M Theory Lesson 21

Apparently kneemo has been reading Tom Leinster’s free book on Higher Operads. This text takes a fresh approach to weak n-categories from the point of view of operads and multicategories. It also has some wonderful quotes, such as, from James R. Brown: some ‘pictures’ are not really pictures, but rather are windows to Plato’s heaven. On this blog, we like to consider simple pictures.

After a meandering introduction, mostly written for topologists, Leinster begins with a picture of an arrow for a multicategory:Of course, this is just the kind of picture that computer scientists like to draw, and indeed many computer scientists work with multicategories. One way of thinking of an operad is as a multicategory with one object.

In operad land, only boundaries really matter. A cylinder, for instance, changes into and a pair of pants into the face diagram

7 Responses so far »

  1. 1

    Mahndisa S. Rigmaiden said,

    03 07 07

    hehehehehehehe A pair of pants. Yea!!!!!

  2. 2

    Anonymous said,

    Yes, let’s have some fun with pants.

    Prove that if A is a Frobenius algebra, then one can define a TFT with A as a state space with multiplication corresponding to the pair of pants and the trace to the cap.

  3. 3

    Kea said,

    Hi folks. Yes, Runkel et al have some amazing papers on the link between Frobenius algebras and TFTs. This is often discussed on blogs, such as in this old post on SCT.

  4. 4

    Doug said,

    Hi Kea, you are BRILLIANT!

    1 – This may be a key post in the attempt to unify everything from QM and GR and in between. If I am correct, this even means dynamic noncooperative game theory.

    I have seen the technique of pairing –
    pants <---> faces
    – before, in other disciplines.
    Note that I think this is a two-way duality.
    I think your insight will improve if you modify your drawings to scale with each other.

    I know that one of the source links was dismantled about two years ago, but with time I can find the reference and maybe a copy of that representation.

    A link to a PDF paper that I tried to make did not work in “M Theory Lesson 20”, RE:
    Mark Morris, [PDF] ‘The Galactic Center Magnetosphere’, figure 2.
    http://ej.iop.org/links/rTjm_8XOS/BhdAfKnM2xGIaYVvav5vpA/jpconf6_54_001.pdf
    1:49 AM

    What did I do incorrectly?

    2 – Urs or one of the other two moderators at the time, actually posted one of about ten comments I submitted to Sci.physics.research regarding one of the TWF (week 221) threads on Andalusian astronomy and Arabic numerals history.
    http://www.lns.cornell.edu/spr/2005-09/msg0071656.html

  5. 5

    Kea said,

    Hi Doug. Yes, dualities abound in this business. There is a lot of fancy category theory underlying all this.

  6. 6

    david said,

    The quotation “some ‘pictures’ are not really pictures, but rather are windows to Plato’s heaven” is taken from a Brown, but it’s not Ronnie. I believe Tom Leinster came across it while reading my book ‘Towards a Philosophy of Real Mathematics’. I quoted it from the philosopher James R. Brown ‘Philosophy of Mathematics: An Introduction to the World of Pictures’, Routledge, 1999.

    James Brown was speaking of those picture proofs such as when you show that the sum of the first n odd numbers is n squared.

    I was referring to his idea in the context of a discussion of the notation of higher-dimensional category theory.

  7. 7

    Kea said,

    Oops! Thanks, David! I’ll correct it.


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