Reformalizing the notion of autonomy as closure through category theory as an arrow-first mathematics

Abstract

Life continuously changes its own components and states at each moment through interaction with the external world, while maintaining its own individuality in a cyclical manner. Such a property, known as "autonomy," has been formulated using the mathematical concept of "closure." We introduce a branch of mathematics called "category theory" as an "arrow-first" mathematics, which sees everything as an "arrow," and use it to provide a more comprehensive and concise formalization of the notion of autonomy. More specifically, the concept of "monoid," a category that has only one object, is used to formalize in a simpler and more fundamental way the structure that has been formalized as "operational closure." By doing so, we show that category theory is a framework or "tool of thinking" that frees us from the habits of thinking to which we are prone and allows us to discuss things formally from a more dynamic perspective, and that it should also contribute to our understanding of living systems.