The n-Category Café

The 2-category implicit in string theory, due to Stolz and Teichner.

We did talk about this way back here. Is there progress on understanding this better?

John wrote :

The closest thing I know to a full treatment is section 4.2 in the paper by Stolz and Teichner cited above. Does anyone know a reference with more details? There are a lot annoyingly tricky issues.

I’ve always been interested in this viewpoint of Stolz and Teichner, because it shows that its possible to interpret nCob (the n-category of points, lines, surfaces, etc.) without using manifolds with corners.

Heresy, you say?

Well, think about it. Let’s strip down their definition, and get rid of all the conformal stuff. Define the 2-category 2Cob as follows:

Objects are 0-dimensional manfiolds $x$.

A morphism $\gamma : x \rightarrow y$ is a 1d oriented manifold with boundary identifications $x \rightarrow \partial \gamma \leftarrow y$.

If $\gamma, \gamma' : x \rightarrow y$ are morphisms, then a 2-morphism $\sigma : \gamma \Rightarrow \gamma'$ is an oriented 2d manifold with boundary (not with corners) together with identifications

$\partial \sigma \cong \gamma \union_{x,y} \gamma'$.

That is, the boundary of $\sigma$ is $\gamma$ glued with $\gamma'$ along $x,y$.

See? No manifolds with corners needed. And the process can be continued to arbitrary $n$. It will always be a case of an $n$-morphism being a $n$-manifold with boundary, whose boundary is identified with two $(n-1)$-morphisms glued over their source and target $(n-2)$-morphisms.

However, there’s a caveat. If you work through the 2Cob example, you’ll see that it doesn’t give you what you might ‘want’ from 2Cob. Its the old old problem of it not distinguishing ‘D-branes’ from ‘open strings’. It should be pointed out, though, that neither does ‘traditional’ nCob - the one with points, lines, surfaces with corners, etc.

I’m not saying that this version of nCob is better than the manifolds-with-corners one. I’m just pointing out (1) it seems to be ‘no problem’ (give or take difficulties with n-categories :-)) to define a version of nCob without using corners - which is often not appreciated - and (2) whichever approach one uses (with corners or without) it is still confusing, and doesn’t give you intuitively what you might expect, or want.

One thing I’ll say about the traditional ‘with corners’ nCob : it’s perfect for tangles, as the Tangle Hypothesis shows!

I’d love to hear other thoughts on this, since its been confusing me for ages.

(In fact, there’s an important picture in this week’s notes that doesn’t really live in the 2-category I was describing — only in the double bicategory. As an exercise, try to spot it!)

I haven’t really read Jeff’s paper, but he’s briefly explained it to me.

I think that the naughty diagram is towards the bottom of page 19 (page 6 in the PDF file). If you compare that with its schematic counterpart directly to its right, you’ll see that the nature of x and x′ are a bit vague. In particular, is x′ the big circle at bottom, or the small one? To be fair, you ought to shrink all three outer big circles, as they approach the top or bottom, to meet the inner small circles, thereby specifying x and x′ precisely. (The next diagram, atop page 20, does this.) As it is, the surfaces linking the big circles to the small circles must be horizontal morphisms in Jeff’s double bicategory.

Things would be even worse if there were holes in those horizontal surfaces, so that it would be impossible to shrink the outer circles to meet the inner circles. To be fair, the picture as given is probably equivalent, in a suitable sense internal to a double bicategory, to the properly drawn picture in the bicategory. But a picture with holes in the horizontal surfaces would be tractable only in the full double bicategory.

Am I right?