Universal Aspects of Probabilistic Automata 3

For lack of composability of their morphisms, probability spaces, and hence probabilistic automata, fail to form categories; however, they t into the more general framework of precategories, which are introduced and studied here. In particular, the notion of adjunction and weak adjunction for precategories is presented and justiied in detail. As an immediate beneet, a concept of (weak) product for precategories is obtained. Thus, universal properties can be used for characterizing well-known basic constructions in the theory of probabilistic automata: The aggregation of two automata is shown to be a weak product, whereas restriction and interconnection of automata are recognized as Cartesian lifts. Finally, we establish that the precategory of decision trees is coreeexive in the precategory of probabilistic automata.