In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (Non-Well-Founded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken

Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and state-transition systems. This paper presents a calculus of terms for operations on such coalgebras, based on a simple type theory, and develops its semantics. The terms admit a single state-valued parameter, but may also have state-valued variables. In a "rigid" term all state-variables are bound. Boolean

Coalgebras of set functors preserving weak pullbacks are particularly well-behaved. Invoking a result by Carboni, Kelly, and Wood (1990), we show that this can be explained by the fact that such functors can be uniquely extended to a relator. This insight next suggests a de nition of metric bisimulation.