. A notion of refinement is defined in the context of coalgebraic specification of classes in object-oriented languages. It tells us when objects in a "concrete" class behave exactly like (or: simulate) objects in an "abstract" class. The definition of refinement involves certain selection functions between procedure-inputs and attribute-outputs, which gives this notion considerable flexibility. The coalgebraic approach allows us to use coinductive proof methods in establishing refinements (via (bi)simulations). This is illustrated in several examples. 1 Introduction Refinement is an important notion in the stepwise construction of reliable software. It is used to express that an abstract description is realised by a concrete one, typically by filling-in some implementation details. This paper concentrates on refinement in an objectoriented setting. What is typical there is re-use of classes 1 : one tries to refine towards existing classes (available in some library). There are two ...

. Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (non-temporal) object-oriented systems). The monoid action captures the influence of time on the state space, where the monoids that we consider are the natural numbers monoid (N; 0; +) of discrete time, and the positive reals monoid (R0 ; 0; +) of real time. Based on this semantics we develop a hybrid specification formalism with timed method applications: it involves expressions like s:meth@ff, with the following meaning: in state s let the state evolve for ff units of time (according to the monoid action), and then apply the (coalgebraic) method meth. In this formalism we specify various (elementary) hybrid systems, investigate their correctness, and display their behaviour in simulations. We furthe...