Gerbes in The Guardian
Posted by John Baez
In his post on Future Gazing, David Corfield invites us to predict the future of $n$-categories over the next year — the second year of the n-Café. I’m seeing lots of signs that $n$-categories are catching on — but it could be happening even faster than I expected. Here’s one article I would not have thought to see in an important British newspaper:
- Marc Abraham, Beyond definition: when I say gerbe I don’t mean grebe, The Guardian, Education Guardian Weekly, August 21, 2007.
Thanks go to Eugenia Cheng for pointing this one out!
Alas, the author of this article is the editor of the bimonthly magazine Annals of Improbable Research and organiser of the Ig Nobel Prize — not a promising sign. And, while he limits his mockery of gerbes to a bit of polite ribbing, he makes no effort to explain them. He mainly explains how hard they are to explain. The article begins thus:
What is a gerbe? A gerbe is a mathematical object. It happens to be pretty obscure. Many obscure concepts are easy to understand. One just needs (A) a little patience, and (B) a reminder that most ideas are built upon other ideas. So it is, perhaps, with the gerbe.
To grasp a new concept, just (C) find a concept upon which it is built, and then (D) grasp that earlier concept. To grasp the earlier concept, just (E) find an appropriate earlier concept, and then (F) grasp it. And so on.
A few years ago, Nigel Hitchin used this technique to explain the concept of a gerbe. He wrote a two-page essay called What Is a Gerbe?, which he published in the Notices of the American Mathematical Society.
Nigel Hitchin is Savilian professor of geometry at Oxford University. Several mathematical concepts bear his name. These include the Hitchin integrable system, the Hitchin-Thorpe inequality, Hitchin’s projectively flat connection over Teichmuller space, Hitchin’s self-duality equations, and the Atiyah-Hitchin monopole metric. Hitchin knows his maths.
In What Is a Gerbe?, he begins by explaining that, to understand the concept of a gerbe, one ought first to understand a simpler concept called an “equivalence class of holomorphic line bundles”. To understand equivalence classes of holomorphic line bundles, Hitchin explains, one needs to know the concept called “transition functions relative to open sets of a covering”. And so on.
Mentally gobbling backwards through a few other, increasingly simpler, mathematical concepts, Hitchen soon comes to the end of page one of his essay. Then it’s on to page two. Eventually, with hardly any digressions, he reaches the essay’s conclusion.
The reader is left with a paralytic grasp on the concept of a gerbe - and, perhaps, also with a burning curiosity to see how this simple concept, the gerbe, can be used as a building block to produce new concepts.
I’m not sure what a “paralytic grasp” is — I imagine someone with their arms locked firmly around the concept, unable to be wrested free, but I somehow don’t think that’s what Abraham meant.
By the way, you don’t really need to understand holomorphic line bundles to understand gerbes!
If the Guardian asked me to explain a gerbe, I’d cheat a little and explain a gerbe with connection, which is actually easier. I’d say it’s a well-behaved recipe that’ll tell you a time of day — by giving a position of a clock hand — if you specify a way of sticking a balloon in a given space. The rules for what counts as “well-behaved” would take a few pictures to explain, but the first rule is that as you move the balloon around in a smooth sort of way, the time of day changes in a smooth sort of way.
Of course, this simple explanation would then open the concept up to further questions, like: how could such a recipe actually be good for anything?
That’s actually a great question, which I’ve answered elsewhere.
Re: Gerbes in The Guardian
Yes “paralytic grasp” is odd here. I could imagine being held by the paralytic grasp of something which prevented me from acting, but submitting a gerbe to this treatment seems odd.
Perhaps he just means the weak grasp of one who is paralysed.