Thursday, January 18, 2007
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.
Thanks for this clear explanation of some of the technical terms regarding extra dimensional categories.
ReplyDeleteHi 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.
ReplyDeleteI didn't understand a word of this.
ReplyDeleteSorry, 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.
ReplyDeleteThat they're commuting diagrams is clear, what's not clear is what F, G, X, Y, a, f, and g are.
ReplyDeleteMy 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.
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.
ReplyDelete