Vol. 4 (4:10) 2008, pp. 1–43 www.lmcs-online.org

We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class of all F-coalgebras.

Abstract. For any set-endofunctor T: Set → Set there exists a largest subcartesian transformation µ to the filter functor F: Set → Set. Thus we can associate with every T-coalgebra A a certain filter-coalgebra AF. Precisely, when T weakly preserves preimages, µ is natural, and when T weakly preserves intersections, µ factors through the covariant powerset functor P, thus providing for every T-coalgebra A a Kripke structure AP. The paper characterizes weak preservation of preimages, of intersections, and preservation of both preimages and intersections by a functor T via the existence of transformations from T to either F or P. Moreover, we define for arbitrary T-coalgebras A a next-time operator ○A with associated modal operators ✷ and ✸ and relate their properties to weak limit preservation properties of T. In particular, for any T-coalgebra A there is a transition system K with ○A = ○K if and only if T weakly preserves intersections. 1.

The functors F: K! H, which are naturally equivalent to every functor G: K! H for which FX is isomorphic to GX for all X, are called DVO functors. We discuss DVO subfunctors of the power-set functor Set! Set and of their factorfunctors. A set-theoretical assumption (EUCE) is introduced and, under (GCH+EUCE), the classes W of cardinal numbers, which have the form W = fjXj; jF Xj = jXjg for some F: Set! Set, are characterized. The presented results solve several

We prove that for any F the category of F -coalgebras is distributive if F preserves preimages, i.e. pullbacks along an injective map, and that the converse is also true whenever has finite products.

Universal coalgebra is a mathematical theory of state based systems, which in many respects is dual to universal algebra. Equality must be replaced by indistinguishability. Coinduction replaces induction as a proof principle and maps are defined by co-recursion. In this (entirely self-contained) paper we give a first glimpse at the general theory and focus on some applications in Computer Science. 1. State based systems State based systems can be found everywhere in our environment -- from simple appliances like alarm clocks and answering machines to sophisticated computing devices. Typically, such systems receive some input and, as a result, produce some output. In contrast to purely algebraic systems, however, the output is not only determined by the input received, but also by some modifiable "internal state". Internal states are usually not directly observable, so there may as well be di#erent states that cannot be distinguished from the input-output behavior of the system. A simple example of a state based system is a digital watch with several buttons and a display. Clearly, the buttons that are pressed do not by themselves determine the output - it also depends on the internal state, which might include the current time, the mode (time/alarm/stopwatch), and perhaps the information which buttons have been pressed previously. The user of a system is normally not interested in knowing precisely, what the internal states of the system are, nor how they are represented. Of course, he might try to infer all possible states by testing various input-output combinations and attribute di#erent behaviors to di#erent states. Some states might not be distinguishable by their outside behavior. It is therefore natural to define an appropriate indistinguishability relation "#...

We study properties of functors on categories of sets (classes) together with set (class) functions. In particular, we investigate the notion of inclusion preserving functor, and we discuss various monotonicity and continuity properties of set functors. As a consequence of these properties, we show that some classes of set operators do not admit functorial extensions. Then, starting from Aczel’s Special Final Coalgebra Theorem, we study the class of functors uniform on maps, we present and discuss various examples of functors which are not uniform on maps but still inclusion preserving, and we discuss simple characterization theorems of final coalgebras as fixpoints. We present a number of conjectures and problems.

ABSTRACT. We survey coalgebras as models of state based systems together with their global and local logics. We convey some useful intuition regarding Set-functors which leads naturally to coalgebraic modal logic where modalities are validity patterns for the successor object of a state. 1.

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