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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.
Equational And Implicational Classes Of CoAlgebras (Extended Abstract)
 In The 4th International Seminar on Relational Methods in Logic, Algebra and Computer Science
"... ) 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 (subcoalgebras) and \Sigma (sums), if and only if it can be defined by a set of "coequations". Similarly, c ..."
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Cited by 3 (0 self)
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) 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 (subcoalgebras) 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 corecursion and define equivalence by coinduction. This view is appropriate in many contexts, prominently when modelling o...
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