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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 38 (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 ...
Products of coalgebras
, 2001
"... 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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Cited by 21 (5 self)
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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.
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 18 (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)
"... If T: Set! Set is a functor which is bounded and preserves weak pullbacks then a class of Tcoalgebras is acovariety, i.e closed under H (homomorphic images), S (subcoalgebras) and (sums), if and only if it can be de ned by a set of "coequations". Similarly, classes closed under H and can ..."
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Cited by 3 (0 self)
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If T: Set! Set is a functor which is bounded and preserves weak pullbacks then a class of Tcoalgebras is acovariety, i.e closed under H (homomorphic images), S (subcoalgebras) and (sums), if and only if it can be de ned by a set of "coequations". Similarly, classes closed under H and 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.
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