Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Monday, December 10, 2007

M Theory Lesson 135

James Dolan speaks of categorification and decategorification, and of information and entropy.

In logos land, these processes have a dimension raising or lowering aspect. It is often said that categorification is ill defined, in comparison to decategorification, but with dual processes it should not be so. Therefore, categorification itself must be defined in some canonical way that generalises the turning of natural numbers into sets or spaces. One way to do this would be to put the heirarchy on a loop, such as the loop labelled by the $q$ parameters at roots of unity. There would be $n$-categories for $n \in \mathbb{N}$ and $r$-categories for $r \in \mathbb{Q}$, and $n \rightarrow \infty$ would look like the limit $q \rightarrow 1$ again, where spaces begin to look like sets.

After all, projective geometry has its horizons, and the cohomology of motives would move left and right, like the mass interaction, or Stokes' theorem, or the Riemann zeta function.


Blogger Kea said...

After all, the absolute point, if it be worthy of topos-like pointlessness, must contain not no structure, but rather all that is.

December 10, 2007 8:13 AM  
Anonymous Anonymous said...

I don't think categorification and decategorification are dual processes. More like the process of quotienting and injecting. You lose a lot when you quotient (decategorify) and it is very difficult to find out where you came from. This, I think, is why categorification is hard.

December 10, 2007 12:26 PM  
Blogger Kea said...

Hi anonymous. Yes, I apologise for using the word 'dual' in too many senses. Too many years of wading through string theory papers...

December 11, 2007 9:26 AM  

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