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Life continuously changes its own components and states at each momentthrough interaction with the external world, while maintaining its ownindividuality in a cyclical manner. Such a property, known as "autonomy," hasbeen formulated using the mathematical concept of "closure." We introduce abranch of mathematics called "category theory" as an "arrow-first" mathematics,which sees everything as an "arrow," and use it to provide a more comprehensiveand concise formalization of the notion of autonomy. More specifically, theconcept of "monoid," a category that has only one object, is used to formalizein 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 frameworkor "tool of thinking" that frees us from the habits of thinking to which we areprone and allows us to discuss things formally from a more dynamic perspective,and that it should also contribute to our understanding of living systems.

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