. Functors preserving weak pullbacks provide the basis for a rich structure theory of coalgebras. We give an easy to use criterion to check whether a functor preserves weak pullbacks. We apply the characterization to the functor F which associates a set X with the set F(X) of all filters on X. It turns out that this functor preserves weak pullbacks, yet does not preserve weak generalized pullbacks. Since topological spaces can be considered as F- coalgebras, in fact they constitute a covariety, we find that the intersection of subcoalgebras need not be a coalgebra, and 1-generated F-coalgebras need not exist. 1. Introduction Coalgebras have been introduced by Aczel and Mendler [AM89] to model various types of transition systems. Reichel [Rei95], and Jacobs [Jac96] show that coalgebras are well suited for modeling object oriented programmming and for program verification. In [Rut96], J.J.M.M. Rutten develops the a fundamental theory of "universal coalgebra" along the lines of univers...

We present two ways to de ne covarieties and complete covarieties, i.e. covarieties that are closed under total bisimulation: by closure operators and by subcoalgebras of coalgebras.

We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint Theorem, and the categorical description of coinduction based on coalgebras. In particular, we examine set-theoretic generalizations of the coinduction proof principle, in the spirit of Milner's bisimulation "up-to", and we discuss categorical counterparts for these. Moreover, we investigate the connection between these and the equivalences induced by T -coiterative functions. These are morphisms into final coalgebras, satisfying the T -coiteration scheme, which is a generalization of both the coiteration and the corecursion scheme. We generalize Rutten's transformation from coalgebraic bisimulations to set-theoretic bisimulations, in order to cover also the case of bisimulations "up-to". A list of examples of set-theoretic coinductive specifications which appear not to be easily expressible in coalgebraic terms are discussed. Introduction Coinductive definitions and ...

) H. PETER GUMM Abstract. If T : Set ! Set is a functor which is bounded and preserves weak pullbacks then a class of T -coalgebras is a covariety, i.e closed under H (homomorphic images), S (sub-coalgebras) and \Sigma (sums), if and only if it can be defined by a set of "coequations". Similarly, classes closed under H and \Sigma can be characterized by implications of coequations. These results are analogous to the theorems of G.Birkhoff and of A.I.Mal'cev in classical universal algebra. 1. Introduction The recently developed theory of coalgebras under a functor T provides a highly attractive framework for describing the semantics and the logic of various types of transition systems. In contrast to the algebraic semantics of abstract data types where data objects are constructed recursively and equality is proven by induction, coalgebras support definitions by co-recursion and define equivalence by coinduction. This view is appropriate in many contexts, prominently when modelling o...

Formal semantics of some representative expressions and statements of sequential Java in the context of the state-based algebraic model of the language proposed by the authors is given.

In 1985 Luca Cardelli and Peter Wegner, my advisor, published an ACM Computing Surveys paper called “On understanding types, data abstraction, and polymorphism”. Their work kicked off a flood of research on semantics and type theory for object-oriented programming, which continues to this day. Despite 25 years of research, there is still widespread confusion about the two forms of data abstraction, abstract data types and objects. This essay attempts to explain the differences and also why the differences matter.