The n-Category Café

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.

how could such a recipe actually be good for anything?

Maybe the following slight variation of your description is just as close to the math, quite faithful to the physical application and still close to everyone’s everyday experience:

Suppose you are a painter. Your job is to go to the newly built conference building and

- 1) paint all the doorknobs

- 2) paint all the handrails

-3) paint all the walls.

But it tuns out that the architect used lots of different materials all over the place. You find that you need different amounts of paint to color these items at different places.

So you sit down and first make a table which lists

- on the left all the doorknobs

- on the right for each doorknob the amount of color needed to paint it (a small amount, right, but suppose there are many many doorknob).

Maybe you recall from your highschool days that such a table is also sometimes called a function: it maps doorknobs to the amount of color coloring them.

What they don’t tell you in high school these days is that

- a function is also called a 0-bundle with connection

- a function is also called a (-1)-gerbe with connection.

So you already know what a $(-1)$-gerbe is! It’s just a very strange name for an assignment of milliliters to doorknobs.

Next, you try to sit down and make a table that has

- on the left all the handrails

- on the right the amount of color needed for them.

But now you run into trouble: the material of the handrails turns out to change every few meters. (It’s very modern architecture ). So instead of making a table which maps entire handrails to milliliters, you make a long, long table which maps

- each meter of handrail

- to the amount of color needed for it.

That takes a while. After pages of pages have been filled, this table is finished.

It doesn’t tell you immediately how much color is needed for painting a given handrail. But using the table it is easy to determine this amount:

for every piece of handrail, you chop it up (mentally) into 1-meter parts, look up the color needed for each of them seperatey and add all this up.

It happens that people have invented a funny name for this procedure: this is called

- a bundle with connection

or

- a 0-gerbe with connection.

These are just words. They mean exactly: a procedure for determining how to color handrails, and how to do it piecewise.

At this point, the attentive painter may alreay be able to go all the way up to inventing the concept of gerbes and even 2-gerbes himselves.

For all other painters out there, here is what a gerbe with connection would be:

next the walls need to be painted. They are made of even more variations of material than the handraisl! Grudgingly, you pull out pen and paper and make a huge list which

- has on the left an entry for each square meter of wall in the building

- and next to it on the right the estimated amount of color needed for that,

Cretaing that takes while. When done, you have a procedure that allows to compute the amount of color needed for an arbitrary wall:

for each square meter of it, look up the corresponding number of milliliters in your table, add that all up.

This procedure is what is called

- a 2-bundle with connection

or else

- a gerbe with connection.

(Or maybe a 2-functor with values in $\Sigma \mathbb{R}$, or maybe a Cheeger-Simons differential 3-character, or maybe…)

See, John, by making mathematics seem accessible and fun, you’re ruining everything.

The idea is supposed to be that math is ‘cool’, in a way – at least we use a bunch of cool-sounding and mysterious words – but please let’s keep it at arm’s length, because you have to be a genius (or weirdo) or something to actually understand it. It’s hard to find the idea in mass culture that anyone does math because it’s fun.

Recently, the actress Danica McKellar (who played the love interest of Fred Savage’s character in The Wonder Years) has been on TV promoting her new book, “Math Doesn’t Suck”. As it turns out, she’s co-authored a scientific paper or two (I think in statistical mechanics) as an undergraduate at UCLA. As she explained, she wrote the book for young teenage girls, to promote the idea that math and science are actually interesting and fun, and not just for geeks.

Anyway, I was watching this segment on her (as Person of the Week on ABC World News Tonight) and her book, and toward the end she was asked what was the title of this scientific paper she wrote. It was some long title with the word ‘percolation’ in it – I couldn’t tell you exactly because she was drowned out by Charles Gibson’s chuckling voiceover, “Well, you’ll just have to take our word for it,” like, heh, get a load of this brainy chick. Precisely undermining the point of her book!

John, you missed out the best bit – the bit where my paper gets cited. Okay, they don’t mention my name and it is sort of in the sense of “Here’s a few weird sounding titles of papers,” but, hey, it’s not everyday I get cited in a national newspaper.

Actually, I’m sure they only chose that paper because they read Urs bigging it up in the $n$-Café.

Perhaps the saddest thing about that silly article is that gerbes are MUCH more general than those that Nigel Hitchin works with. If I was to attempt the difficult task of describing them I would probably go for the route via sheaves and bundles rather than that mentioned in the article. To get a plug in: if you do not know of the Bangor website entry you might be amused at trying to build a fibre bundle in wood. (Later it was done in bronze.) Now I challenge anyone to build a gerbe out of cardboard!!!