  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 the actual values
distances take, thus covering, e.g., two-valued equivalences, (pseudo-)metrics,
and probabilistic (pseudo-)metrics. Coalgebraic behavioural distances have been
based either on liftings of SET-functors to categories of metric spaces, or on
lax extensions of SET-functors to categories of quantitative relations. Every
lax extension induces a functor lifting but not every lifting comes from a lax
extension. It was shown recently that every lax extension is Kantorovich, i.e.
induced by a suitable choice of monotone predicate liftings, implying via a
quantitative coalgebraic Hennessy-Milner theorem that behavioural distances
induced by lax extensions can be characterized by quantitative modal logics.
Here, we essentially show the same in the more general setting of behavioural
distances induced by functor liftings. In particular, we show that every
functor lifting, and indeed every functor on (quantale-valued) metric spaces,
that preserves isometries is Kantorovich, so that the induced behavioural
distance (on systems of suitably restricted branching degree) can be
characterized by a quantitative modal logic.
