Back at the start of our M Theory lessons, when we were discussing twistor amplitudes, associahedra, motivic cohomology and Kepler's law, we were always thinking about ternary analogues of Stone duality. Recall that a special object in the (usual) category of topological spaces is specified by a two point lattice, because every space contains the empty set (a zero) and the whole space itself (a one).

In a sense, this is the most basic arrow ($0 \rightarrow 1$) of category theory, since much mathematical industry is motivated by classical topology. The requirements of M Theory*, however, are different. Recall, once more, that if we cannot begin with space, which after all is an emergent concept, we cannot begin with the classical symmetries that act upon space, or with simple generalisations such as traditional supersymmetry. The imposition of such symmetries from the outset simply does not make any sense. The groups that physicists like, such as $SU(2)$ and $SU(3)$, are very basic categories, with one object and nice properties, so we should not fear that they will disappear into the mist, never to be recovered.

There are a number of ternary analogues for the arrow, but what really is the ternary analogue of its self dual property (ie. true triality)? We have drawn triangles and cubes and globule triangles. We could draw Kan extensions. As a minimum, we expect three dualities for the sides of the ternary triangle (let us call them S, T and U), but what truly three dimensional element appears for a self ternary arrow? For a start, the Gray tensor product of Crans will be busy generating higher dimensional arrows for us. Since a classical space, properly described, ought to be an infinite dimensional category, we would like to make use of arrow generation in its definition.

But M Theory* requires even more. The dualities S, T and U do not merely reflect the properties of a classical space. Already, quantum information takes precedence. We let our lone arrow stand for the noncommutative world, and look further into the nonassociative one for the meaning of ternary. And here at last, categorical weakness is forced upon us, unbid. The associahedra provide the simplest possible object (a $1$-operad) for describing nonassociativity (of alphabets). There exists a vast collection of such combinatorial structures, of higher and higher information dimension.

*This term, as always, will refer here to the correct form of M Theory, and not to some offshoot of crackpot stringy physics.

8 years ago

This is great stuff. You remind me of Newton making his greatest discoveries while exiledfrom Cambridge by the plague, or Einstein in the patent office. Keep your spirits up. These posts on M theory could make a fine book.

ReplyDeleteHave Stone-type duality theorems been given for quantales?

ReplyDeleteWell, naturally this is a big industry with the (logical branch of the) Noncommutative Geometry crowd. See for instance P. Resende's work. There is probably something there we could use, but I believe the mathematics should follow the physics (ie. have more respect for finitism and constructivism) so I have not kept up to date with this area of research.

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