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Elements Of The General Theory Of Coalgebras
, 1999
"... . Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intend ..."
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Cited by 30 (7 self)
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. Data Structures arising in programming are conveniently modeled by universal algebras. State based and object oriented systems may be described in the same way, but this requires that the state is explicitly modeled as a sort. From the viewpoint of the programmer, however, it is usually intended that the state should be "hidden" with only certain features accessible through attributes and methods. States should become equal, if no external observation may distinguish them. It has recently been discovered that state based systems such as transition systems, automata, lazy data structures and objects give rise to structures dual to universal algebra, which are called coalgebras. Equality is replaced by indistinguishability and coinduction replaces induction as proof principle. However, as it turns out, one has to look at universal algebra from a more general perspective (using elementary category theoretic notions) before the dual concept is able to capture the relevant ...
Functors for Coalgebras
 Algebra Universalis
"... . 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 t ..."
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Cited by 19 (5 self)
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. 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 1generated Fcoalgebras 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...
Coalgebraic Structure From Weak Limit Preserving Functors
, 1999
"... Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and ..."
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Cited by 13 (7 self)
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Given an endofunctor F on the category of sets, we investigate how the structure theory of Set F , the category of F coalgebras, depends on certain preservation properties of F . In particular, we consider preservation of various weak limits and obtain corresponding conditions on bisimulations and subcoalgebras. We give a characterization of monos in Set F in terms of congruences and bisimulations, which explains, under which conditions monos must be injective maps.
TYPES AND COALGEBRAIC STRUCTURE
"... We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class of all Fcoalgebras. ..."
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Cited by 6 (4 self)
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We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class of all Fcoalgebras.
From Tcoalgebras to filter structures and transition systems
 Algebra and Coalgebra in Computer Science
, 2005
"... Abstract. For any setendofunctor T: Set → Set there exists a largest subcartesian transformation µ to the filter functor F: Set → Set. Thus we can associate with every Tcoalgebra A a certain filtercoalgebra AF. Precisely, when T weakly preserves preimages, µ is natural, and when T weakly preserve ..."
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Cited by 4 (2 self)
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Abstract. For any setendofunctor T: Set → Set there exists a largest subcartesian transformation µ to the filter functor F: Set → Set. Thus we can associate with every Tcoalgebra A a certain filtercoalgebra 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 Tcoalgebra 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 Tcoalgebras A a nexttime operator ○A with associated modal operators ✷ and ✸ and relate their properties to weak limit preservation properties of T. In particular, for any Tcoalgebra A there is a transition system K with ○A = ○K if and only if T weakly preserves intersections. 1.
State Based Systems Are Coalgebras
 Cubo  Matematica Educacional 5
, 2003
"... 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 corecursion. In this (entirely selfcontained) pap ..."
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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 corecursion. In this (entirely selfcontained) 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 inputoutput 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 inputoutput 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 "#...
5. Bisimulations andsimulations 33
"... 8.3. Weakpullbacks and their preservation 53 8.4. Preservation theorems 55 ..."
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8.3. Weakpullbacks and their preservation 53 8.4. Preservation theorems 55
© Birkhäuser Verlag, Basel, 2001 Products of coalgebras
"... We prove that the category of Fcoalgebras is complete, that is products and equalizers exist, provided that the type functor F is bounded or preserves mono sources. This generalizes and simplifies a result of Worrell ([Wor98]). We also describe the relationship between the product A × B and the lar ..."
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We prove that the category of Fcoalgebras is complete, that is products and equalizers exist, provided that the type functor F is bounded or preserves mono sources. This generalizes and simplifies a result of Worrell ([Wor98]). We also describe the relationship between the product A × B and the largest bisimulation ∼ A,B between A and B and find an example of two finite coalgebras whose product is infinite. 1.