Causal Unfoldings
Résumé
In the simplest form of event structure, a prime event structure, an event is associated with a unique7causal history, its prime cause. However, it is quite common for an event to have disjunctive causes8in that it can be enabled by any one of multiple sets of causes. Sometimes the sets of causes may be9mutually exclusive, inconsistent one with another, and sometimes not, in which case they coexist10consistently and constitute parallel causes of the event. The established model ofgeneralevent11structures can model parallel causes. On occasion however such a model abstracts too far away12from the precise causal histories of events to be directly useful. For example, sometimes one needs13to associate probabilities with different, possibly coexisting, causal histories of a common event.14Ideally, the causal histories of a general event structure would correspond to the configurations of its15causal unfolding to a prime event structure; and the causal unfolding would arise as a right adjoint16to the embedding of prime in general event structures. But there is no such adjunction. However, a17slight extension of prime event structures remedies this defect and provides a causal unfolding as a18universal construction. Prime event structures are extended with an equivalence relation in order19to dissociate the two roles, that of an event and its enabling; in effect, prime causes are labelled20by a disjunctive event, the equivalence class of its prime causes. With this enrichment a suitable21causal unfolding appears as a pseudo right adjoint. The adjunction relies critically on the central22and subtle notion of extremal causal realisation as an embodiment of causal history.