Tree-functors, determinacy and bisimulations

We study the functorial characterization of bisimulation-based equivalences over a categorical model of labeled trees. We show that, in a setting where all labels are visible, strong bisimilarity can be characterized in terms of enriched functors by relying on reflection of paths with their factorizations. For an enriched functor F , this notion requires that a path (an internal morphism in our framework) π going from F (A) to C corresponds to a path p going from A to K, with F (K) = C, such that every possible factorization of π can be lifted in an appropriate factorization of p. This last property corresponds to a Conduché property for enriched functors and a very rigid formulation of it has been used by Lawvere to characterize determinacy of physical systems. We also consider the setting where some labels are not visible and provide characterizations for weak and branching bisimilarity. Both equivalences are still characterized in terms of enriched functors that reflect paths with their factorizations: for branching bisimilarity, the property is the same as the one that characterized strong bisimilarity when all labels are visible; for weak bisimilarity, a weaker form of path factorization lifting is needed. This fact can be seen as an evidence that strong and branching bisimilarity are strictly related and that, differently from weak bisimilarity, they preserve process determinacy in the sense of Milner.