The n-Category Café

I hadn’t known of the mysteries of sphere eversion before. Watching the video was a surprising amount of fun.

Now I am trying to see what is going on here. On the machine in my current office any attempt to download the 50MB file linked to above fails. I am lacking any information that might be hidden there.

So let me just try to understand what the entry here is saying:

Consider the 2-category given as follows. The objects correspond to a (possibly empty) set of dots along a vertical axis. Each dot has an associated sign: − will indicate that a 1-morphism emanating from the dot should be a left pointing strand; + indicates a right pointing strand. The orientation information will be completely suppressed here. The 1-morphisms are generated (in the sense of composition and tensor products) by $\subset$, $\supset$, and $X$.

So we have a bunch of arrows drawn on the line, touching each other at their endpoints (but not necessarily target-source but also source-source and target-target), and we want to generate a monoidal category from them, I suppose.

Do I read the above as saying that $\subset$ is to denote concatenation of arrows, while $\supset$ is to denote their monoidal tensor product? Why, then, that unusual notation. And what is $X$? Or is that a typo for $\times$?

I am afraid I didn’t understand the message of this first paragraph. Can anyone help me?

I really am sorry about the file size!

Chris has the definitions correct. I am mimicing the category of tangles, but using
left/right instead of up down. The zig-zag move that ordinarily defines a duality (or the existence of a non-degenerate bilinear form in the vector space case) is now a 2-isom. Schematically, the 2-isom is a cusp:
Think of the surface (x, x^3 - a x , a) for
x between -1 and 1 and a between -1 and 1.
That it is a 2-isom are the lips and the beak-to-beak moves.
The vertical lines on which the objects live are parallel, and a 1-morphism is an immersed curve in the plane with a non-degenerate critical points at distinct critical levels.

On a related note, does anyone have any good ideas for getting 3-d graphics on the web?

I tend to use maple to do a lot of visualation things: curves and surfaces in 3d, often animated. That’s good because I can code up animations in maple very quickly. The problem comes when I want to show them to other people. I can’t find a sensible way to export such graphics from maple and get them viewable on the web.

Here’s a very low-bandwidth (4KB) animated gif showing the trajectory of some pairs of particles in the plane being created and anihilated, so that the trajectory forms a trefoil.

The full version in maple looks a lot more impressive and can be rotated by hand as the animation is running.

Animations can be exported as animated gifs, but these come out very large and have to be compressed in some other software, it would seem more sensible to be able to save as some sort of mpeg file, or possibly something else more appropriate.

As for 3-d things, well one possibility is to do an animation of the view rotating and save it as an animated gif, but that is not very satisfactory. Another possibility is to try to export as a vrml file, but I have had absolutely no success with this – trying to view the resulting files with the cortona browser plug-in gave no joy – I get the impression that this is not well supported. The other options for exporting are POV and DXF, but I have no idea how to put those up on the web.

So any suggestions? Can I do anything sensible with maple or do I have to learn something completely new like flash? How do you do your pictures Scott? Greg?

I cut the file into smaller pieces, but made a mistake on the 4th piece. I will fix that tonight. Meanwhile, smaller files are found at Unapologetic.