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The question “What is nLab?” is incomplete without first asking “What is the nCommunity?”
The nCommunity is a collection of researchers interested in examining their field from (or whose field is) the nPOV.
How does the nLab fit into the nCommunity?
First and foremost, the nLab is a reference. A reference for things viewed from the nPOV. It is both formal and, in some cases, informal. But in all cases, it is intended to be a reference we can point to as we do our jobs. It is mostly a selfish endeavor. We, for the most part, don’t put stuff on the nLab solely for the benefit of others. We put stuff on the nLab for the benefit of ourselves (although their are wonderful exceptions of angels as well).
The nCafe is the nCommunity journal. It is a collection of articles of interest to the nCommunity written by a group of regular (?) contributors.
The nForum is the place to discuss things of interest to the nCommunity from the nPOV and hopefully a resource for generating material for the nLab.
PS: I was never very fond of the “lab book” analogy. “Note book” isn’t quite right either, but is closer. As I said, it is first and foremost a reference.
The nLab is a lab book written by people who like it when their lab book looks like a collection of expository articles. That’s why it looks like a collection of expository articles.
But it’s not really a collection of expository articles. It only looks that way. It’s really a mess. (^_^)
Seriously, the collaborative editing process is what makes something look like an expository article that started out as a collection of notes. Also, many of us are moving notes from previous lab books into the nLab, and straightening them out as we go.
I pretty much stopped making any notes anywhere except on the Lab.
my original research I type into my private lab, currently busily working on this entry
if for my research I need some lemma or theorem from the literature, I put a link to the corresponding nLab page and create it if necessary. Recently I needed the cosimplicial Eilenberg-Zilber theorem, so there it is now on the nLab: Eilenberg-Zilber theorem.
when I feel my original research is somehow good enough that I myself count as a “secondary source”, then I add observations to corresponding main nLab entries. For instance at interval object only the first bits are material from the literature, then further below comes original research from Todd and then from me, with the help of others.
when I teach or run a seminar, I strictly prepare all notes on the nLab now. What is useful for a small number of people should be useful for more. This is where much of the Lurie-transcription comes from that Todd mentioned, because we are currently running a seminar on this.
finally, now that it is there, I start having parental feelings towards the nLab. So sometimes (too often for my own good, I think) I spend half an hour just trying to make it look better. Looking better also involves that an entry that mentions some standard concept also says somenthing about that standard concept. As a result, we do spend some time just writing expositions of very standard stuff. I think the point is that an exposition of cutting-edge stuff is all the more useful the better it is embedded into expositions of standard stuff.
Eventually, all good math in the end looks like an exposition, right? Math is about understanding the book of nature, so page by page we try to extract transcripts of that book.
There will inevitably be more expository articles than research articles because each research article spawns multiple expository ones. For example, in working on manifolds of mapping spaces then I decided that I needed some space to be sequentially compact. That space didn’t exist, so I created it and listed the property that I needed. As I work on manifolds of mapping spaces then these little buds will shoot off and grow (as others add to them). Also, someone coming to the nLab from outside is more likely to search for something known than unknown and so the expository ones are the more likely to be seen.
We have no problem with exposition on the nLab, far from it! The key point is we work here.
Also, there’s probably an effect from knowing that this stuff is viewable by the public so we are probably a little more careful in our writings that if we were doing it all privately. So in my office, I might scribble something about sequential compactness with a note “Look this up”. Whereas here, I might take the time to look it up first, or (if I can’t find it) put it in a query box and ask about it here, whereupon someone else who knows will take the trouble to clear the matter up for me (example coming up).
...when I teach or run a seminar, I strictly prepare all notes on the nLab now.
This is one big benefit of having an online, linked, editable knowledge base, one of my motivations is to provide a source that people could use to prepare seminars and classes, sort of salvage material from offline sources, which are often hard to come by (like books where no reprint has been done for ages). Do the attendees of your seminars use the nLab, too, to look up stuff that you told them?
This is one big benefit of having an online, linked, editable knowledge base,
Yes! If I think about all the time I spent in my life digging through books, through articles, etc. on the search for information about particular topics – or worse and more often: not yet knowing what it is I am actually looking for. I imagine if all that information were instead available in one huge hyperlinked edifice, the amount of time wasted would be immensely reduced.
Do the attendees of your seminars use the nLab, too, to look up stuff that you told them?
Yes, they do, at least a good number of them. In parts that’s because this happen to be the notes that I provide, but of course then they follow the links and roam around. Recently I started to explain something on the board, when the person I was talking to informed me that he haad already learned this from my nLab article.
But the thing is: even if all other people in the seminar or course would not look at the nLab entries, already just for me having my notes there is a big advantage for me, over having them on paper or in some Latex file. It has a multitude of advantages:
I can more easily and more quickly find my own stuff. Google is on my side when the stuff is on the nLab, and all the hyperlinks are there.
The stuff I put on the nLab tends to automatically improve over time. This is really the most astonishingly pleasant aspect of the Lab: I put something here, and next day somebody has added a link to a topic that I hadn’t known was related, or spotted a mistake and complained about it, or even fixed that mistake. All this happens regularly here.
Finally, as Andrew already mentioned, the fact alone that I put these a priori private notes in a publuc area makes me be much more careful about writing them, and makes me write them in a much more re-usable form.
I have added to What is… the nLab? this quote from Diderot on the need for an Encyclopédie:
As long as the centuries continue to unfold, the number of books will grow continually, and one can predict that a time will come when it will be almost as difficult to learn anything from books as from the direct study of the whole universe. It will be almost as convenient to search for some bit of truth concealed in nature as it will be to find it hidden away in an immense multitude of bound volumes. When that time comes, a project, until then neglected because the need for it was not felt, will have to be undertaken.
(The surrounding paragraphs deserve to be improved/reorganized. But no time now.)
There’s a certain unresolved tension expressed on the page between an encyclopaedic outlook and the nPOV. Encyclopaedias have tended to present themselves as the accumulation of facts, such as in the 1889 ninth edition of the Encyclopaedia Britannica:
The available facts of human history, collected over the widest areas, are carefully coordinated and grouped together, in the hope of ultimately evolving the laws of progress, moral and material, which underlie them, and which help to connect and interpret the whole movement of the race,
as though facts are just position-neutral objective reports to be arranged under alphabetically ordered headings, and the theories that will unite them will just come into the minds of readers.
Hegel’s use of the term to present a system in Enzyklopädie der philosophischen Wissenschaften im Grundrisse is untypical. For one thing, it has an essential order,
I. Logic: the science of the Idea in and for itself.
II. The Philosophy of Nature: the science of the Idea in its otherness.
III. The Philosophy of Mind: the science of the Idea come back to itself out of that otherness.
In the nLab spirit, I see this last quotation doesn’t appear at Encyclopedia of the Philosophical Sciences, so will add it.
A little further down Diderot has this impressive paragraph (now added to the entry, too):
Thanks to encyclopedic ordering, the universality of knowledge, and the frequency of references, the connections grow, the links go out in all directions, the demonstrative power is increased, the word list is complemented, fields of knowledge are drawn closer together and strengthened; we perceive either the continuity or the gaps in our system, its weak sides, its strong points, and at a glance on which objects it is important to work for one’s own glory, or for the greater utility to humankind. If our dictionary is good, how many still better works it will produce.
I am taking this from this nice webpage here which does point out the striking resemblance to the idea of hypertext.
If our dictionary is good, how many still better works it will produce.
I gave Denis Diderot a stub entry.
Did you see just before that final paragraph his comparison of the totality of knowledge with mathematics itself?
One page always presents something different from the preceding or subsequent page. The need of a proposition, a fact, an aphorism, a phenomenon, a system, requires no more than a single citation in an encyclopedia, just as in geometry. The geometrician refers from one theorem or problem to another, and the encyclopedist from one article to another. And so it is that two types of opus, which seem so very different in nature, come by the same means to create a most dense, tightly knit, and continuous whole. What I say is so precisely true that the method by which mathematics is treated in our dictionary is the same followed for other topics. From this point of view there is no difference between an article on algebra and an article on theology.
What is… the nLab (schreiber) looks to me more of a manifesto and less of an intro about what it is for a newcomer. That is OK, of course, but just to give you some feedback.
The intro for the newcomer is supposed to be at About.
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