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17
Algebras versus coalgebras
 Appl. Categorical Structures, DOI
, 2007
"... Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of the ..."
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Cited by 12 (10 self)
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Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 70’s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between
Coalgebras of Bounded Type
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2001
"... Using results of Trnková, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal Fcoalgebra. Several equivalent characterizations of boundedness are provided. ..."
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Cited by 8 (4 self)
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Using results of Trnková, we first show that subcoalgebras are always closed under finite intersections. Assuming that the type functor F is bounded, we obtain a concrete representation of the terminal Fcoalgebra. Several equivalent characterizations of boundedness are provided.
Factorization systems and fibrations: Toward a fibred Birkhoff variety theorem
, 2002
"... It is wellknown that a factorization system on a category (with sufficient pullbacks) gives rise to a fibration. This paper characterizes the fibrations that arise in such a way, by making precise the logical structure that is given by factorization systems. The underlying motivation is to obtain g ..."
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Cited by 4 (0 self)
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It is wellknown that a factorization system on a category (with sufficient pullbacks) gives rise to a fibration. This paper characterizes the fibrations that arise in such a way, by making precise the logical structure that is given by factorization systems. The underlying motivation is to obtain general Birkho results in a fibred setting.
Modal Predicates and Coequations
, 2002
"... We show how coalgebras can be presented by operations and equations. We discuss the basic properties of this presentation and compare it with the usual approach. ..."
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Cited by 4 (2 self)
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We show how coalgebras can be presented by operations and equations. We discuss the basic properties of this presentation and compare it with the usual approach.
A Comonadic Account of Behavioural Covarieties
 Mathematical Structures in Computer Science
, 2002
"... A class K of coalgebras for an endofunctor T : Set ! Set is a behavioural covariety if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras). K may be thought of as the class of all coalgeb ..."
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Cited by 2 (1 self)
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A class K of coalgebras for an endofunctor T : Set ! Set is a behavioural covariety if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras). K may be thought of as the class of all coalgebras that satisfy some computationally signi cant property, or properties.
Presentation of set functors: a coalgebraic perspective
"... Abstract. Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations. 1. ..."
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Cited by 1 (1 self)
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Abstract. Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations. 1.
From grammars and automata to algebras and coalgebras
, 2013
"... Abstract. The increasing application of notions and results from category theory, especially from algebra and coalgebra, has revealed that any formal software or hardware model is constructor or destructorbased, a whitebox or a blackbox model. A highlystructured system may involve both construct ..."
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Abstract. The increasing application of notions and results from category theory, especially from algebra and coalgebra, has revealed that any formal software or hardware model is constructor or destructorbased, a whitebox or a blackbox model. A highlystructured system may involve both constructor and destructorbased components. The two model classes and the respective ways of developing them and reasoning about them are dual to each other. Roughly said, algebras generalize the modeling with contextfree grammars, word languages and structural induction, while coalgebras generalize the modeling with automata, Kripke structures, streams, process trees and all other state or objectoriented formalisms. We summarize the basic concepts of co/algebra and illustrate them at a couple of signatures including those used in language or compiler construction like regular expressions or acceptors. 1