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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.

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    $\begingroup$ There is a way to display direct sums in semisimple tensor categories. But it's basically introducing a basis. It's used for example in Sec .3.1 in arxiv.org/abs/hep-th/0204148 and it's sometimes useful in calculations in tensor categories. $\endgroup$
    – Marcel Bischoff
    Commented Oct 16, 2017 at 18:17
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    $\begingroup$ @Peter Heinig thanks, I've changed the title. I originally had "string" in quotes because I was unsure if the graphical calculus for rig categories would involve strings or perhaps some sort of surfaces. $\endgroup$
    – ಠ_ಠ
    Commented Oct 16, 2017 at 19:44

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There isn't anything like a graphical calculus really expressly dedicated to rig categories, and well-documented and proved to be coherent, but there are 'approximations' to what you are looking for:

1. Proof nets. The most well-known graphical calculus for categories with two 'operations' are

proof nets

though it seems not quite 'canonical' how they relate to string-diagrams in the most usual sense (graphical calculus for 2-categories).

2. Graphical calculi for ologs.

A recent preprint goes some way in this direction:

Evan Patterson: Knowledge Representation in Bicategories of Relations. arXiv:1706.00526

Therein one finds the following passage relevant to your questions:

enter image description here

(source: op. cit. p. 44; colors added)

Please note that 'distributive monoidal category' is more special than 'rig category' (the monoidal structure is assumed to by symmetric, and the symmetric monoidal structure is assumed to conincide with the category's coproduct), but it is relevant in that every instance of the former is an instance of the latter.

  1. Distributive laws via monads. This is something I am currently only beginning to really understand, but I'll mention it here briefly: there is a notion of 'distributive law' formalized via

    • two specified monads on the same specified category,

    • one more specified natural transformation,

and such a situation can of course be formalized via 'usual' string diagrams; whether this is relevant to what you are looking I do not know.

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  • $\begingroup$ In view of Cole's answer, it might be worth reconsidering whether this should be the 'accepted' answer. $\endgroup$
    – Todd Trimble
    Commented Nov 2, 2020 at 0:09
  • $\begingroup$ @Todd I agree, though I guess ಠ_ಠ would need to contact you to ask for it... $\endgroup$
    – David Roberts ♦
    Commented Nov 2, 2020 at 1:03
  • $\begingroup$ @DavidRoberts To ask for what? He/she has the ability to unaccept and accept a different answer. $\endgroup$
    – Todd Trimble
    Commented Nov 2, 2020 at 1:29
  • $\begingroup$ @Todd Hmm, I thought old accepted answers couldn't be unaccepted! If I'm wrong, then that's good to know! $\endgroup$
    – David Roberts ♦
    Commented Nov 2, 2020 at 2:09
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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:

controlled-not gate

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    $\begingroup$ Thanks for adding this, Cole, and welcome to MO! $\endgroup$
    – David Roberts ♦
    Commented Nov 1, 2020 at 20:51
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    $\begingroup$ Thanks for the welcome, David! $\endgroup$
    – Cole Comfort
    Commented Nov 1, 2020 at 23:30
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    $\begingroup$ Very cool! Thanks for sharing. $\endgroup$
    – ಠ_ಠ
    Commented Nov 3, 2020 at 11:25

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