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.

This content is only available as a PDF.
This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a full description of the license, please visit