M Theory Lesson 35
One day, many years ago, I wanted to catch a bus from Sikkim down to the plains. It was my first lesson in the theory of order-under-chaos. A major landslide had blocked the only road out of the mountains, and yet in no time at all a relay of buses was setup. Travel between buses simply required a short walk along the goat track over the landslide debris.
It looks like M Theory needs to discuss the concepts of operadification and cooperadification. Don't worry, I am not suggesting that we adopt this cumbersome terminology. Instead, let's name the 2 functors that describe these processes entropy and information respectively. Entropy describes the process of leaping up to the next quantum level, which is far more complex and intricate. Information is the dual process of dropping down whilst investigating a question. It might not be humans asking questions. The galaxy might want to ask Computer Earth a question now and again.
Baez et al have been describing a program of groupoidification. This lives in the realm of n-category theory, and we expect that such categorical structures will arise as algebras of master operads.
It looks like M Theory needs to discuss the concepts of operadification and cooperadification. Don't worry, I am not suggesting that we adopt this cumbersome terminology. Instead, let's name the 2 functors that describe these processes entropy and information respectively. Entropy describes the process of leaping up to the next quantum level, which is far more complex and intricate. Information is the dual process of dropping down whilst investigating a question. It might not be humans asking questions. The galaxy might want to ask Computer Earth a question now and again.
Baez et al have been describing a program of groupoidification. This lives in the realm of n-category theory, and we expect that such categorical structures will arise as algebras of master operads.
posted by Kea | 8:25 AM
3 Comments:
Kea
I really admire Baez's groupoidification program, but in certain contexts, a groupoid is too restrictive. This is mainly because composition of morphisms is still associative.
Recently I've been thinking it would be helpful to categorify the Cayley-Dickson process. For at each step of this process, we see we lose properties such as: every element is its own conjugate, commutativity, associativity, and the division property. If we were taking an n-Cat approach, by the third Cayley-Dickson doubling we couldn't even make a 3-groupoid! This would force us to resort to using quasigroupoids or other weaker structures to build our higher "*-algebroids" (Sedenions, etc).
What's cool is that when we go higher than the 3-Cat level, our n-cells include zero-divisors, i.e., we can have fg=0 for non-zero n-cells f and g. I've seen multiplication tables for the *-algebras higher than the octonions (sedenions, 32-ions, 64-ions, 128-ions, etc.) and they display fractal self-similarity after a while. It would probably be fun to explore these using n-Cats.
Yes, kneemo, we are trying to describe metacategories. But there are many details to fill in.
Ah ok, JB is working at the level of incidence geometry in Week 248, where the groupoid we'd likely use has objects that are lines in projective space and morphisms that are collineations.
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