  We study different behavioral metrics, such as those arising from both
branching and linear-time semantics, in a coalgebraic setting. Given a
coalgebra $\alpha\colon X \to HX$ for a functor $H \colon \mathrm{Set}\to
\mathrm{Set}$, we define a framework for deriving pseudometrics on $X$ which
measure the behavioral distance of states.
  A crucial step is the lifting of the functor $H$ on $\mathrm{Set}$ to a
functor $\overline{H}$ on the category $\mathrm{PMet}$ of pseudometric spaces.
We present two different approaches which can be viewed as generalizations of
the Kantorovich and Wasserstein pseudometrics for probability measures. We show
that the pseudometrics provided by the two approaches coincide on several
natural examples, but in general they differ.
  If $H$ has a final coalgebra, every lifting $\overline{H}$ yields in a
canonical way a behavioral distance which is usually branching-time, i.e., it
generalizes bisimilarity. In order to model linear-time metrics (generalizing
trace equivalences), we show sufficient conditions for lifting distributive
laws and monads. These results enable us to employ the generalized powerset
construction.

Logic in Computer Science