### Origin of Species II

Todd Trimble kindly made the following comment:

Note also that from the logos perspective (ie. M theoretic higher topos ideas) the category of finite sets (resp. vector spaces), as it sits in the topos Set (resp. Vect), plays an important role in attempting to define the generalised logic behind the simple operators that we associate to measurement algebras. The one major alteration to these structures that M theory requires, which comes up again and again in many guises, is the relaxation of the monoidal condition. This happens naturally with higher dimensional structures, as illustrated by Batanin's tower of coherence laws. For QFT, this forces a complete change of mathematical language, since too many concepts are only defined in categorical terms. Fortunately, everybody can understand simple diagrams of ribbons and trees.

Aside: Post written with wireless connection from my heavenly warm room amidst the fresh snow at the observatory. Can't seem to find any old skis lying about ...

I hope I'm not being too obnoxiously anticipatory by also remarking that the usual notion of permutative V-operad can be defined very concisely as a monoid in the monoidal category of V-species, where the monoidal product is species substitution.This is far from being obnoxious, Todd! Struggling physicists appreciate perceptive comments from knowledgeable category theorists who can see where we are trying to go with M Theory. Indeed, although we tend to talk about operads loosely as one object multicategories, the species definition comes very close to what Rivasseau et al have in mind for redefining quantum field theory.

Note also that from the logos perspective (ie. M theoretic higher topos ideas) the category of finite sets (resp. vector spaces), as it sits in the topos Set (resp. Vect), plays an important role in attempting to define the generalised logic behind the simple operators that we associate to measurement algebras. The one major alteration to these structures that M theory requires, which comes up again and again in many guises, is the relaxation of the monoidal condition. This happens naturally with higher dimensional structures, as illustrated by Batanin's tower of coherence laws. For QFT, this forces a complete change of mathematical language, since too many concepts are only defined in categorical terms. Fortunately, everybody can understand simple diagrams of ribbons and trees.

Aside: Post written with wireless connection from my heavenly warm room amidst the fresh snow at the observatory. Can't seem to find any old skis lying about ...

## 3 Comments:

If only I knew more about M theory... [yes, you have over 200 lessons on the subject! -- can you recommend a crash course for a categorically-minded mathematician who doesn't know much M theory or quantum field theory for that matter? :-) ].

Can you point me to the relaxation of the monoidal condition you mentioned?

Also, you're not the only one in the blogosphere taken with species: A Neighborhood of Infinity had a nice post on the categorification of Faa di Bruno's formula, using species and their derivatives.

-- Todd

Glad to hear you're tucked in.

I seem to have found a nice parameterization of the 3x3 "doubly magic" matrices. Since I'm prone to error, let me verify it by computer program and then I'll blog it.

Also, I've finally noticed that the CP violation that is built into the CKM matrix can be written as a Berry-Pancharatnam phase. This is so beautiful that I will have to blog this too, but first, I think the parameterization of the doubly magic 3x3 matrices is more important. (And from there, I think it's not that far to writing the CKM matrix as an exact doubly magic matrix, which will give us an exact prediction for the CKM values similar to how the tribimaximal predicts the MNS matrix.)

Hi again Todd. As you can see from Carl's comment, we are talking about new physics here, but published references are in short supply, because it's not stringy or loopy or in any other way trendy. Although there are large numbers of related papers, one has to wade through the mud for years to find a coherent picture in the mess. I wish I had the time to write up more for the benefit of category theorists, but it is a low priority since I need to earn enough to eat and real physics takes precedence.

More traditional M theorists are mostly not too fond of this more quantum information theoretic approach, because the physics disagrees rather violently with the stringy prejudices of SUSY particle spectra and the like. Needless to say, we view any post SM predictions that rely solely on classical geometry to be highly dubious and unreliable, especially since they have no concrete numerical predictions, and we do, and ours agree well with experiment so far.

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