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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
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′:y→z and g,g′:x→y and two 2-morphisms α:f⇒f′ and β:g⇒g′
∂(α∘β)=∂α∘f′∘∂β∘f′−1.To interpret this, we can think of ∂(α∘β) 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 f′−1, then traverse ∂β, and finally carry the result back to z along f′. Neat :)
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