, 1995.

    There are many other books including Goldblatt or Barr and Wells, Category Theory for Computing Science or Herrlich and Strecker, Category Theory: an Introduction, etc. As already mentioned, the classic source is Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer.
    Last edited: Dec 8, 2007
  6. Dec 8, 2007 #5
    I'm sorry, I should not write any posts after not sleeping for almost 30 hours (I barely remember writing the first one, but it seems to be right before I collapsed)

    The book that I was trying to work though is the online edition of Adámek, Herrlich, and Strecker's Abstract and Concrete Categories. Very early on (page 37), they give a proof that the existence of fully faithful, isomorphic dense functors between categories is an equivalence relation. In this proof, they define a mapping between categories G and say that by the way they have defined it, a certain diagram commutes. I got lost here before because although I suspected what it meant for the diagram to commute, I didn't know for sure. Now it seems to be fairly obvious once you can justify the existence and uniqueness of each of the morphisms in the diagram. Then they present another diagram and say that G preserves composition because another diagram commutes (which is obvious since it is just two of the previous diagrams put together)

    Anyway, leaving out the explanation of something that they intend to use throughout the book makes me skeptical of the rest of the book, so I'll look into some of the others. Thank you for the suggestions.

    Also, has anyone heard anything about Steve Awodey's book? Like I said, he teaches a course on Category Theory here (Carnegie Mellon). Unless I happen to teach myself satisfactorily before then, I hope to take it when it is offered next. Though... on second though, since he gives handouts from the book, it might be beneficial to read a different book so that I'll hear everything explained in multiple ways instead of entirely from him.
  7. Dec 8, 2007 #6


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    Well, I think it's best to buy maclane's book, I myself intend to buy it next year.
    there's also a handbook on categorical algebra (by a french guy, search in amazon it's called handbook of categorical algebra), I don't think it's a better introduction than maclane's cause handbooks rarely are good as introductions, but perhaps after reading maclane's it will shed some light on the active research being done on category theory.
  8. Dec 8, 2007 #7

    Chris Hillman

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    The book by Mac Lane is actually quite readable (provided one has the assumed "mathematical maturity" and background knowledge shared by most well-prepared first year graduate students). The book by Lawvere and Schanuel is a really nice introduction which assumes far less background (but demands far more intelligence than might appear from the inviting and even somewhat "corny" appearance); Lawvere and Mac Lane are legendary figures in this field, btw. (Mac Lane and Eilenberg founded category theory, and Lawvere is commonly regarded as the founder of topos theory, one of the most lovely subjects to grow out of category theory. Another notable figure is Andre Joyal, who founded the theory of structors, or combinatorial species, which are special functors which provide a powerful and natural foundation for the elementary theory of enumerative combinatorics.)

    Er, Luke, not to get all paternalistic or anything like that :wink: but I'd urge you to avoid staying up for 30 hours in future, and certainly to avoid posting when you are so short of sleep. Mathematics and lack of sleep make a poor mixture!
    Last edited: Dec 8, 2007
  9. Dec 8, 2007 #8
    Oh, trust me, I don't like it either. I'm desperately trying to break my habit of procrastination.

    I should mention that I'm an undergrad. So you would recommend avoiding MacLane then? Are there any Category Theory books that would be appropriate or would you recommend avoiding the subject altogether until I've had more algebra?

    I've so far only had about half a semester worth of both group theory and linear algebra. I am taking an honors math course with Algebra and Analysis, and we stopped halfway through with group theory so that we could learn enough Linear Algebra so that we can do Multivariate Calculus in Analysis next semester. Once we finish up with the Linear Algebra, we'll be going back to Group Theory. A little backwards, I know... and I'm worried about not remembering Group Theory as well once we get back into it.

    I only started reading the book on Category Theory because I was "interested" in it, but had no idea what it was and wanted some idea. From the description, introduction, and first section or so of Abstract and Concrete Categories, it seemed like a cursory knowledge of Category Theory would make a lot of other algebra topics more intuitive.
  10. Dec 8, 2007 #9

    Chris Hillman

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    I read Mac Lane when I was an undergraduate. From what you say you might get something out of the first few chapters and its certainly a great book to have lying around so you can consult it at will. I like all the books I mentioned for various reasons and probably no one is "the one and only" book for any one individual. If you're feeling insecure, though, Lawvere and Schanuel would be a natural choice.
  11. Dec 9, 2007 #10
    I'm thinking of getting Conceptual Mathematics because it is clear and covers a lot of ground. I read several pages of it with Amazon's Search Inside. Unfortunately, I really do not like the style that it is written in. I also read a few pages of Mathematical Physics, which is just as clear but without the childishness. However, Mathematical Physics only covers enough of Category Theory to develop the structures useful in Physics.

    Of the rest of the books, I haven't been able to get a feel for them at all (except for Category Theory for Computer Science, which does not seem to be the book for me because the author says that he did not include many proofs because they are often not useful to Computer Science students), and for most of them, I cannot find a good enough price that they would fit on my budget for textbooks that I am reading in my leisure.

    MacLane's book can be found for a good price, and I know that it covers everything that any other book would. However, while the first few pages are clear to me, after that, I fear that I would have a hard time working through it. Also, (just like Conceptual Mathematics and Mathematical Physics) I can read a copy of it essentially for free online if I want, so I think if I spend any money, I'd rather get the book that I know I'll be able to understand.
  12. Dec 9, 2007 #11
    Then again... looking at it again, I'm thinking that I won't get much out of the book on the whole, so it might be better to just read the sections that I'll learn something from online and maybe slowly work my way through MacLane's book.

    I've also just realized that Goldblatt's book can be read for free online, and it has fairly good reviews, so I'll check that one out.

    Also, Chris Hillman, I read a little of your expository paper. It's very well written. Unfortunately, I do not have a good enough background in algebra yet to appreciate your examples, but the paper is now saved on my computer, and I intend to read it once I know more.
  13. Dec 10, 2007 #12


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    the basic work is homological algebra, by cartan and eilenberg. they treated modules without using elements, just maps in diagrams. abelian category theory is just their treatment of modules, but without saying the objects are modules.
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