This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...

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An introduction to coalgebraic specification is presented via examples. A coalgebraic specification describes a collection of coalgebras satisfying certain assertions. It is thus an axiomatic description of a particular class of mathematical structures. Such specifications are especially suitable for state-based dynamical systems in general, and for classes in object-oriented programming languages in particular. This paper will gradually introduce the notions of bisimilarity, invariance, component classes, temporal logic and refinement in a coalgebraic setting. Besides the running example of the coalgebraic specification of (possibly infinite) binary trees, a specification of Peterson's mutual exclusion algorithm is elaborated in detail.

This paper investigates the notion of dialgebra, which generalises the notions of algebra and coalgebra. We show that many (co)algebraic notions and results can be generalised to dialgebras, and investigate the essential dierences between (co)algebras and arbitrary dialgebras.

Coalgebras for endofunctors C --> C can be used to model classes of object oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C --> C . This extension allows the incorporation of binary methods into coalgebraic class speci cations. The paper also discusses how to define bisimulation for coalgebras of extended polynomial functors and proves some standard results.

The aim of this short note is to give an impression of the use of coalgebras in specification and verification for object-oriented languages. Particular emphasis will be given to the rôle of coalgebraic operations in describing state-based systems. At the end some active research topics in coalgebra will be sketched, together with pointers to the literature.