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nLab > Latest Changes: ordinal sum

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  1.  
    • 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!

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

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

    Anonymous

    diff, v9, current

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

    @ Anonymous #3 ???? There are no natural isomorphisms but for finite ordinals, [i]⊕[j][i]\oplus [j] is the same ordinal as [j]⊕[i][j]\oplus [i] as both are [i+j+1][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.)

    diff, v11, current

    • 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 (Δ a,+)(\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.

    diff, v12, current

    • 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.

    diff, v15, current

1 to 9 of 9

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