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James Gilles (Feb 14 2026 at 18:16):I have a question about how to draw surface diagrams in a rig category (that's symmetric in both operations, not sure what that's called). I'm thinking about this isomorphism:
I have a commutative diagram:
Apologies if I've gotten my left and right distributors mixed up.
I'm trying to turn it into a surface diagram:
You could draw in colored wires here representing A,B,C,D, but I've left them out because I'm curious about the topology of the underlying surface. How do you draw the seams on it?
Here's a blown up picture of the question:
Simon Burton (Feb 15 2026 at 19:09):image.png
I don't think there is a standard agreed upon definition of surface diagram for rig categories. Anyway, this was what I came up with, which is maybe more appropriate to think of as a (2-)matrix over a 2-rig... i wrote a paper about this stuff here
James Gilles (Feb 15 2026 at 19:55):Oh that's a nice picture. Also a nice paper, I ran into this a while ago and got a lot of intuition from your diagrams, but I was kind of new to category theory at the time and didn't fully understand the formal setup. It makes more sense now.
I was just noticing how some of the surfaces I want to draw don't have names in the rig category language. E.g. precisely the top of your diagram there has no name built from tensors and sums:
PXL_20260215_194908829.jpg
This is part of what the 2-rig structure provides right? I will reread in detail.
James Gilles (Feb 15 2026 at 20:00):Oh also, what are you using to make those diagrams?
Simon Burton (Feb 16 2026 at 11:16):This is part of what the 2-rig structure provides right?
Yes, something like that. You should also check out tape diagrams which are very similar. I make these diagrams using a python package which is here...