Quantalic Behavioural Distances

Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to behavioural distances. In particular, coalgebraic methods capture variations in the system type (nondeterministic, probabilistic, game-based etc.), and the notion of quantale abstracts over actual values distances take, thus covering, e.g., two-valued equivalences, metrics, and probabilistic metrics. Coalgebraic behavioural distances have variously been based on liftings of $\mathsf{Set}$-functors to categories of metric spaces; on modalities modeled as predicate liftings, via a generalised Kantorovich construction; and on lax extensions of $\mathsf{Set}$-functors to categories of quantitative relations. Every lax extension induces a functor lifting in a straightforward manner. Moreover, it has recently been shown that every lax extension is Kantorovich, i.e. induced by a suitable choice of monotone predicate liftings. In the present work, we complete this picture by determining, in coalgebraic and quantalic generality, when a functor lifting is induced by a class of predicate liftings or by a lax extension. We subsequently show coincidence of the respective induced notions of behavioural distances, in a unified approach via double categories that applies even more widely, e.g. to (quasi)uniform spaces.