MSC: 03B45 03B70Keywords: Predicate liftings Behavioral equivalence Hennessy-Milner Theorem Stochastic relations Coalgebraic and modal logic Measurable selections a b s t r a c t We generalize stochastic Kripke models and Markov transition systems to stochastic right coalgebras. These are coalgebras for a functor F · S with F as an endofunctor on the category of analytic spaces, and S is the subprobability functor. The modal operators are generalized through predicate liftings which are set-valued natural transformations involving the functor. Two states are equivalent iff they cannot be separated by a formula. This equivalence relation is used to construct a cospan for logical equivalent coalgebras under a separation condition for the set of predicate liftings. Consequently, behavioral and logical equivalence are really the same. From the cospan we construct a span. The central argument is a selection argument giving us the dynamics of a mediating coalgebra from the domains of the cospan. This construction is used to establish that behavioral equivalent coalgebras are bisimilar, yielding the equivalence of all three characterizations of a coalgebra's behavior as in the case of Kripke models or Markov transition systems.

Categories of lax (T, V )-algebras are shown to have pullback-stable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax (T, V )-algebras, with T preserving inverse images. Moreover, we show that any such category of (T, V )-algebras has a concrete, coproduct preserving functor into the category of topological spaces.

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