• April 22, Michael Shulman, Star-autonomous envelopes.

Abstract. Symmetric monoidal categories with duals, a.k.a. compact monoidal categories, have a pleasing string diagram calculus. In particular, any compact monoidal category is closed with [A,B] = (A* ⊗ B), and the transpose of A ⊗ B → C to A → [B,C] is represented by simply bending a string. Unfortunately, a closed symmetric monoidal category cannot even be embedded fully-faithfully into a compact one unless it is traced; and while string diagram calculi for closed monoidal categories have been proposed, they are more complicated, e.g. with “clasps” and “bubbles”. In this talk we obtain a string diagram calculus for closed symmetric monoidal categories that looks almost like the compact case, by fully embedding any such category in a star-autonomous one (via a functor that preserves the closed structure) and using the known string diagram calculus for star-autonomous categories. No knowledge of star-autonomous categories will be assumed.

This subject is especially interesting to me since Mike Stay and I introduced string diagrams for closed monoidal categories in a somewhat ad hoc way in our Rosetta Stone paper—but the resulting diagrams required clasps and bubbles:

This is the string diagram for beta reduction in the cartesian closed category coming from the lambda calculus.

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Will a recording of this be available? Seems relevant to linguistics (for glue semantics=semantic composition by linear logic for ling. structures that aren’t trees), but don’t fancy my chances of effecting a Zoom capture at 3AM (Eastern Aus Time)

Yes, you can see it on YouTube now—look at the blog article again. I should have said that. Every lecture in the ACT@UCR seminar series will show up on YouTube later that same day.

A key issue is how to express the bijection fundamental to the lifting property (used to express Kan extensions 2-categorically) using string diagrams.
An interesting discussion of this issue, well-motivated by the lambda-calculus, is at:
“Kan Extensions for Program Optimization, Or: Art and Dan Explain An Old Trick” by Ralf Hinze

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You need the word 'latex' right after the first dollar sign, and it needs a space after it. Double dollar signs don't work, and other limitations apply, some described here. You can't preview comments here, but I'm happy to fix errors.

Will a recording of this be available? Seems relevant to linguistics (for glue semantics=semantic composition by linear logic for ling. structures that aren’t trees), but don’t fancy my chances of effecting a Zoom capture at 3AM (Eastern Aus Time)

Yes, you can see it on YouTube now—look at the blog article again. I should have said that. Every lecture in the ACT@UCR seminar series will show up on YouTube later that same day.

A key issue is how to express the bijection fundamental to the lifting property (used to express Kan extensions 2-categorically) using string diagrams.

An interesting discussion of this issue, well-motivated by the lambda-calculus, is at:

“Kan Extensions for Program Optimization, Or: Art and Dan Explain An Old Trick” by Ralf Hinze