States as Morphisms

Using elementary categorical methods, we survey recent results concerning D-posets (equivalently effect algebras) of fuzzy sets and the corresponding category ID in which states are morphisms. First, we analyze the canonical structures carried by the unit interval I = [0,1] as the range of states and the impact of “states as morphisms” on the probability domains. Second, we analyze categories of various quantum and fuzzy structures and their relationships. Third, we describe some basic properties of ID and show that traditional probability domains such as fields of sets and bold algebras can be viewed as full subcategories of ID and probability measures on fields of sets and states on bold algebras become morphisms. Fourth, we discuss the categorical aspects of the transition from classical to fuzzy probability theory. We conclude with some remarks about generalized probability theory based on ID.