### M Theory Lesson 52

The logical necessity of weakening distributivity in logos theory forces a study of pseudomonads, not just monads. Steve Lack has shown that a good theory for pseudomonads really requires Gray categories, our favourite tricategorical toys. This is the primary reason that a quantum analogue for a topos must go higher than 2-categorical structures.

The first kind of distributivity that we learn about is that of ordinary multiplication over addition. This is fully described by monads (in particular + and x) in a (causal) square involving the categories Set, Ring, Monoid (for multiplication) and Ab (for addition). The category of rings is where the numbers actually live. Now by characterising Set as a ground 2-logos, we begin to see that a very fundamental axiomatisation of M Theory should be possible, in terms of pseudomonads for 3-logoses.

Hopefully by now it has occurred to our readers that the term M Theory does not merely refer to an 11 dimensional supergravity.

The first kind of distributivity that we learn about is that of ordinary multiplication over addition. This is fully described by monads (in particular + and x) in a (causal) square involving the categories Set, Ring, Monoid (for multiplication) and Ab (for addition). The category of rings is where the numbers actually live. Now by characterising Set as a ground 2-logos, we begin to see that a very fundamental axiomatisation of M Theory should be possible, in terms of pseudomonads for 3-logoses.

Hopefully by now it has occurred to our readers that the term M Theory does not merely refer to an 11 dimensional supergravity.

## 2 Comments:

Kea

I found a nice paper by Mulvey and Pelletier on quantales, a quantum generalization of locales. In the paper, an involutive quantale is constructed using a functor from the dual category of C*-algebras to the category of quantales. Given such a functor and an involution, we can conceivably talk about the self-adjoint parts of the C*-algebras, containing the observables. Such observables do not form a C*-subalgebra but do form a Jordan-Banach (JB) algebra, even in the infinite dimensional case.

Over the years, the problematic case has been the Jordan algebra of 3x3 Hermitian matrices over the octonions J(3,O), as it cannot be seen as the self-adjoint part of a C*-algebra. However, in 1977 Wright showed that J(3,O) is the self-adjoint part of a certain Jordan C*-algebra, thus proving the general result that

each JB-algebra is the self-adjoint part of a unique Jordan C*-algebra.So it may be helpful to extend the work of Mulvey and Pelletier and consider the spectrum of a Jordan C*-algebra which determines a functor from the dual of the cat of Jordan C*-algebras to the cat of quantales.

Great idea kneemo. Gee, yet another research project to sort out! Check out also the papers of Resende (who was at the Streetfest). He has thought a lot about the C* picture and he really knows Mulvey's work.

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