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nLab > Latest Changes: discrete object classifier

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  1. Page created, but author did not leave any comments.

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeFeb 27th 2021
    • (edited Feb 27th 2021)

    Am I misunderstanding something, or isn’t *⨿** \amalg * the discrete object classifier in the (2,1)-category of groupoids? More generally, in ∞Gpd{\infty}Gpd, the characterization of fully faithful functors as being the insertion into a coproduct show that *⨿** \amalg * is the discrete classifier there.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 28th 2021

    I assume that the author meant to write “faithful” instead of “fully faithful”: The text does correctly start out speaking about 0-truncated morphisms; and between groupooids these are the faithful functors (I have added a pointer to the discussion there).

    I have briefly edited the entry accordingly, but I suppose this deserves to be explained in more detail.

    We have ancient discussion in this direction at pointed sets – As the universal set bundle and scattered remarks elsewhere. One day all this ought to be polished up. Maybe this entry here could be the seed.

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeMar 2nd 2021
    Hello, I'm the anonymous author this discussion is referring to. Sorry about that; I've been making contributions to the nLab on and off for the past year anonymously, the vast majority of them being trivial edits, which do not need to have comments. However, because my edits have been trivial edits for a long time and thus needed no comments, I had forgotten that comments are needed for major edits and page creations. The nForum also happens to be a part of this wiki that I've never had to deal with before. I'll try to keep both in mind in the future.
    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMar 2nd 2021
    Regarding faithful vs fully faithful functors, yeah, I meant ordinary faithful functors in the article. Fully faithful functors should be classified by the subobject classifier of a (2,1)-category, which as Hurkyl pointed out is the coproduct of the point with itself.
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2021

    Thanks for confirming! And thanks for all your edits, much appreciated.

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeMar 2nd 2021

    I’m currently blocked from editing the nLab, but David Corfield says at this blog post here that the role of the terminal object ** in the definition of a (-1)-truncated object classifier is played by Set *Set_* instead of the terminal object ** in the definition of a 0-truncated object classifier, and in a (1,1)-category it just so happens that the object Ω *\Omega_* of pointed truth values just happens to be equivalent to the terminal object **.

    He then goes on to talk about (1,1)-truncated object classifiers, which implies a notion of (n - 1, r)-truncated object classifier in an (n + 1, r + 1)-category.

  8. Fixed definition and added references

    Anonymous

    diff, v7, current

  9. Added details about the classifying morphism

    Anonymous

    diff, v8, current

  10. The definition of the category of pointed set relies on interval objects and internal homs, so we might as well use those instead.

    Anonymous

    diff, v9, current

  11. adding link to class object

    Anonymous

    diff, v10, current

    • CommentRowNumber12.
    • CommentAuthorvarkor
    • CommentTimeJan 3rd 2024
    • (edited Jan 3rd 2024)

    Clarified wording regarding the discrete object classifier in Grpd.

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorvarkor
    • CommentTimeJan 3rd 2024

    This page could really do with some references, but as far as I can tell this concept does not exist in the literature yet…

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