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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2011

    I noticed that discrete object used to redirect to discrete morphism, where I expected it to take me at least to discrete space, if not to its own entry.

    We should eventually disambiguate here and add some comments. For the moment I made it redirect to discrete space and added there a remark “to be merged with discrete morphism”.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeDec 9th 2011

    It’s discrete morphism that actually defines the term ‘discrete object’. Are the two concepts (discrete morphisms and discrete spaces) actually related?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeDec 9th 2011

    discrete morphism and the discrete objects which it defines are talking about 0-truncated objects (categorical homotopy). By contrast, discrete space is talking about cohesively-discrete objects (geometric homotopy). They are orthogonal concepts.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2011

    I’d rather that we eventually stop using “discrete” for “0-truncated”, and always use if for “cohesively-discrete”. That seems more systematic to me.

    Notice that we can recover “0-truncated” from “cohesively discrete” to some extent by devising a suitable cohesion. For instance the category of simplicial sets is a cohesive 1-topos, and its discrete objects are the simplicially constant simplicial sets, hence, up to equivalence, the 0-truncated Kan complexes.

    This is not fully satisfactory for talking about 0-truncated/h-level 2 generally, but then, if you want to talk about it generally, why not say “0-truncated” instead.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeDec 12th 2011

    I think I can go along with that. But we should make sure to point out that many people use “discrete” to mean 0-truncated. (I also like the noun “0-type” to mean “0-truncated object”.)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2011
    • (edited Dec 12th 2011)

    I have added a discussion along these lines to discrete space, in the Examples-section. (Similar discussion already existed at discrete category all along.)

    I have also made discrete object in a 2-category a redirect to discrete morphism, and I would be inclined to actually rename the latter by the former term, the remarks about “discrete morphisms” there notwithstanding.

    Then I have added there in a new Examples-section a brief pointer to discrete groupoids etc.

    Finally I have also added pointers to discrete object in a 2-category to discrete category and maybe elsewhere.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeDec 12th 2011

    I actually think the discussion of 0-truncatedness as a type of discreteness using simplicial sets is confusing. I have enough trouble keeping the two concepts straight as it is. Is there a real reason to keep that remark at discrete space?

    I would be inclined to actually rename the latter by the former term, the remarks about “discrete morphisms” there notwithstanding.

    Can you explain why?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2011

    Is there a real reason to keep that remark

    Hm, I’d think that with two conflicting usages of a given terminology, it is worthwhile to say in which sense they are compatible after all.

    I have put it in parenthesis now. :-) Is it so bad that we really need to remove it? It feels strange to remove information if its not wrong.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 12th 2011
    • (edited Dec 12th 2011)

    Can you explain why?

    I guess I am prejudiced by the groupoidal case, where the “N.B.”-remark in the entry becomes automatic.

    How about we at least create separate entries “discrete morphism in a 2-category” and “discrete morphism in a (2,1)-category”? Or the like. Or “(0,1)-truncated morphism”?

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 13th 2011

    It feels strange to remove information if its not wrong.

    I remove information from my papers all the time even though it is not wrong. My first impulse is always to write down everything that is true; then I have to pare away to make it vaguely readable.

    Of course, the nLab has a somewhat different purpose than a paper, so that doesn’t necessarily apply. I put that information in a labeled Remark to offset it some more; I think I’m happy with that. I also realized the organization on discrete space needed a lot of work (examples were scattered throughout the Definition and Idea sections, with a lot of duplication) so I reorganized it.

    I guess I am prejudiced by the groupoidal case, where the “N.B.”-remark in the entry becomes automatic.

    Indeed! Can you try to overcome your prejudice? (-: The page is not about the groupoidal case (or at least not primarily about it). What’s wrong with leaving it as it is?