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    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeAug 12th 2010
    • (edited Aug 12th 2010)

    Hi,

    I’ve been excited around here lately as I learn about functors and forms, higher transport, 2-groupoids, and 2-categories.

    As usual though, I need to “find my own way”, which is understandably frustrating. Anyway, I think I’ve found my own way to understand 2-categories and it involves cooking up a construction I call the “boundary of a 2-morphism”.

    I’ve written up some of my thoughts on my personal web:

    I’d love to hear your thoughts. Is it nuts? Is it useful? Is it interesting? I thought the calculation involving the interchange law was particularly cute. What do you think?

    Best regards,

    Eric

    • CommentRowNumber2.
    • CommentAuthorEric
    • CommentTimeAug 15th 2010
    • (edited Aug 15th 2010)

    I’ve added a little more material and have also tried asking a question on MO.

    By the way, thinking in terms of parallel transport provides a very nice picture for the boundary of horizontally composed 2-morphisms in a 2-groupoid, i.e. with four morphisms f,f:yz and g,g:xy and two 2-morphisms α:ff and β:gg

    (αβ)=αfβf1.

    To interpret this, we can think of (αβ) as a group element located at object z, α as a group element also located at z, but β is a group element located at y. To “compare” α and β we need to “transport the transport”, i.e. transport the group element β at y to z. The above formula says that to do this, we backtrack from z to y along f1, then traverse β, and finally carry the result back to z along f. Neat :)