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• CommentRowNumber1.
• CommentAuthorTim_Porter
• CommentTimeApr 25th 2012

I have split off ordinal sum from the entry on joins as it is needed in other entries as well. I have not revised the entry just doing a cut and paste, so it needs more work!

1. I added a note here on Lawvere’s definition of the ordinal sum of categories, from “Ordinal sums and equational doctrines”.

2. Previous edit said there were isomorphisms from $[i] \oplus [j]$ to $[j] \oplus [i]$, this was false and has been corrected.

Anonymous

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeJan 3rd 2019
• (edited Jan 3rd 2019)

@ Anonymous #3 ???? There are no natural isomorphisms but for finite ordinals, $[i]\oplus [j]$ is the same ordinal as $[j]\oplus [i]$ as both are $[i+j+1]$, so is that what you ment?

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJan 3rd 2019

Further clarified that addition of finite ordinals is symmetric, but not infinite ones. Actually this page needs some more serious work in clarifying the finite/infinite distinction; it reads kind of as if whoever wrote it thought that “ordinal” meant “finite ordinal”. (Not that I think anyone actually thought that, I’m just saying the wording is confusing.)

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeJan 3rd 2019

Wait a minute: anonymous must have been saying that there is no symmetry isomorphism (even for finite ordinals) for the ordinal sum monoidal product that is natural with respect to ordinal maps. That’s a correct statement! The current page suggests that $(\Delta_a, +)$ carries symmetric monoidal structure, but that’s wrong.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeJan 3rd 2019
• (edited Jan 3rd 2019)

I have tried to correct the entry a bit. There is probably a lot more that should be done however as it still is largely centred on finite ordinals and the application of ordinal sum in that context.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJan 3rd 2019

Thanks Todd; that’s probably a better guess as to what Anonymous had in mind.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeMay 2nd 2020

Added a redirect for ordinal sum of categories.