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I have to do some messy calculations with weak 2-functors between bicategories, and I know the most efficient way to do it would be via some sort of string diagram methods. Also, it means that I can put pretty calculations in my paper, unlike some of this paper's 'ancestors' (written by others) where they simply can't put in all the diagrams. To me this seems to be a blow to the reader, because not only does the reader have to figure out what on earth is going on, they are stuck with inelegant methods for doing so.

I know exactly what I should be doing if I was working in a single bicategory (just a simple extension of string diagrams for monoidal categories) but I don't know if anyone has written out a version with weak 2-functors. I'm sure someone has. Any pointers?

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Is there a reason to have both tags "string-diagrams" and "diagrammatic-calculus"? What about a single tag "graphical-calculus" (to include surface diagrams, but clearly distinguish from ordinary diagrams)? –  Mike Shulman Mar 30 '12 at 17:06
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up vote 5 down vote accepted

I imagine you could easily apply Micah McCurdy's graphical calculus for monoidal functors (see this paper or these slides) by just labeling the regions between strands as one usually does in the graphical calculus for bicategories.

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I like all these colors :) –  Martin Brandenburg Mar 30 '12 at 9:04
Yes, Kate and I did exactly that in arxiv.org/abs/0910.1306 (sections 8 and 9). We used cross-hatching for the functor regions, so that we could use color for labeling the regions. –  Mike Shulman Mar 30 '12 at 16:56
@Mike - thanks for the reference! –  David Roberts Apr 2 '12 at 3:03
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FWIW, not exactly sure how relevant this is, but have a look at

Peter Sellinger's - A survey of graphical languages for monoidal categories (or on his homepage)

This gives a nice overview of many constructions used in graphical languages.

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