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• CommentRowNumber1.
• CommentAuthorTapo
• CommentTimeDec 30th 2020
I was looking at the unit conditions for Trimble's tetracategory and encountered a problem when fiddling around trying to extend to U51 and U55. I noticed that the whiskered product cell of the pentagonator modification was the only product cell that would add an "interior vertex" and that each unit condition had a single pentagonator in both the positive and negative paths. When extending Trimble's trees to U51 and U55, it seemed to define unit conditions that would lack a pentagonator in the positive and negative paths, resp.
• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeDec 30th 2020

I can have a look, but it may be a day or so before I find the time.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeDec 30th 2020

Before I really drill down and try to produce $U_{51}$ and $U_{55}$ myself (my experience is that it takes at least a couple of hours to do these things), maybe I could ask: are you seeing behavior that you don’t see in the other unit conditions? I’m having a slightly hard time picking up the thread from your description.

• CommentRowNumber4.
• CommentAuthorTapo
• CommentTimeDec 30th 2020
• (edited Dec 30th 2020)

Precisely, the other unit conditions contain a single copy of $\Pi$ in both paths of product cell and constraint compositions, which doesn’t seem to happen for $U_{51}$ or $U_{55}$. Also, it seemed that all the product cells and constraints take away a vertex, except for $\Pi$ which adds a vertex by creating a pentagon at the top of the cell. For $U_{51}$, the trees produced a positive path (containing $K_{5}$) that has $U_{41}$ where the $\Pi$ “should” be, but the negative path does have a $\Pi$. In regards to diagrams instead of trees, $U_{51}$ seems to have an arrow from I((xy)(yz)) to (xy)(yz) sort of blocking the pentagon from forming. I’m not sure if these missing vertexes could appear from a product cell or constraint that isn’t $\Pi$.

• CommentRowNumber5.
• CommentAuthorTapo
• CommentTimeDec 30th 2020

$U_{55}$ has a similar problem, except with the negative path missing the $\Pi$.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeDec 30th 2020

Well, I think there’s no help for it but to be able to share actual pictures. I haven’t finished writing down $U_{51}$ today, but I hope to get to that. Might you be able to reach me by email?

Incidentally, if you’ve managed to produce these things correctly (by my lights) based on the sparse hints in those notes, then I’m impressed. :-)

It happens occasionally that I meet someone who wants to discuss such things in detail, but it’s pretty rare. So, thanks for your interest.

• CommentRowNumber7.
• CommentAuthorTapo
• CommentTimeDec 30th 2020
Thank you for reaching out as well!

Hopefully what I've produced isn't too far off, but I'll go ahead and contact you through email.