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Algebra Categories for the Working Mathematician

  1. Aug 20, 2016 #1
    Hello. I am about to start learning category theory. I keep hearing mixed opinions on the book Categories for the Working Mathematician, by Sanders MacLane (I am aware he is one of the founders of the theory). Some say it's a "must read", and others have called it "outdated." What would seem outdated about this book? What would be the pros and cons of using it? Is there a book or a collection of books that you feel cover the same (or more) material but better? IF any of my background is needed, I have a fair amount of mathematical maturity. I can read Rudin, Lang (Grad algebra), and other such terse books and fill in or construct my own such proof fairly well when things are missing. I might just be shaky in terms of knowledge, such as definitions etc. For example, I haven't learned algebraic topology yet.
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  3. Aug 20, 2016 #2


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    At least I have the impression that you can read it. The question is, why do you wish to read it? Do you have a special goal, e.g. to learn some tools for cosmology, in which case I would recommend a book that is more (co-)homological and less categorical. If you are interested in anything in the sphere of logic, I'd say read it. So the answer to your question depends pretty much on where you want to arrive at.

    The fact it is some decades old doesn't matter a second. It's a book on fundamental conceptions and these haven't changed. (And I doubt they will at any time.)
  4. Aug 20, 2016 #3


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    i would suggest that you mainly want to know what a functor is, and a natural transformation, a representable functor, and then Yoneda's lemma. That's about it, as far as I am concerned. Oh and I guess you want to know the categorical definitions of isomorphisms, products and sums (coproducts), as well as inverse and direct limits.

    edit: My viewpoint is that of an algebraic geometer who is not a category theorist. So to me most of the stuff in the free book you linked is totally unnecessary verbiage. As a youngster I recall thinking category theory was a lot of fun, but as a practicing mathematician, it seemed like (to quote one somewhat cranky and opinionated algebraic geometer, Miles Reid, p.116, Undergraduate Algebraic Geometry) "surely one of the most sterile of all intellectual pursuits".

    This in the vein of the earlier question of what are your goals. I.e. if your goal is to be a mathematician in a field other than category theory, you will not need all this technical terminology. But if you enjoy this pursuit, then wonderful. Go for it.

    I would actually recommend reading the original paper that started the theory, at least the introduction:

    Last edited: Aug 22, 2016
  5. Aug 21, 2016 #4
    Well, here are more details on my current situation. I am taking a directed independent study in Category Theory. My professor (he is one of the few calling it outdated) wants me to use the book Category Theory: An Introduction by Herrlich and Strecker. I am aware that these authors have another free online popular book Abstract and Concrete Categories. So, my plan so far for this independent study course is to use Herrlich and Strecker's introduction book, supplement it with Awodey's book, and then Abstract and Concrete Categories. I guess a better question would be, if I read all of those books, would it cover everything MacLane does? Would there be any incentive to do so after the independent study, say on my own next semester?
  6. Aug 22, 2016 #5


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    Sorry, but your details left me with even more questions.
    What is directed + independent? Directed by whom and what for? Independent from whom and what for? How is your professor related to these questions? In general, I would simply recommend to follow his advice, for he knows best about your situation and has probably more insights as well in the subject as in your personal development and goals than anyone here on PF.
    Which leaves us again at the starting point of my previous post: How do you measure incentive and what's going on in your next semester?
    Why do you wish to read so many different books on this subject?

    However, I'm probably not the one to ask anyway. But what you call details aren't any (IMO). I thought, you might want to know.
  7. Aug 25, 2016 #6
    What's your background? What are you studying category theory for? Have you taken homological algebra?
    As far as I know, Categories for the working mathematician draws example from algebraic topology (where the subject naturally arose). If you have no idea what algebraic topology is, the book might go over your head. If your interest lies in applying category theory to other mathematics subjects (or CS, as a matter of fact) you most likely won't need to go through a whole book. Without more information, we won't be able to help you much.
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