  Using the simulation paradigm in information theory, we define notions of
quantum hypergraph homomorphisms and quantum hypergraph isomorphisms, and show
that they constitute partial orders and equivalence relations, respectively.
Specialising to the case where the underlying hypergraphs arise from non-local
games, we define notions of quantum non-local game homomorphisms and quantum
non-local game isomorphisms, and show that games, isomorphic with respect to a
given correlation type, have equal values and asymptotic values relative to
this type. We examine a new class of no-signalling correlations, which witness
the existence of non-local game homomorphisms, and characterise them in terms
of states on tensor products of canonical operator systems. We define jointly
synchronous correlations and show that they correspond to traces on the tensor
product of the canonical C*-algebras associated with the game parties.
