Ka Tiritiri O Te Moana
namely U-|V-|W-|X-|Y where Y is the Yoneda embedding. There is both a monad and a comonad hidden in here. To deal with annihilation and creation operators we are going to need both structures.
M-theorists would like to have a similar characterisation for the quantum topos of M-theory. By its higher dimensional nature, this is not simply a 2-category or 3-category. The dimension raising aspects of Gray type products are crucial to explaining the physical combination of systems. If the basic object is modelled on projective geometry, such as in twistor space, then one expects the combination of two particles to yield two copies of twistor space, more or less as noted long ago by Hughston and Hurd, who used the Kunneth formula to study mass generation in the twistor setting.
Witten's study of N=4 SYM showed that certain amplitudes were localised on curves. Clearly something interesting is happening here. If one steps back and considers supersymmetry from a Machian perspective, then one finds it is related to the Real-Symplectic duality of Mulase et al in matrix models.
This manifests T-duality, but what about S-duality? Could this arise from including the octonion case? One shouldn't be bothered by nonassociativity. After all, it comes up in the K-theory studies of T-duality, and in a simple monoidal categorical way. Moreover, the association of Jordan algebras and projective geometry means that one can make the connection to twistor string theory very explicit. In doing so, one ends up studying moduli spaces as orbifolds, with cell decompositions. It is well known that both these decompositions, and Deligne-Mumford type compactifications, can be described by operad like polytopes and their duals. In this higher categorical language, the web of dualities is a kind of Poincare duality in higher cohomology.