M Theory Lesson 11

Let’s try another picture formatting option on squashed cube diagrams, which are what we get when we start thinking about higher dimensional categories. A natural transformation is an arrow between functors between 1-categories. What if we had a kind of 2-functor between 2-categories? These can have pseudonatural transformations between them, and then of course there has to be yet another level of arrow, and these are called modifications.

6 Responses so far »

  1. 1

    nige said,

    Thanks for this clear explanation of some of the technical terms regarding extra dimensional categories.

  2. 2

    Kea said,

    Hi Nigel. Unfortunately the picture isn’t very clear – with my current options I can only guess how it will come out! Thanks for your comment the other day. I agree we need to get the ‘Standard Model’ straight. We are trying our best.

  3. 3

    CarlBrannen said,

    I didn’t understand a word of this.

  4. 4

    Kea said,

    Sorry, Carl. I was assuming that the reader has spent a little time thinking about what a functor is, and has also seen diagrams with 2-arrows before. These are ‘commuting diagrams’ so instead of equations they are really different sides of a cube (one could put in identity arrows to fill out the cube). It’s not supposed to be obvious why this definition is a good one – actually it isn’t obvious at all.

  5. 5

    CarlBrannen said,

    That they’re commuting diagrams is clear, what’s not clear is what F, G, X, Y, a, f, and g are.

    My book on “sets for mathematics” does not include notation “F(X)”. Are “F” and “X” both functors? Or are X and Y objects in a category?

    As an alternative to explaining your notation, you could also consider putting in a link to a paper or book that explains your notation.

    I mean if I use a symbol like [tex]\sigma_x[/tex] I say that it means a Pauli spin matrix, I don’t leave my readers guessing.

  6. 6

    Kea said,

    Hi Carl. Yes, the convention is that F and G (and H etc) are functors, and X, Y etc are objects, whereas f and g are arrows. The letter ‘a’ represents the new transformations defined by such diagrams.

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