Monoidal categories, symmetries, and compound physical systems

The conceptually non-trivial problem of relating the notion of a compound physical system and the mathematical descriptions of its constituent parts is dramatically illustrated in standard quantum physics by the use of the Hilbert tensor product of the spaces representing the subsystems, instead of the more familiar cartesian product, as it is the case for classical physical systems. Aspects of the general structure of this relationship can be explained by endowing suitable categories that arise in the mathematical descriptions of classical systems and of quantum systems with their natural monoidal structures, and constructing a monoidal functor, relating the monoidal structures of the domain and codomain categories in a coherent way. To highlight some of the structural aspects involved, I will confine myself in this paper to the simple case of finite sets or finite-dimensional Hilbert spaces, on which finite groups act.