We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra $\alpha\colon X \to FX$ in Set, where the functor $F$ specifies the branching type, we define a framework for deriving pseudometrics on $X$ which measure the behavioral distance of states.
A first crucial step is the lifting of the functor F on Set to a functor in the category 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.
Then a final coalgebra for $F$ in Set can be endowed with a behavioral distance resulting as the smallest solution of a fixed-point equation, yielding the final coalgebra in PMet. The same technique, applied to an arbitrary coalgebra $\alpha\colon X \to FX$ in Set, provides the behavioral distance on $X$. Under some constraints we can prove that two states are at distance $0$ if and only if they are behaviorally equivalent.