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nLab > Latest Changes: virtual double category

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    • CommentRowNumber1.
    • CommentAuthormaxsnew
    • CommentTimeJul 11th 2017
    • (edited Jul 11th 2017)

    I added some examples of virtual double categories that do not have composites described in Crutwell-Shulman.

    • CommentRowNumber2.
    • CommentAuthormaxsnew
    • CommentTimeAug 22nd 2018

    @Mike Shulman, on this page you said virtual double categories can be defined as

    a generalized multicategory, in the sense of Hermida, Cruttwell-Shulman, and others, relative to the monad T on graphs-internal-to-Cat whose algebras are double categories

    did you mean “categories internal to Grph” rather than “graphs internal to Cat”? That seems to be how y’all defined it in Cruttwell-Shulman. I have a suspicion they turn out to be the same thing, but when you define the monad it is easier to think of it as categories internal to Grph because the monad is a lifting of the free category monad on Grph.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeAug 22nd 2018

    They are the same thing. I find it easier to think of graphs internal to Cat, because then the monad is just the special case of the free internal-category monad in the topos of graphs.

    • CommentRowNumber4.
    • CommentAuthormaxsnew
    • CommentTimeMay 17th 2022

    Example of a “nullary” 2-cell

    diff, v14, current

    • CommentRowNumber5.
    • CommentAuthormaxsnew
    • CommentTimeMay 17th 2022

    Hm, I tried to use tikzcd but it is not rendering properly. Am I doing something very wrong?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2022

    Am I doing something very wrong?

    Where should we look to find out?

    Let’s take it to the Sandbox.

    • CommentRowNumber7.
    • CommentAuthormaxsnew
    • CommentTimeMay 17th 2022
    I copied it to the sandbox, here's the code in case it gets overwritten:


    \[\begin{tikzcd}
    & X \\
    & {\Downarrow \alpha} \\
    {Y_0} && {Y_1}
    \arrow["f"', from=1-2, to=3-1]
    \arrow["g", from=1-2, to=3-3]
    \arrow["q"', "\shortmid"{marking}, from=3-1, to=3-3]
    \end{tikzcd}\]
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2022

    You need to remove the enclosing \[ \].

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2022

    (This is an issue to be handed to an ambitious sysadmin as soon as we have a new one: So far the tikzcd-support here is completely alien to the ambient Instiki: It interacts neither with display-math environments, nor with equation numbering nor with bullet listst nor with any other form of indentation.)

    • CommentRowNumber10.
    • CommentAuthormaxsnew
    • CommentTimeMay 17th 2022
    Thanks, Urs!
    • CommentRowNumber11.
    • CommentAuthorvarkor
    • CommentTimeDec 17th 2022

    Add original reference.

    diff, v17, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2022

    Let’s copy all references also to the author’s page, with a link back to the entry, such as here

    • CommentRowNumber13.
    • CommentAuthorvarkor
    • CommentTimeMar 9th 2024
    • (edited Mar 9th 2024)

    Mention that virtual double categories are 2-comonadic over double categories.

    diff, v19, current

    • CommentRowNumber14.
    • CommentAuthorBryceClarke
    • CommentTimeMay 10th 2024

    Added the example of multicategories.

    diff, v20, current

    • CommentRowNumber15.
    • CommentAuthorMatt Earnshaw
    • CommentTimeOct 31st 2024

    Multi-sources need not be composable. Mention covirtual double categories.

    diff, v21, current

    • CommentRowNumber16.
    • CommentAuthorvarkor
    • CommentTimeOct 31st 2024

    Use the tight and loose terminology to avoid ambiguity.

    diff, v22, current

1 to 16 of 16

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