A view on implementing processes : categories

We construct a category of circuits: the objects are alphabets and the morphisms are deterministic automata. The construction diiers in several respects from the bicategories of circuits appearing previously in the literature: it is parameterized by a monad which allows exibility in the emergent notion of process. We focus on the circuits which arise from a distributive category and the exception monad. These circuits are partial in that they may, based on their state, choose to abort on some inputs. Consequently, certain circuits determine languages, and safety and liveness properties with respect to these languages are captured by circuit equations. Actually, the notions of safety and liveness arise abstractly in any copy category. Extracting the category of circuits which are both safe and live corresponds to the extensive completion of a distributive copy category. Partial circuits coincide with elements of the terminal coalgebra of a speciic datatype. The co{induction principle provides mechanisms for the construction of circuits, the normalization of circuit expressions and for the proof of safety and liveness properties.