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• CommentRowNumber1.
• CommentAuthorSam Staton
• CommentTimeJan 3rd 2018

After this discussion at string diagrams for linearly distributive categories with unit = counit, I finally got round to having a go at making star-polycategory. Still much more to do. Hopefully I didn’t make any major mistakes so far.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJan 3rd 2018

Looks great to me, thanks! Two comments:

• When creating a new page, it’s a good idea to make links to it from all reasonable related pages so that people can find it. This sometimes requires a little thought as to where on such pages to put the link, perhaps making a new section somewhere for a few sentence discussion of the relationship, or if the relationship isn’t that important it can just be linked without explanation in the “Related pages” section. In this case it would be good to have links at least from star-autonomous category and polycategory, and maybe other places such as cyclic multicategory (oops, that doesn’t exist yet either, but we have cyclic operad). Would you like to try adding some links?

• It would be nice to actually have a precise definition of the compatibility conditions between composition and the exchange/negation structure. Are these axioms actually written down anywhere, or does everyone always just say “they’re obvious”? If the latter, maybe we can remedy that situation by working out what they should be.

• CommentRowNumber3.
• CommentAuthorSam Staton
• CommentTimeJan 4th 2018

Thanks Mike. I added some links for star-autonomous categories and polycategories. I am not quite sure how to make the link from cyclic multicategories. I agree it would be good to work out the compatibility conditions.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJan 5th 2018

Looks good, thanks. For the plural link star-polycategories (which you wrote at polycategory) to work, you need to add

[[!redirects star-polycategories]]


to the bottom of the page star-polycategory. Probably it would be good to also redirect other spellings like *-polycategory and *-polycategories.

To make a link from cyclic multicategory someone would have to first create the page cyclic multicategory. I don’t have time to do that now, but anyone who wants to could just make a stub to record links.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJan 30th 2018

I added to star-polycategory a bit more detail about what I think the compatibility conditions should be.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 11th 2019

I feel like I’ve seen somewhere in the literature an explicit construction of the free (symmetric) $\ast$-polycategory on a (symmetric) polycategory $P$. It adds formal duals $A^\bot$ for each object $A$ of $P$, and a morphism $(\Gamma,\Pi^\bot) \to (\Delta,\Sigma^\bot)$ therein (where each greek letter is a list of objects of $P$) is a morphism $(\Gamma,\Sigma) \to (\Delta,\Pi)$ in $P$. Composition is as in $P$, perhaps in reversed order. But right now I can’t find what paper this was in; does anyone know?

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeOct 11th 2019

Would you be satisfied with the corresponding construction of the free $\ast$-autonomous category on a symmetric linearly distributive category? Because that was given in the paper by Blute, Cockett, Seely, and Trimble.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeOct 12th 2019

Where specifically in that paper? That free functor is mentioned in section 5.2, but I don’t see an explicit description of it.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeOct 13th 2019

I guess you’re right that that specific construction is not mentioned in the paper. I must have been thinking of the description of the free $\ast$-autonomous category on a polygraph given in 5.1.