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at connected space I have started a section Properties with statement and proof that connected components are always closed subsets.
at locally connected space in the Definition section I used this to write out the proof that the equivalent characterizations of local connectedness are indeed equivalent.
Cf. also Result 3.3 which appears above at connected space.
First I wanted to write out the proof of this more general statement, but then I was being lazy. But I’ll add a cross-pointer anyway.
It’s not more general.
Do you want me to change anything?
Well, I’ll report having mixed feelings about the inclusion of so much elementary material, but I’m not sure I want to argue about that right now. (I do generally favor the recent custom of giving these little results their own pages, to help prevent page bloat on the larger pages.)
But you might want to change the proof of 5.1, so as to refer to disjoint closed sets (or open sets, depending on taste).
you might want to change the proof of 5.1, so as to refer to disjoint closed sets (or open sets, depending on taste).
Ah, right, thanks. Fixed now.
I’ll report having mixed feelings about the inclusion of so much elementary material
And I was thinking that if only we had more of this kind of stuff, various problems with the public perception of the Lab would disappear. Also I am happy that it catalyzed you adding so much beautiful material.
In any case, my activity in this direction will be over in just a few more weeks. In winter then I will be instead be teaching mathematical quantum field theory. That’s guaranteed not to be elementary.
You are probably far more aware of the public perceptions of the nLab than I am, but is it really on the order of needing more elementary material of the type introduced in undergraduate classrooms? I’m genuinely curious: what is it that you hear?
I am actually in agreement that much of the nLab is very hard-core in places (I for one have trouble reading quite a bit of it), and my guess is that you’re hearing similar things. But my own inclination would be to shoot for exposition at (roughly) a 2nd or 3rd year graduate student level.
By the way, I should emphasize that my feelings really are mixed. I applaud the hard work you’re putting into good, solid, honest exposition for undergraduates, and I think drawing up course notes like you’re doing has a very definite value. I think if it were me, I’d be more inclined to keep sets of undergraduate course notes on my own nLab page, but as I say I think having these little proofs on separate pages is arguably not a bad way to go. Some of these things are matters of personal taste, but I think what I myself would like for concepts of central interest like “connected space” is a main page with a strong nPOV narrative (like the main blackboard), with supporting pages of little lemmas and such relegated off to smaller articles (which are like side boards, to give the readers an option whether to look at them and pursue/recall the fine details, or ignore that and just get the main line/gist from the main article).
BTW: if there is clutter on nLab pages, then I am probably – almost certainly – as guilty as anyone for that.
There is a ton of work to be done in improving the nLab! I’m eager to hear your thoughts. (And I’m sorry if I sound harsh sometimes – I really just mean to be honest.)
I’d be more inclined to keep sets of undergraduate course notes on my own nLab page
My feeling is the opposite, I wish you would move more of the material on your private page to the main Lab.
There is all this re-hashing going on, everyone writing his or her own sets of notes. This seems outdated to me. We should all collaborate to work on one body of text, continuously improving it.
We are speaking of elementary material here. Nevertheless, in the course of preparing these notes, I find myself digging through dozens of texts, none of which have all the good stuff. I’d much rather not waste my time with all this work, would much rather simply open a good wiki from which I could collect everything I need. This doesn’t exist yet. Maybe it’s at us to make it happen.
a main page with a strong nPOV narrative (like the main blackboard), with supporting pages of little lemmas and such relegated off to smaller articles
Yes, by all means, we should separate material this way.
As opposed to text on paper, here we have unlimited space. As long as some material is not actually wrong or misleading, there should be no reason not to include it here. The reader who wants something elementary should be able to get it, the reader who doesn’t should be able to ignore it.
Regarding reception of the Lab: As with many good things, nothing will convince the old senior guard. If we want to spread the word, we need to cater for the young, the unexperienced, the open minded. Those that are the future.
In winter then I will be instead be teaching mathematical quantum field theory.
Excellent news! I have a chance of understanding it at last.
Urs, can I ask you again what it is that you hear about public perception of the nLab? I just figured that those you’re hearing from are themselves relatively advanced, i.e., mostly professionals or advanced graduate students.
I got feedback from all kinds of people. A common complaint is that it is not readable enough. Where it is, people express surprise:
The nlab article on rigid analytic geometry is actually somewhat lucid…
(here)
That’s okay, we are not being paid for helping out with exposition, instead we make notes for our own sake. But conversely I think it should not be actively discouraged if articles try to bow down to pick up their readers from where they are.
For instance we have lots of articles on synthetic and higher geometry, but we had almost no working exposition of ordinary geometry. There is a PhD student working on coding higher Cartan geometry for étale -stacks into Homotopy Type Theory (thesis out in just a few weeks) and for his sake and that of his future readers, I wish for the ordinary concepts of Cartan geometry etc. I could point to more and more expositional Lab entries.
I agree with the last comment, but by my estimate of where people are, who are likely to be reading the nLab, they won’t need a great deal of elementary undergraduate-level exposition. And that embedding a lot of that type of thing in the main articles clutters them up and makes them harder to read. (That’s certainly my experience, as a reader.) That’s why I want to keep such material off to the side, for the most part. But it seems we are agreeing on that anyway:
Yes, by all means, we should separate material this way.
At the same time, I think if you had a long article “Introduction to Topology” which maintains such a level throughout, then that’s good too. Whether it’s on your own labweb or it’s on the main nLab, I don’t see it as making too much difference for undergraduate-level readers (as long as it’s within easy reach). Although: the overall selection and emphasis and arrangement is yours, Urs, and I think maybe it should bear your authorship, and without having too many future editors messing with it once it’s been hammered into shape and you’re more or less satisfied. That might be a reason for maintaining such a thing on your web. Just a thought.
As for the stuff on my web: I quite often do move stuff over to main, if I’m sort of happy with it. I’d rather avoid putting wrong or half-baked stuff on main and making more people suffer for it. (I also have a second web which is hidden from the public, with even more scribblings and incomplete thoughts, with stuff I occasionally move over to my publicly viewable web or to main.) There are a few articles on my public web that I really do want to have my name on as the author, at least for now.
embedding a lot of that type of thing in the main articles clutters them up and makes them harder to read.
Okay, do you have some specific examples in mind? Let’s break them apart.
Just a thought.
I know what you mean. But the same is true for any one contribution to the Lab. Would it not be interesting to go all the way and have all this material be editable by anyone?
But the main reason for doing this is more mundane:
My first lecture notes (you will remember) I did on my personal web: Introduction to Homological Algebra (schreiber). But it was too much of a pain prefixing every single hyperlink with
nLab:
Okay, do you have some specific examples in mind? Let’s break them apart.
Unfortunately I haven’t tallied such instances, but I’ll try to remember to report them as/when I spot them. Thanks.
Would it not be interesting to go all the way and have all this material be editable by anyone?
Obviously it could be. It’s very commendable that you are so open to that. But you deserve a ton of credit! (And I know what you mean about consulting dozens of sources.) Your students are very lucky.
But it was too much of a pain
Ah, very reasonable point.
I agree that introductory material is fine to have, and also that too much of it on “main” pages can be cluttering; I think our pages already have a tendency to be cluttered. I don’t, though, think that it is “outdated” to have multiple expositions of the same material written by different people; there is scope for many different ways to look at the same subject and it’s not always possible to accomodate all points of view in a single place.
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