M Theory Lesson 7
Craig Pastro says it should be called $M^2$ Theory (after the two conspirators) but the conventional name will suffice. Recall that in Lesson 6 we discussed points and how one really should worry about one's concept of point in thinking about quantum gravity. The star student charged ahead to think about generalised idempotents in relation to parity.
So let's go back to the relation $T^2 = T$. The reason for the capital $T$ (besides our swanky new latex capabilities) is that rather than sources or targets for arrows in a category, we would now like to weaken the relation and talk about monads.
A monad is a functor $T: C \rightarrow C$ with natural transformations $\mu: TT \rightarrow T$ and $\eta: 1 \rightarrow T$. Think of these as multiplication and unit. They satisfy an associativity and unit law. The square that represents associativity may have its vertices labelled by signs --, -+, +- and ++ where the source -- is the composition TTT before bracketing. Such parity cubes appear naturally in higher categorical contexts.
So let's go back to the relation $T^2 = T$. The reason for the capital $T$ (besides our swanky new latex capabilities) is that rather than sources or targets for arrows in a category, we would now like to weaken the relation and talk about monads.
A monad is a functor $T: C \rightarrow C$ with natural transformations $\mu: TT \rightarrow T$ and $\eta: 1 \rightarrow T$. Think of these as multiplication and unit. They satisfy an associativity and unit law. The square that represents associativity may have its vertices labelled by signs --, -+, +- and ++ where the source -- is the composition TTT before bracketing. Such parity cubes appear naturally in higher categorical contexts.
posted by Kea | 12:57 PM
4 Comments:
12 16 06
Hey Kea:
Thanks for the link. I started thinking about idempotence again and realized that the extra DeMoivre phase that shows up when we take P^N and operate on an arbitrary ket is a result of a GAUGE CHOICE. hehehehehe Cool stuff.
Thanks for the categorical lessons. I anxiously await diagrams:)
Hi Mahndisa
Unfortunately diagrams might be a while in coming. I don't at present have the software (even basic Office stuff) for putting in my own latex generated pictures. I think downloading ML fonts is easy enough for most people, so the inline maths we have now should not be too prohibitive for readers.
HI Kea and Mahndisa: Great that Gebar's help is spreading around. I will contunue to follow your posts for progress. Concerning Higgs, it has three phases:
1) Of course, we knew about it all along
2) It might be true, but not worth following
3) Don't bother me with such nonsense!
This may be a discovery in time-asymnetry!
Hi gals!
The growth rate for week 4 of monitoring this blog is 64% (over only 1 week)!
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