  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.
