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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!
I added a note here on Lawvere’s definition of the ordinal sum of categories, from “Ordinal sums and equational doctrines”.
@ 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?
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.)
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.
Thanks Todd; that’s probably a better guess as to what Anonymous had in mind.
Added a redirect for ordinal sum of categories.
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