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Oops, we need to harmonize here: this overlaps/connects with infinitesimal object, subsection atomic objects. Also with amazing right adjoint.
Given the current definitions, is the right convention this one:
(-)^D preserves small colimits ==> D is tiny
(-)^D has a right adjoint ==> D is atomic
atomic ==> tiny
?
At least I'll quickly add cross-links to this extent.
I added a ref to tiny object and infinitesimal object also now to compact object.
I had missed the bit about finite objects there. So if I understand correctly, as least inside a topos we have the following implications:
atomic => tiny => finite => compact .
Which is kind of fun, as it does make sense. I'll add that list of implications to these entries.
<div>
<blockquote>
atomic => tiny => finite => compact .
</blockquote>
<p>sorry, wait, this is wrong...</p>
</div>
I created compact object - contents and added it as a floating table of contents to the correspondding entries
Mistakenly started entry 'tiny'. How do I get rid of that?
Change the page name to tiny > history and change its contents to
< [[tiny object]]
(which I have done).
Then add [[!redirects tiny]] to tiny object if you think that are likely to be links to tiny that are meant to go to tiny object. (I did not do this part, since it requires judgement.)
Owen Biesel had left a question at tiny object, which I answered in the affirmative by incorporating it into the text.
H’m, my comment #6 above is no longer formatted correctly, and I can’t figure out how to fix it!
ad 6: Word “history” for outdated pages is, in retrospective, a bad choice. When I saw it first time, nearly 2 years ago, I thought that it was about history of that notion in mathematics. Now if I want to search for history articles, I will mainly get a list of outdated non-history articles. On the other hand, for long entries I would sometimes like to split off terminological and historical issues into a separate entry not to interfere with -hypermodern-point of view, but no good alternative key word for modifying the title does not come into my mind, nor we have a category: for that. Suggestions are required from elf-thinktank :)
The “history” here is supposed to remind one of the History link at the bottom of the regular page. Both contain (in principle) the history of past edits to the article.
We could use another name for that, or for your articles on historical aspects. (Maybe there’s some way to use “historical” instead? That just came to me, maybe not a very good idea.)
I was thinking on historical as a category: class, but it seems less suited for a part of the title. Say we have derivator - historical notes would be reasonable except that I and most of the others have problem with hyphen versus dash, emptyspaces and other URL features for such names. Most entris I work on I still use the habit to type the end part of the URL by hand. You may have some advice.
The obvious name is history of derivators, but since that causes clashes in searches, one could use instead historical aspects of derivators. But I’m really not sure that this works well.
Categories might be more useful to search on instead. Or we could rename the > history articles in some less mnemonic way. In principle, we could even use case-sensitive searching to distinguish history (in titles of articles on historic matters) and History (in titles of pages with stored edit history), but the default search in instiki is case-insensitive (which is good for other reasons).
Now the whole history topic is still a marginal issue but with growth of Lab in future I think it will be really convenient to create/separate many historical articles as stand alone entries. For a category:historical I think we can decide right now. It looks unquestionable.
To change the labelling of obsolete entries, I think it is too big a task and change to go through.
I like your title template historical aspects though notes sound more complete (and sounds reminding Bourbaki’s terminology). Something like historical notes on derivators (do we need plurals? probably yes). Whatever we decide there should be clear link on the derivator page to historical notes on derivator, early up in the page.
To change the labelling of obsolete entries, I think it is too big a task and change to go through.
I can do this pretty well automatically if we decide how to change them.
Something like historical notes on derivators (do we need plurals? probably yes).
On ’derivators’, definitely; on ’notes’, probably.
finally added the pointer to Toposes of laws of motion to tiny object, which is an early (maybe the earliest?) appearance of the term “atomic object” (or rather “a.t.o.m”).
Added to tiny object a short note that the only tiny object in the category of modules over some ring is the zero module. I’m sure that there is a more elegant proof than the following very ad hoc one:
If is a tiny object in a category of modules, the map
is bijective and in particular surjective. Therefore the diagonal embedding can be written as for some linear map , where is one of the inclusions . This shows that any element of is zero.
Thanks. I have cross-linked your addition with category of modules.
Well, this example seems slightly misleading to me since it is usually more appropriate to think of module categories as -enriched, not -enriched. There is it certainly true that a finitely generated projective module is tiny.
Please add a sentence with that remark to the entry!
But if is zero then is a singleton, so anyway. So there are no tiny objects at all in an additive category (considered as a -category).
@Zhen Lin: good point.
Thank you for the additions and fixes!
I have added more hyperlinks.
(As a rule of thumb, I think every technical keyword deserves to be hyperlinked. I don’t want to be a pain about this, but just think about your own younger self and how cool if would have been if you had learned maths from texts where whenever you came across any term you’d wanted to see more details on, you could have just click on it!)
Every instance of every technical keyword?
I might not have linked (in my most recent edit) every keyword that hadn’t linked prior, but often I follow the custom in Wikipedia that the first occurrence of a keyword in an article is to be linked, but later occurrences in that article are optional.
Yes, I know. I tend to want to link every occurence, because one never knows at which point a reader enters a text. While I see why that may seem too much. I’d still prefer it.
Added to tiny object a nice observation which I learned from Qiaochu Yuan’s blog: In any Set-enriched category with a zero object, there cannot be any tiny objects.
I took the liberty of removing the proof that there are no tiny objects in (originally added by me in #16 and refined by Zhen Lin in #20), since the argument for the more general observation is much simpler.
Thomas Holder alerts me of the presently last edit at tiny object (here). User Paolo Capriotti simply removed remark 1.5
If is a sheaf topos, then tiny objects and atomic objects coincide, by the adjoint functor theorem.
without any replacement or comment.
This is occasion for me to appeal to all users to stick to the request that non-trivial edits be announced here on the Lab. In any single case this may seem of little use, but examples like this show that it is important in order to preserve the culture of joint editing, without which the nLab cannot work.
What I don’t understand is why it was removed. The statement seems absolutely correct; I’m pretty sure I can prove it (using for example the special adjoint functor theorem).
Edit: so in fact I’m going to roll back, and provide a proof, after I get back from an errand.
Edit: done.
Re #27: I think it would not be too much work to generate a message on the nForum when an edit is made. I will try to implement this when I get the chance. Probably something like a checkbox to tick or untick when one makes an edit on the nLab for announcing the message on the nForum, together with a text field to describe what is done, with the checkbox ticked by default, and with instructions to untick it only for trivial edits.
Todd, thanks! I gave your proof a prop/proof-environment: here.
Hi Richard,
you write:
it would not be too much work to generate a message on the nForum when an edit is made. I will try to implement this when I get the chance. Probably something like a checkbox to tick or untick when one makes an edit on the nLab for announcing the message on the nForum, together with a text field to describe what is done, with the checkbox ticked by default, and with instructions to untick it only for trivial edits.
To my mind something like this would be excellent to have! A box with “Describe your edit in a few words”.
Hi, sorry for not writing a comment here. I figured that statement originated from past differences in terminology and was somehow left over. That’s why I thought it was ok to just remove it. According to the page, tiny means that the hom functor into Set preserves colimits, whereas atomic means that the internal hom has a right adjoint. So even if preserving colimits and having a right adjoint are the same thing in those special circumstances, tiny does not imply atomic, because they are about different functors. For example, any representable in a presheaf category is tiny, but very rarey atomic (e.g. the interval in simplicial sets is not atomic).
I’ll make sure to drop a comment in the forum next time.
Re #29, I agree that would be nice. What thread on the nForum would you post the message to, though? Standard procedure is that it should go in a “Latest Changes” thread named after the page being edited, and if there isn’t such a thread yet then one should be created. Some pages, though, already have (for whatever reason) more than one such threads - would it just pick one randomly to use?
Thanks very much for explaining, Paolo.
Todd, Paolo,
I should look into this in more detail, but from just your exchange I am unsure now what to make of it: Is there a remaining issue with the terminology in the entry?
I think there is: the way that “atom” or “atomic object” is defined and used in say Bunge’s thesis, and which is the referent of the proposition 1.5, is clearly different from the way it’s defined within the entry – as clearly pointed out by Paolo, and something I had unfortunately overlooked. So Paolo was justified in removing it, but rather than removing it again, I think the entry should be expanded to explain the discrepancy in terminology, and some sort of enhancement of the article should be made to take into account the “amazingness” of the stronger notion of atom (or a.t.o.m. = “amazing tiny object model” to use Lawvere’s acronym), especially as it arises in SDG.
Okay, I performed some edits to bring in the usual internal/external distinctions which are relevant here. I also added a proposition that in a category of presheaves on a category with finite products, internally atomic objects coincide with externally atomic objects; probably a stronger result should be included that reflects the situation with the types of sheaf toposes that crop up in SDG, but I haven’t found the time yet to look into this (maybe Paolo would like to give it a go?).
Oh, I forgot to mention that I saw, in the proof of the current proposition 2.5, a dangling reference which reads \ref{CauchyComplIsFullSubcatOnTinyObjects}. I’m not sure what this refers to.
Thanks, Todd!!
By the way, speaking of non-announced edits: Lately Thomas Holder has been making lots of edits to entries related to topos theory and topology, as you can see from the logs. Myself, I don’t feel that I have time to look into this, and my request to say what the edits are about got rejected.
I took a look at some of Thomas’s recent edits, which look good to me. For example, he recently created lex total category, which has been discussed elsewhere but really deserved a page to itself. (Actually, it had been “created” before, but as a spam page, which was then emptied out before Thomas gave it proper shape.)
In case Thomas is reading: the recent edit at total category which mentions non-cototality of Grp was revised to be more than a reference to Wood’s paper. The examples of and of commutative rings are now treated in more detail, and I took the opportunity to add the example of the category of compact Hausdorff spaces, as being both total and cototal.
(And oh yes, I created a stub Richard J. Wood.)
Since Urs puts me in a spot here: “the lots of edits to entries related to topos theory and topology” that I’ve made recently are mostly waived by the triviality convention on edit announcements and consist almost entirely of interconnecting to or dropping isolated references by copy-and-paste from the few pages that I’ve created around and closely related to the page exponentiable topos that was announced on the forum, namely, the pages injective topos, lex total category and Scott topos. The only edit interferring in any way non-trivially with already existing material is the addition of a short remark on cartesian-closedness at total category, a rather minor and conservative extension of the page while I was at interconnecting it. Even the impression that I would have done plenty here recently is, maybe unfortunately, maybe fortunately, an optical illusion due to the number of trivial edits triggered by the creation of these three new pages which I felt were ’implicitly co-announced’ with ’exponentiable topos’.
waived by the triviality convention
No, creation of new entries ought to be announced.
This is not about putting you in a spot, this is about giving the rest of us a chance to participate and to profit.
it had been “created” before, but as a spam page, which was then emptied out before Thomas gave it proper shape.)
While I am at it, I’ll say again that we should not recycle spam pages this way. That this keeps being done seems to be based on a misunderstanding of how the Wiki software works: While the idea is probably that this recycling serves to “overwrite” and hence remove the spam, in fact the opposite is true, it serves to anchor and archive the spam in our poor database.
Instead, the way to completely remove something from the nLab database is to not have any links to it, and in particular to not keep it in the history of another entry. The server has a trash removal command which deletes all “orphaned” pages, which are those that have no links whatsoever to them.
Hence, when creating a new entry, please never re-use an existing page.
If someone has already used the name you want, then you can rename that page as empty nnn (for the next number), empty the page and remove redirects.
Test.
Seeing if this works.
Yes, standard practice seems to be to announce all new page creations, even nearly-empty stubs. Perhaps that should be written down somewhere?
Re #31: thanks for the feedback! I’m now working on implementing it.
Re #33: thanks for the feedback and for the question, which was useful for me to consider! I’ll reply properly once I have an implementation up and running with some suggestion for this, and then we can discuss whether to change it. Basically my idea would be: use the last used thread with the same title as the nLab page, and create a new one if one doesn’t exist. Also have a single thread for all edits, regardless of whether the author of the edit chooses to announce them, and regardless of whether they are trivial.
Re #46: I can enforce that in the new tool.
43 > Instead, the way to completely remove something from the nLab database is to not have any links to it, and in particular to not keep it in the history of another entry. The server has a trash removal command which deletes all “orphaned” pages, which are those that have no links whatsoever to them.
Are you kidding ? All ? It is too harsh to automatically remove the entries without a human approval. Do we know which non-spam pages have been removed so far for this reason ? While I agree that we should all tend to plan the pages non-orphaned as much as it is possible and especially as the entries grow, there is lots of good pages which are still orphaned and this is often convenient in an early phase. One has to record something quickly and has no time to think where would that be classified or does not want to mess with a bigger page before a page or a whole circle of related new stubs is created together (say when studying a new topic). Should we live in fear of deletion whenever the page is started from scratch ?
For example, there are other legitimate situations say you want to record the homepage of a collaborator or somebody (even a student) you heard a talk and the person is not important as a contributor to some subject to be listed under big title like topology. Or pages of the type of sandbox or say emptypage which is useful to me at least as a start or test point whenever I am on bad internet connection or extra slow device.
47 > Basically my idea would be: use the last used thread with the same title as the nLab page, and create a new one if one doesn’t exist.
I assume you mean: a thread inside Latest Changes subsection of Forum ? I think it would be counterproductive if the discussions in other sections of the Forum would be interspersed with the latest changes messages just because the title matches. The Latest Changes by the default should fit in (though to me this looks even better something like news feed category).
29 > I think it would not be too much work to generate a message on the nForum when an edit is made.
Is there a reasonable delay ? I mean, when we edit a page there is some 20 minutes or so (I do not know exactly), if you are idle more than that and continue editing then you generate a new version. Maybe the author wants to report in more sensible way and could get the same grace period before the editing is finished. I think that it could be more sensitive if the machine posts a message to the forum once a idle-grace period of non-editing, or stepping in of a second new contributor, occurs.
All ? It is too harsh to automatically remove the entries without a human approval.
Hi Zoran, don’t worry, I agree completely that we cannot use this criterion for removing pages, and would never use it, and I’m sure Adeel wouldn’t either. I don’t know if it has ever been used before, perhaps when Andrew Stacey was administrator, but I would be surprised if it has been used on any occasion in the recent past.
But I also agree completely with the essential point that Urs is making, that recycling the pages helps to anchor the spam, and that it is easier to remove the spam if the pages are left as untouched as possible.
I assume you mean: a thread inside Latest Changes subsection
Yes, exactly, sorry for not being clearer.
Is there a reasonable delay ? I mean, when we edit a page there is some 20 minutes or so (I do not know exactly), if you are idle more than that and continue editing then you generate a new version. Maybe the author wants to report in more sensible way and could get the same grace period before the editing is finished. I think that it could be more sensitive if the machine posts a message to the forum once a idle-grace period of non-editing, or stepping in of a second new contributor, occurs.
Interesting, thanks for raising this. For now I was just planning to make the announcements immediately on submission of an edit/page creation. But we can certainly change it to something like you suggest if people would prefer that. Let’s discuss once we have an initial implementation up and running.
Indeed, for years Zoran has been creating orphaned pages, and for years I have been asking to cross-link them and have done so myself on a myriad of occasions. I don’t think there are a large number left.
Irrespective of garbage collection, I find it clear that if a page is not cross-linked to anything else on the nLab, it could as well not be part of the nLab. Not cross-linking new entries is similar to not announcing them here, in that it defeats the purpose of contributing to a public Wiki.
We have never used the “Delete Orphane Pages”-command, but it happens to be one of the few functionalities that Instiki provides. See at Edit Web under “Other administrative tasks”.
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