Abstract

Systems across science and engineering naturally arrange into levels of abstraction: reaction networks treat chemical compounds as placeholders with no internal structure, whereas the molecular structure is considered when modelling the compounds as graphs; the molecular structure can further be seen as an abstraction of quantum chemical models; similarly, higher level descriptions in computer architecture must ultimately be implemented as microelectronic circuits.

In this talk, I will introduce layered monoidal theories as a mathematical framework for studying the levels of abstraction. While monoidal theories provide a well-developed and intuitive graphical syntax for many scientific theories via their representation as string diagrams, they lack a uniform way of incorporating translations between theories. Categorical models, therefore, deal with translations – such as adjunctions or functors refining a more abstract theory by translating it to a more detailed one – on a case-by-case basis. Layered monoidal theories remedy this by “glueing together” several monoidal theories as well as translations between them, while retaining the recursively defined syntax and semantic interpretability. After defining layered monoidal theories, I will demonstrate how they can be used to reason about various systems via three examples: digital circuits, electrical circuits and chemical reactions.