@Qiaochu : I don't think that someone struggling to understand the basics would be enlightened by a discussion of topi and higher categories. In fact, most experts get by without ever studying these things...

@Qiaochu : That's pretty meta. In any case, I want to remark (largely for the OP) that the people at the nlab have an amazing ability to take things which I use every day and understand at a fairly deep level and write about them in such a way that I have no idea what is going on. It's probably best not to spend too much time there unless you have a really good reason to.

@Qiaochu: Probably the "non-technical" (but still incredibly technical) motivation page might be better: ncatlab.org/nlab/show/…

@Adam: If there are things there that you use every day and understand deeply, but you think aren't explained the right way...then add to it! They encourage different explanations and want many ways to view the same thing. Take some initiative. Don't just wait for someone else to write it the way YOU want.

@Matt : But I don't agree with their underlying viewpoint (that the best way to understand things is via higher category theory). There are a couple of people who prove good theorems that way, but if you look at the work of most of the exponents of this viewpoint, then you will see a lot of pointless generalization, but few theorems that are interesting to people outside their little world. In fact, if I were in a bad mood, then I might describe things like the ncatlab as damaging to mathematics -- they lead smart but impressionable young people away from the core areas of mathematics.

In summary : there are a couple of geniuses (eg Grothendieck or Jacob Lurie) that can prove good theorems via excessive generality. However, a mediocre imitator of Grothendieck is much worse than an average mediocre mathematician.

@Adam: So surely you would be benefiting mathematics by writing up your viewpoint...as I said they aren't really pushing this viewpoint. They encourage the down-to-earth perspective to be present on all pages. Alright, this isn't the place to get into this argument, I just wanted to let the poster know that the nlab people are really smart and willing to answer any questions you have about anything on the pages including giving you a non-categorical way of thinking about it. So cautioning to not spend much time there isn't great advice in my mind.

@Matt : This is straying offtopic, so it will be my last response. Are you honestly claiming that the ncat people are not pushing a point of view on how one should think about mathematics? They certainly claim to be pushing one -- see their page ncatlab.org/nlab/show/nPOV. Since I strongly disagree with this point of view, why would I want to contribute to their website? And anyway, a website is a pretty lame place to spend a lot of time writing things up. If I decide to write an expository account of something, I'll write a book.

I respect your opinion, and so I see absolutely no need to quibble over this anymore, but all of your concerns have been brought up numerous times in conversations at the nForum for instance see here: math.ntnu.no/~stacey/Mathforge/nForum/…

Regarding "why would I want to contribute to their website?" That's pretty simple. You claim to have valuable things to contribute, and considering you disagree with their point of view you could get your point of view represented.

Regarding "a website is a pretty lame place to spend a lot of time writing things up" ... well, clearly you don't think that or else you wouldn't be on stackexchange.

But even if you do think that, it is becoming a pretty outdated viewpoint. You say you'd write a book, but writing it on a wiki allows you to do it in much less time and have it be polished and expanded upon by other people. Way less time than writing a book.

There are lots of examples of successful ways this is being done, and in all honesty it is probably going to be one of the main ways things are done in the not too distant future.

I've never used this chat before, so I have no idea whether you'll see this or not, but I certainly didn't want to rudely veer off topic on that question.

If you feel intimidated by that huge discussion on the forum I sent you I'd draw your attention to post 21 and the few after that. They very clearly have people with no categorical background contributing and they want it. There are also parts that make arguments why writing something at nLab is a far better use of time than answering someone's question at MathOverflow (or stackexchange).

I don't really care if my view is represented on their website. I don't think there is anyone that contributes there that I have any interest in interacting with mathematically. I've got many more important things to do.

As far as publishing online goes, excitable young people often claim that publishing on webpages will supplant physical publishing, but I see no sign of that happening. Every few years there is a new thing, and then it passes. This fad will too.