Relational Presheaves as Labelled Transition Systems

We show that viewing labelled transition systems as relational presheaves captures several recently studied examples. This approach takes into account possible algebraic structure on labels. Weak closure of a labelled transition system is characterised as a left (2-)adjoint to a change-of-base functor. A famous application of coalgebra [3, 4] is as a pleasingly abstract setting for the theory of labelled transition systems (LTS). Indeed, an LTS is a coalgebra for the functor P(A × −), where A is the set of labels. LTSs are thus objects of the category of coalgebras for this functor. The arrows, in LTS terminology, are functional bisimulations. This category of coalgebras (modulo size issues) has a final object that gives a canonical notion of equivalence, although other approaches are available in general, see [34]. Ordinary bisimulations can be understood as spans of coalgebra morphisms. The coalgebraic approach has been fruitful: amongst many notable works we mention Turi and Plotkin’s elegant approach to structural operational semantics congruence formats via bialgebras [36]. In another influential approach, Winskel and Nielsen [38] advocated the use of presheaf categories as a general semantic universe for the study of labelled transition systems. Morphisms turn out to be functional simulations, functional bisimulations can be characterised as open maps with respect to a canonical (via the Yoneda embedding) choice of path category [20]. Ordinary bisimulations are then spans of open maps, with some side conditions. Both the coalgebraic approach and the presheaf approach have generated much subsequent research and have found several applications that we do not account for here. Concentrating on the theory of labelled transition systems, there are some limitations to both approaches. For example both take for granted that the set of labels A is monolithic and has no further structure. In fact, several labelled transition systems have “sets” of labels that are monoids [9] or even categories [14,24,25]. Such examples are more challenging to capture satisfactorily with the aforementioned approaches but some progress has been made—for instance, Bonchi and Montanari [7] captured labelled transition systems on reactive systems (in the sense of Leifer and Milner [24]) as certain coalgebras on presheaves. There is also a certain mismatch between notions typically studied by concurrency theorists or researchers in the operational semantics of concurrent languages and the morphisms in categories of coalgebras or in presheaves. From