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Stream: learning: questions

Topic: string diagrams for coproducts of endofunctors

Ralph Sarkis (Aug 22 2021 at 17:45):

Is there a string diagram syntax to describe coproducts of endofunctors?

Ralph Sarkis (Aug 22 2021 at 17:45):

I am presenting my paper at a conference and I was thinking it would be great to present the results with string diagrams, but I don't remember seeing a graphical notation for coproducts.

Matteo Capucci (he/him) (Aug 27 2021 at 09:50):

Mmh surely string diagrams for moncats work for cocartesian cats as well

Matteo Capucci (he/him) (Aug 27 2021 at 09:50):

So I'm probably not understanding your question

Ralph Sarkis (Aug 27 2021 at 11:04):

I need to handle both. So vertical and horizontal composition of natural transformations are still represented as juxtaposition, and there is a way to denote $\phi + \eta$ and $[\phi, \eta]$ that obeys some visually obvious equations.

Jules Hedges (Aug 27 2021 at 11:09):

Perhaps I can sell you sheet diagrams for distributive monoidal categories? https://arxiv.org/abs/2010.13361

Jules Hedges (Aug 27 2021 at 11:10):

Screenshot-2021-08-27-at-12.10.31.png

Ralph Sarkis (Aug 27 2021 at 11:26):

I think it would be very hard for composition to distribute over coproduct or vice versa, but that is exactly the kind of syntax I had in mind (though I was only thinking in 2d).

Jules Hedges (Aug 27 2021 at 12:13):

My gut feeling is I'd be pretty surprised if you don't get some nice interaction between the 3 operators, and being a distributive monoidal category is the first thing I'd check

Ralph Sarkis (Aug 27 2021 at 12:48):

Thanks, I'll keep thinking about it. (Probably not for the coming presentation but for another one.)

Antonin Delpeuch (Sep 17 2021 at 15:12):

Since you wrote that paper you already have a mental representation of how these things compose. How about first drawing the pictures that make sense to you, and then trying to see how they could be recast as string diagrams of some sort?

Antonin Delpeuch (Sep 17 2021 at 15:14):

(I am very keen to try to help you in that process, it's a fun game!)