Categorical systems theory

Category theory for systems.

Myers (2023):

Categorical systems theory is an emerging field of mathematics which seeks to apply the methods of category theory to general systems theory. General systems theory is the study of systems — ways things can be and change, and models thereof — in full generality. The difficulty is that there doesn’t seem to be a single core idea of what it means to be a “system”. Different people have, for different purposes, come up with a vast array of different modeling techniques and definitions that could be called “systems”. There is often little the same in the precise content of these definitions, though there are still strong, if informal, analogies to be made accross these different fields. This makes coming up with a mathematical theory of general systems tantalizing but difficult: what, after all, is a system in general?

Category theory has been describe as the mathematics of formal analogy making. It allows us to make analogies between fields by focusing not on content of the objects of those fields, but by the ways that the objects of those fields relate to one another. Categorical systems theory applies this idea to general systems theory, avoiding the issue of not having a contentful definition of system by instead focusing on the ways that systems interact with eachother and their environment.

These are the main ideas of categorical systems theory: 1. Any system interacts with its environment through an interface, which can be described separately from the system itself. 2. All interactions of a system with its environment take place through its interface, so that from the point of view of the environment, all we need to know about a system is what is going on at the interface. 3. Systems interact with other systems through their respective interfaces. So, to understand complex systems in terms of their component subsystems, we need to understand the ways that interfaces can be connected. We call these ways that interfaces can be connected composition patterns. 4. Given a composition pattern describing how some interface are to be connected, and some systems with those interfaces, we should have a composite system which consists of those subsystems interacting according to the composition pattern. The ability to form composite systems of interacting component systems is called modularity, and is a well known boon in the design of complex systems.

In a sense, the definitions of categorical systems theory are all about modularity how can systems be composed of subsystems. On the other hand, the theorems of categorical systems theory often take the form of compositionality results. These say that certain facts and features of composite systems can be understood or calculated in terms of their component systems and the composition pattern.

See also Capucci (2025).

Why would we bother with such a thing? I saw Martin Biehl present some interesting work (Biehl and Virgo 2023; Virgo, Biehl, and McGregor 2021) that used the “string diagrams” formalism to do interesting computations over Bayesian networks, and he made the case that it might be simpler to work with complex systems using this formalism.