Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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”.
It’s discrete morphism that actually defines the term ‘discrete object’. Are the two concepts (discrete morphisms and discrete spaces) actually related?
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.
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.
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”.)
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.
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?
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.
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”?
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?
1 to 10 of 10