### Triple the Music

The cubist Picasso (1881-1973) painted Les Trois Musiciens in 1921. These days another kind of cubism has influenced music, namely Mazzola's book The Topos of Music (Birkhauser 2002). This mighty tome, first shown to me by Dr Fauser, puts to rest any idea that it is impossible to relate toposes to the real world.

But traditional topos theory only describes classical spaces. Recall that a typical example of a topos is a category of sheaves on some topological space. Since we prefer to study logoses, we might begin simply with the example of Set, a topos with Boolean logic. However, other categories of sheaves are important in twistor theory and it is therefore necessary to generalise all of topos theory in a higher categorical setting.

On the other hand, the sheaf of germs for the celestial 2-sphere is a particular functor on the topological space. The quaternionic analogue uses the 4-sphere and the octonions require the 8-sphere. Only the octonions force us into a consideration of ternary logic, but the quaternions also require non-commutative spaces. So let us begin with the sequence of spheres $S^0$, $S^{2}$, $S^{4}$, $S^{8}$ of dimension $2^n$. A study of generalised sheaves on these spaces should tell us about n-logoses.

But traditional topos theory only describes classical spaces. Recall that a typical example of a topos is a category of sheaves on some topological space. Since we prefer to study logoses, we might begin simply with the example of Set, a topos with Boolean logic. However, other categories of sheaves are important in twistor theory and it is therefore necessary to generalise all of topos theory in a higher categorical setting.

On the other hand, the sheaf of germs for the celestial 2-sphere is a particular functor on the topological space. The quaternionic analogue uses the 4-sphere and the octonions require the 8-sphere. Only the octonions force us into a consideration of ternary logic, but the quaternions also require non-commutative spaces. So let us begin with the sequence of spheres $S^0$, $S^{2}$, $S^{4}$, $S^{8}$ of dimension $2^n$. A study of generalised sheaves on these spaces should tell us about n-logoses.

## 3 Comments:

Kea

Stanford has a nice little page on quantum logic and projection operators.

Yes, the Stanford encyclopedia is a great website. Douglas Bridges from here is one of the contributors.

Reminds one of Steve Martin's play "Picasso at the Lapin Agile," about Picasso meeting Einstein. The simplicity of the drawings has great beauty.

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