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New stub order category (redirecting also the more general case of preorder category).
I would just redirect this term to poset.
Or perhaps (0,1)-category.
This would be for ultracategorists, not for ordinary mathematicians who would like to bridge the hi entrance barrier to understand the $n$Lab lingo and use portions of material without becoming specialists in category theory. Besides the term is widely used in meaning viewed there differently from informationally equivalent (pre)ordered set.
But the nPOV is that a preorder is no different from the corresponding category. We should certainly have such a redirect, and a discussion of the terminology at the page redirected to, but the goal should be to inculcate the nPOV in the reader, not assist them in maintaining a distinction without a difference. I agree that pointing to (0,1)-category is maybe a bridge too far, but I think redirecting it to poset is entirely reasonable.
Absolutely. If there is energy to improve the general readability of nLab pages, that is best done by explaining differences of perspectives right there on the relevant pages, instead of creating spurious synonym pages not linked with the material that they are trying to disambiguate.
Even I find that searching the nLab for a term and being redirected to a page in which that term is barely mentioned is very off-putting. Perhaps we should try to ensure that some explanation of the linkage between the intuitions behind the terminology is provided, thus at (0,1)-category the definition is quite daunting and a fairly down-to-earth discussion would help. For instance, the proof that any (0,1)-category is really a poset given on that page is not 100% user friendly.
Although the lab is a ’lab book’ for us, it also has a use (for the wider mathematical community, and especially for the debutant’ higher category theorists) as a place for learning the nPOV. At some places it succeeds in that extremely well, but not everywhere. ’Poset’ is an accepted generally understood term, so the wording here should be saying ’posets from the viewpoint of $(n,r)$-category theory are essentially just (0,1)-categories’, and that suggests a whole wealth of generalisations of ideas to try out in a search for higher categorical analogues of properties. As it is, the page reads more as saying ‘you may call them posets but we know they are really (0,1)-categories,’ and I don’t think that shows higher category theory as an enabler to revealing structure.
I followed up the link to poset and find just:
The 2-category $Poset$ is the full sub-2-category of Cat on the posets.
Oh joy, oh rapture! what clarity. It even contains a self reference and hence I could spend a happy infinity or two clicking on the link to poset in the entry poset to be directed to the entry on poset. (I will try and make some changes to these entries, to see if I can make them a bit more user friendly. If I don’t manage to improve them to everyone’s liking we can always revert.)
I followed up the link to poset and find just:
Oops, that’s a technical error somewhere! poset is supposed to redirect to partial order, but for some reason it appears that instiki is sending it instead to Poset, the (rather stubby) page about the category of posets. Did instiki’s case-sensitivity change at some recent point in time? I couldn’t swear to the history of poset/Poset, but I’m pretty sure that in addition to the page topos about toposes we used to have a page called Topos about the 2-category of toposes, whereas right now the latter seems to redirect to the former.
Aha! That does clarify things somewhat as what was going awry. Can it be fixed simply? Perhaps replacing the link to poset by one to partially ordered set? That leaves the user-friendliness of the page on $(0,1)$-categories. (It seems to me that there might be a place to help an understanding of $(n,r)$-categories so might need a different type of wording recalling there what an $(n,r)$-category is then being kind to the reader and taking it apart (avoiding the use of ’truncated’ until a bit later) in quite simple terms all within the idea section.
Regarding posets, we have Pos as well as Poset. Anyone have a preference for the main name?
poset goes to Poset, while posets goes to ’partial order’.
I guess there aren’t so many cases of the capitalized name of a category misleadingly redirecting. Any other cases than Topos, Set, Poset? Normally there’s an abbreviation involved.
Oh, did we not have Spectrum once? Yes, searching show there is such a page, but Spectrum directs to ’spectrum’.
I’m glad we noticed the capitalization issue and got it fixed.
I suppose the page (0,1)-category could be expanded with more discussion of negative thinking, but maybe an even better place for that would be negative thinking. (I just added a link from the former to the latter.) In any case I don’t think the userfriendliness of (0,1)-category is relevant for the question of whether to redirect order category to poset (i.e. partial order), since the latter is much more user-friendly.
My point was to use (0,1)-category as an opportunity to help explain (n,r)-category.
I created PreOrd (which to my surprise seems not to have been created before under any name, although I didn’t search for long).
This discussion got somewhat sidetracked, but since Urs seemed to agree with me, no one else specifically expressed an opinion, and Zoran didn’t continue the discussion, I went ahead and redirected order category to poset (and preorder category to preorder). But if anyone disagrees (including Zoran, if you’re still reading and want to continue to argue for your POV), speak up!
Thanks. I have equipped your “see below” with a hyperlink, and mentioned the term “order category” again in the paragraph where it is described (here).
At which address I can find the old page preorder category/order category ?
Thanks!
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