TITLE: String diagrams for bimonoidal categories (a.k.a. rig categories)?
QUESTION [22 upvotes]: I'm having some fun playing around with string diagrams for monoidal categories, expressing familiar constructions from Riemannian geometry and linear algebra in terms of elegant string diagrams.
I've been wondering if there is a nice extension of this diagram calculus to bimonoidal categories (also known as rig categories)? This would allow us to work with direct sums (of bundles, or representations etc.) diagrammatically as well.

REPLY [13 votes]: This question is answered in the affirmative in the following preprint:

Cole Comfort, Antonin Delpeuch, Jules Hedges, Sheet diagrams for bimonoidal categories, arXiv:2010.13361

The morphisms are represented by so-called proof sheets.
Here is an example of such a proof sheet, representing the controlled-not gate using the program sheetshow which Antonin created to typeset the diagrams in the preprint: