Arcadian Functor

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Marni D. Sheppeard

Sunday, May 06, 2007

M Theory Lesson 49

Recall that internalisation turns one square into four. This occurs in the category of (stable) trees, discussed in Leinster's book Higher Operads, Higher Categories from page 230. For example, consider the square in the diagram. This square is one of the three squares on the 9 faced Stasheff polytope, which one obtains by reducing the 5-leaved tree quadruple squares to ordinary faces, yielding a kind of classifying space. This diagram makes the contraction and expansion moves on trees more explicit. Recall that the entire polytope is labelled by a simple 1-level tree as befits a 1-operad polytope.

Observe that the total number of squares describing the Stasheff polytope is 6x5 (for the pentagons) plus 3x4 (for the squares) which comes to 42!

4 Comments:

Blogger Doug said...

Hi Kea,

John Baez TWF week 251 is interesting.

My favorite is this reference to Bob Coecke, 'Kindergarten Quantum Mechanics' and the discussion of "picture calculus".
[Some pictures may worth more than a thousand words?]

http://arxiv.org/PS_cache/quant-ph/pdf/0510/0510032v1.pdf

"Picture calculus" appears to relate to your thread topic.

From my perspective these are all consistent with some type of mathematical game theory?

May 06, 2007 11:22 PM  
Blogger Kea said...

Yes, the Coecke papers are great reading, and 'picture calculus' is exactly what we're trying to do here.

May 07, 2007 9:22 AM  
Blogger kneemo said...

Isn't 42 also the number of vertices for the 6-factor associahedron?

May 07, 2007 10:54 AM  
Blogger Kea said...

kneemo, YES, that's the point! Louis Crane and I once had a giggle over the fact that it was 42!

May 07, 2007 11:30 AM  

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