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Category Theory.

  1. Feb 13, 2007 #1
    Essentially I'd just like to learn more about it in my own spare time. Is there any particularly good introductory books or articles?

    I'd particularly like anything that uses examples from physics, but it isn't essential.
     
  2. jcsd
  3. Feb 13, 2007 #2
    There's alway wikipedia. Dummitt and Foote has a decent intro to category theory as an appendix.
     
  4. Feb 13, 2007 #3
    John Baez is one of the experts in category theory who often posts to his blogs 'n-category Cafe' or 'This Week’s Finds in Mathematical Physics'.

    Week 245 has talks in this theory from a U-Toronto workshop at the Fields Institute January 9-13, 2007.
    http://math.ucr.edu/home/baez/week245.html
     
  5. Feb 13, 2007 #4

    matt grime

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    John Baez is not an expert in category theory.

    Mac Lane is the standard book, but it is very hard going.
     
  6. Feb 13, 2007 #5

    mathwonk

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    i liked peter freyd's little book in the 60's and 70's.

    hungerford ahs a short little section that helps too.

    but the amin point is to make yourself always focus on the maps betwen objects instead of just the obejects.

    e.g. from a pair of spaces one can form a product space. but also from maps of two pairs of spaces, obe derives maps of their products.

    ]this si thew whole point. which constructions of spaces allow comparable constructions of maps between those spaces?

    the fundamental group assigns to a space maps from a circle into that space. but also think abiout how a map of the spaces indiuces a map of their fundamental groups. now you are thinking "categorically".
     
  7. Feb 13, 2007 #6

    mathwonk

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    the fact that the product of determinants is the determinsnt of the product matrix says that a determinant isa functor from amtrices to numbers.

    the chain rule says that the derivative is a fucntor from pointed functions to numbers.

    at some point people realized that amny important constructions were fucntors. but to=defione functors, betwen categories, one ahd to define categories.

    i recomend the original apper by maclane and who?

    matt, what is the original paper defining exact sequences and so on, and especially natural transformations, with the double dual as the basic example? was it by eilenberg and maclane?

    is this it?

    General Theory of Natural Equivalences
    S Eilenberg, S MacLane - Transactions of the American Mathematical Society, 1945 - JSTOR
    General Theory of Natural Equivalences. Samuel Eilenberg. Saunders MacLane.
    Transactions of the American Mathematical Society, Vol. 58, No. 2, 231-294. ...

    it seems to be available free on the internet.

    do you think this is a good one matt?
     
    Last edited: Feb 13, 2007
  8. Feb 14, 2007 #7
    I learnt some category theory from a book called "The Joy Of Cats" by Adamek, Herrlich and Strecker. Its a fairly decent book. But then i also had a good instructor to go along, so i am not sure, how good it might be for self study.

    -- AI
     
  9. Feb 14, 2007 #8

    mathwonk

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    categories are nothing. functors are something. natural transformstions are more interesting still.
     
  10. Feb 15, 2007 #9
    So I guess you really love categories of functors with natural transformations as the morphisms.

    Incidentally, in a homological algebra course, it was after talking about this and bi-, tri, ... , infinity-categories, that my professor remarked about people calling category theory abstract nonsense.
     
  11. Feb 15, 2007 #10

    matt grime

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    Even us category theory advocates use that term. It just means: true for elementary formal reasons. Often things are proven in some concrete case in a very inobvious way, but are true for for other simpler reasons in some manner. I.e. in the more general case the proof is actually easier.
     
  12. Feb 15, 2007 #11
    I got out Mac Lane on your suggestion. I must say this is an incredibly interesting branch of mathematics. I think it's a really "clean" way to view concepts from other areas.

    Thanks for that example. Solidified the concept a bit for me.

    I was just wondering, in what other areas of mathematics has category theory been useful. (I'd assume it has been useful in Algebraic Topology.)
     
  13. Feb 15, 2007 #12

    matt grime

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    All parts of algebra (so that is geometry and topology too). Some parts of analysis. Theoretical physics. Computer Science. Is that enough? Few parts of mathematics are not touched by category theory.

    Silly examples:

    a group is a category with one object and all morphisms invertible. A representation of a group is a functor from this category to the category of vector spaces. An isomoprhism of representations is a natural transformation of functors.

    completion of a normed vector space is left adjoint to the forgetful functor from banach spaces to normed vector spaces.

    classifications of things are functors (moduli spaces). There is a functor from the genus 2 curves to abelian varieties of dimension 2 (an equivalence, right, roy? curve to jacobian, jacobian to the theta divisor or some other such slogan).

    The representation theory of a field is essentially the same as the representation theory of nxn matrices over that field - Morita equivalence.
     
    Last edited: Feb 15, 2007
  14. Feb 15, 2007 #13
    Yeah. Computer Science surprised me to be honest. Of course though, it doesn't matter how "useful" it is. I was just idly wondering how much it had seeped into the general language of mathematicians.

    Thanks for the examples. I really like the representation theory example.
     
  15. Feb 15, 2007 #14
    Hi matt grime

    Could you clarify your comment in your post of 02-13-2007, 06:41 PM?

    I am confused because why would so many university math / physics workshops employ a non-expert to discuss category theory.

    Counterparts of yours in ‘Beyond the Standard Model’ [Physics] and others who comment in that forum appear to regard the individual as an expert in category theory as it relates to physics.
     
  16. Feb 15, 2007 #15

    Hurkyl

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    You know how you denote the domain and range of a function with the notation [itex]f : D \rightarrow R[/itex]? (AFAIK) Before category theory, people wrote [itex]f \subseteq D \times R[/itex].
     
  17. Feb 16, 2007 #16

    matt grime

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    knowing about some aspect of something doesn't make you an expert in the subject as a whole

    You have just answered your own question.

    John is very knowledgeable about n-categories and how they pertain to physics. Most mathematicians are under your rules therefore experts in categories since they use them 'expertly' in their own field. That does not in any reasonable sense as far as I am concerned make them experts in category theory.

    I know how string theory relates to algebra. That does not make me an expert in physics, or even string theory.
     
  18. Feb 16, 2007 #17

    Gib Z

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    Not to mention, a member posting in the Physics section is more likely to be biased to the physicist.

    Hurkyl: My Teacher still uses that. He either even more ancient than I thought, or just old fashioned.
     
  19. Feb 17, 2007 #18
    Introduction to Category theory

    I have written a website introducing category theory using examples in physcis.

    Hope you find it useful: http://topos-physics.org

    If you have any questions use the comments on the site or message me
     
  20. Feb 17, 2007 #19
    matt grime

    1 - Thank you for the clear and concise explanation of expertise in your post of 02-16-2007, 04:55 AM.

    I agree that it is necessary and likely sufficient for expertise to be qualified as specifically as possible.

    2 - I have become very interested in Game Theory because of what I perceive to be an extremely powerful analytic tool, particularly as used by Basar [engineer] and Olsder [mathematician].

    Elements of Game Theory are used “... general search algorithm for movement strategies based on the detection of sporadic cues and partial information ...” in the Editor's Summary, 25 January 2007 of Nature 25 January 2007 Volume 445 Number 7126, pp339:
    Letter: 'Infotaxis' as a strategy for searching without gradients
    Massimo Vergassola, Emmanuel Villermaux and Boris I. Shraiman
    doi:10.1038/nature05464
    http://www.nature.com/nature/journal/v445/n7126/edsumm/e070125-10.html

    I am somewhat surprised that the only mathematics ever to win a Nobel Prize is not listed in within the category of Mathematics. Even though the Nobel category was Economics, only a minor tweak should be needed for use in Energy Economics. The ‘Set Theory, Logic, Probability, Statistics’ appears to be an appropriate forum.

    I have posted this thread on this forum:
    Game Theory - applied mathematics - how powerful is it?
    https://www.physicsforums.com/showthread.php?t=154996

    Are you able to refer me to a website or other textbook [preferably with expertise noncooperative theory] so that I might learn more about this mathematical tool?
     
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