Saturday, August 10, 2013

Proudly presenting the »nopetope«!

(This is is jotted down, raw posting, that may never get finished. I am publishing it anyway, as it pretty much reflects my current mood.)

I am about to do some research with coverings of trees, and it was only natural to look at Baez-Dolan metatrees and the corresponding notion of opetopes. The paper contains a famous 5-minute definition and it is a brain teaser worth reading. They introduce trees and nestings (actually two sides of the same coin) and their superposition, called a constellation, which has to follow some rules, but drawing spheres is a creative process.

Then there are zooms which connect constellations as long as the left nesting and the right tree are morally identical.

My lambda graphs are basically search trees and a nesting would add the operational notion of evaluation. Since we can freely choose our evaluation strategy (confluence?), the latter corresponds to a nesting. It remains to find out what the degenerate zooms are under this aspect.

I am just reading the "Polynomial functors and opetopes" paper ( and it asserts that it's starting constellation is a one leafed tree: But I wonder if this is fundamental, and since leaves are stripped by zooms, any number will do.
So I'll suggest starting with zero leaves, and calling the resulting zoom complex(es) the nopetopes. This might be the first mathematical term I have coined :-)

For the mathematically-challenged, a nopetope is just a lollipop wrapped in cellophane, while an opetope is the two-stick version thereof.

It is a funny coincidence that "one" and "ope-" contain the same vocals. Going on, we could also have twopetopes and thropetopes, fouretopes etc. But I doubt these are significant in any way.

And now back to the paper and then to an Ωmega implementation of nopetopes...

PS: while writing this my imagination went though... Are trees and nestings compatible with the famous correspondences energy-mass, wave-particle of physics? Looks like I have to start some more research.

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