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1. Definiton
Let $(C, \otimes, I, a, l,r)$ be a (not necessarily symmetric) monoidal category. A (planar) star-autonomous structure on the monoidal category $C$ consists of an adjoint equivalence $D \dashv D’: C^{op} \xrightarrow{\sim} C$ together with bijections $\phi_{X,Y,Z}: Hom_C(X \otimes Y,DZ) \xrightarrow{\sim} Hom_C(X, D(Y \otimes Z))$ natural in $X,Y,Z$.

2. Problem
When trying to prove certain properties of star-autonomous categories, I often have to show that some horrendous (extremly large) diagram in a star-autonomous category $C$ (involving the natural isomorphism $\phi$ as well as the application of the contravariant functors $D$ and $D'$ to morphisms in $C$) commutes. If it weren't for the star-autonomous structure, I would translate those diagrams into string diagrams (in monoidal categories) since the graphical calculus to me is more intuitive to handle.

3. Question

  • Is there a graphical calculus for star-autonomous categories as defined above? How would I pictorally represent the natural transformation $\phi$ and the functors $D$ and $D'$?
  • While reading on star-autonomous categories the terms "proof net" and "Kelly-Mac Lane graph" kept coming up. Are they what I am looking for?
Max Demirdilek
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    This talk by Mike Shulman proposes one: https://www.youtube.com/watch?v=5CjSu5hLtcw – N. Virgo Apr 08 '22 at 13:37
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    Thank you. That talk is certainly interesting. However, Shulman does not define star-autonomous categories as above. Firstly, he only considers the symmetric case (Todd Trimble even asks about the non-symmetric case in the end). Secondly, he defines a star-autonomous category as a linearly distributive category with negation. I am actually trying to prove that a star-autonomous category (as above) carries a linearly distributive structure. To verify that the diagrams required commute, I was hoping for a graphical calculus (which cannot presuppose a lin. distributive structure). – Max Demirdilek Apr 08 '22 at 14:53
  • Besides, he writes in the abstract of the talk: "In this talk we obtain a string diagram calculus for csm categories, by […] using the known string diagram calculus for star-autonomous categories." It would be nice to learn how this calculus looks like. – Max Demirdilek Apr 28 '22 at 11:03

1 Answers1

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A number of references to different approaches to graphical calculi for *-autonomous categories are given in the introduction to Shulman's *-Autonomous Envelopes and Conservativity (which is the paper corresponding to the talk mentioned in the comments, which has since become available).

varkor
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