Layered Monoidal Theories I: Diagrammatic Algebra and Applications

We develop layered monoidal theories – a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams , monoidal theories provide a graphical formalism to reason algebraically about information flow in models across different fields of science. Layered monoidal theories allow mixing several monoidal theories (together with translations between them) within the same string diagram, while retaining mathematical precision and semantic interpretability. We develop the mathematical foundations of layered monoidal theories, as well as providing several instances of our approach, including digital and electrical circuits, quantum processes, chemical reactions, concurrent processes, and probability theory.