A coalgebraic characterization of behaviours in the linear time - branching time spectrum

The paper outlines an approach for characterizing several kinds of behaviours for transition systems in coalgebraic terms and illustrates the approach with some behaviours in the linear time -- branching time spectrum, namely, traces, ready-traces and failures. The approach is based on an abstract notion of ``behaviour object'' that can be defined in any (concrete) category and enjoys a uniqueness property similar to the uniqueness of morphisms to final objects. That property makes behaviour objects final in a suitable extension of the given category with additional morphisms, which allows to define the behaviours of arbitrary objects by the unique morphisms to the behaviour objects. The main purpose of the paper is to show how trace, ready-trace and failure semantics can be characterized in terms of behaviour objects.