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nLab > Latest Changes: category of cubes

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTim Campion
    • CommentTimeFeb 3rd 2019

    Added a few additional descriptions of □ ≤1\Box_{\leq 1}, which is the same as Δ ≤1\Delta_{\leq 1}.

    diff, v18, current

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 3rd 2021

    Our pages category of cubes, cubical set, model structure on cubical sets, and so on, seem to all be exclusively about non-symmetric cubes. Given the increasing importance of more structured cube categories (symmetric, cartesian, de Morgan, etc.), especially in relation to HoTT, it would be nice to be more general. What opinions do people have about organization? Should these generically-named pages all be disambiguation pages? Should they include the different kinds of cubes as subsections? Should the generic pages remain about the non-symmetric version, with pointers to separate pages about other versions? In general, should the unqualified term “cubical set” on the nLab continue to refer to non-symmetric cubes, or should it be ambiguous unless qualified?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2021

    Go by the amount of available material: As long as you dont’ have much to say, write it into the next best existing entry. If and when the new material grows larger than is sensible for a subsection, split it off to its own stand-alone entry.

    • CommentRowNumber4.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 3rd 2021
    • (edited Feb 3rd 2021)

    Good question! I have thought a bit about this myself. As usual, I think there are advantages and disadvantages to all approaches. One perspective I like is that something is a ’category of cubes’ if one can construct it as I do at ’category of cubes’: as the free strict monoidal category on a ’structured interval’ category. I think this encompasses all examples. One way to proceed would be to write this down formally (this is not completely trivial as soon as one has 2-dimensional structures involved) and then make all the specific categories of cubes examples of this formal construction for specific choices of structured interval category. They could be listed on the formal/generic ’category of cubes’ page, but have (or the corresponding cubical sets pages could have) their own pages where they are significant enough, e.g. if they have specific applications or special properties.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 9th 2021

    Added an Idea section mentioning that there are many kinds of cube category.

    diff, v19, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 9th 2021

    Added brief mention of some other cube categories.

    diff, v19, current

  7. adding section on the semicartesian cube category

    Anonymous

    diff, v20, current

  8. The symmetric and semicartesian cube categories were listed separately but are the same.

    Evan Cavallo

    diff, v21, current

1 to 8 of 8

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