  We show how to reconstruct a process theory of local systems starting from a
global theory of reversible processes on a single global system, by using the
purification principle. In such a process theory, local systems are not given,
but rather `emerge' as the global system is decomposed into subsystems. Local
systems thus have specific identities and their composition is naturally
limited by structural constraints, a behaviour which we formalise by defining
symmetric partially-monoidal categories. We reconstruct quantum theory from the
global theories of unitary groups acting on projective Hilbert spaces.
