Decision making is a vital function in the age of machine learning and artificial intelligence; however, its physical realizations and their theoretical fundamentals are not yet known. In our former study, we demonstrated that single photons can be used to make decisions in uncertain, dynamically changing environments. The multi-armed bandit problem was successfully solved using the dual probabilistic and particle attributes of single photons. Herein, we revolutionize how decision making is comprehended via a category theoretic viewpoint; we present the category theoretic foundation of the single-photon-based decision making, including quantitative analysis that agrees well with the experimental results. The category theoretic model unveils complex interdependencies of the entities of the subject matter in the most simplified manner, including a dynamically changing environment. In particular, the octahedral structure and the braid structure in triangulated categories provide clear understandings and quantitative metrics of the underlying mechanisms for the single-photon decision maker. This is the first demonstration of a category theoretic interpretation of decision making, and provides a solid understanding and a design fundamental for machine learning and artificial intelligence.
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