The concept of affordance, proposed by James J. Gibson as an opportunity for action offered by the environment to the organism, has been adopted in various fields, including psychology, neuroscience, and robotics. However, different interpretations exist as to whether it is a feature of a relation between the environment and the organism and therefore cannot exist independently of the organism, or a “resource” that exists in the environment independent of the organism’s presence and is waiting to be used, or both, or neither. In this paper, we defend the position that affordances are both relational and resources using a category-theoretic approach. This idea is formalized by the concept of “natural transformations” in category theory, which are structure-preserving transformations between “functors” – mathematical expressions representing “seeing from a particular perspective.” We propose that formalizing the realism of affordance in terms of natural transformations offers a more rigorous and lucid understanding of this concept. Furthermore, our formalization enables us to relate the reality of affordances to a broader context, especially the shift in the meaning of “reality” in modern physics. Our category-theoretic approach offers a potential solution to the problems and limitations associated with existing set theory-based frameworks for affordances, paving the way for a future theory that better accounts for the openended interplay between organisms and their environments.

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.