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.

翻译：本文发展了分层幺半群理论——这是幺半群理论的一种推广，它结合了对系统在不同抽象层次上的形式化描述。通过其弦图表示，幺半群理论提供了一种图形化形式体系，用于跨科学不同领域对模型中的信息流进行代数推理。分层幺半群理论允许在同一弦图中混合多个幺半群理论（及其间的转换），同时保持数学精确性和语义可解释性。我们建立了分层幺半群理论的数学基础，并提供了该方法的多个实例，包括数字与电路、量子过程、化学反应、并发过程以及概率论。