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.
Arcadian Functor 
kneemo said,
May 10, 2007 @ 3:46 pm
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 3×3 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.
Kea said,
May 10, 2007 @ 9:23 pm
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.