  Process theories combine a graphical language for compositional reasoning
with an underlying categorical semantics. They have been successfully applied
to fields such as quantum computation, natural language processing, linear
dynamical systems and network theory. When investigating a new application, the
question arises of how to identify a suitable process theoretic model.
  We present a conceptually motivated parameterized framework for the
construction of models for process theories. Our framework generalizes the
notion of binary relation along four axes of variation, the truth values, a
choice of algebraic structure, the ambient mathematical universe and the choice
of proof relevance or provability. The resulting categories are
preorder-enriched and provide analogues of relational converse and taking
graphs of maps. Our constructions are functorial in the parameter choices,
establishing mathematical connections between different application domains. We
illustrate our techniques by constructing many existing models from the
literature, and new models that open up ground for further development.
