  By considering a generalisation of the CPM construction, we develop an
infinite hierarchy of probabilistic theories, exhibiting compositional
decoherence structures which generalise the traditional quantum-to-classical
transition. Analogously to the quantum-to-classical case, these decoherences
reduce the degrees of freedom in physical systems, while at the same time
restricting the fields over which the systems are defined. These theories
possess fully fledged operational semantics, allowing both categorical and
GPT-style approaches to their study.
