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11
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
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.
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.
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.
Modal Operators for Coequations
, 2001
"... this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree ..."
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this paper, we develop the theory of coequations from a logical viewpoint. To clarify, let G = #G, #, ## be a comonad on E , where G preserves regular monos and E is "coBirkho #" (see Definition 2.1). A coequation # over a set C of colors is a regular subobject of GC, the carrier of the cofree coalgebra # C : GC ## G 2 C over C. Hence, we can view # as a predicate over GC. In particular, we can form new coequations out of old by means of the logical connectives #, #, etc. Furthermore, we have available a modal operator taking a coequation # to the (carrier of the) largest subcoalgebra # contained in the coequation. As we will see, arises as the formal dual of a familiar operation on sets of equations in categories of algebras. Explicitly, the operator is dual to the closure operation taking a set E of equations over X (i.e., E # UFX UFX , where UFX is the carrier of the free algebra over X) to the least congruence containing E. Hence, is dual to the closure of sets of equations under the first four rules of inference of Birkho#'s equational logic (Birkho#, 1935). Thus, we see that closure under these rules of inferences is dual to the "coalgebra interior" of a set of elements. We introduce a modal operator that is dual to closure under Birkho#'s fifth rule of inference, i.e., substitution of terms for variables. We confirm that is an S4 operator and show that, under certain conditions, commutes with . We then prove the invariance theorem in terms of and . In this way, we develop the coequationsaspredicates view by augmenting the predicates over GC with two modal operators and and show that the partial order of covarieties definable by arbitrary coequations over C is isomorphic to the partial order of predicates # over GC such th...
Talk I: Final Coalgebras
, 2003
"... For the remainder of this exposition assume that T: Set! Set is an endofunctor. Unless otherwise stated, the results and proofs of the material presented is taken from Rutten [10]. 1 Preliminaries Deo/nition 1.1. (i) A Tcoalgebra (Tsystem) is pair (C; fl) where C is a set and fl: C! T C. (ii) If ( ..."
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For the remainder of this exposition assume that T: Set! Set is an endofunctor. Unless otherwise stated, the results and proofs of the material presented is taken from Rutten [10]. 1 Preliminaries Deo/nition 1.1. (i) A Tcoalgebra (Tsystem) is pair (C; fl) where C is a set and fl: C! T C. (ii) If (C; fl) and (D; ffi) are Tsystems, then f: C! D is a Tcoalgebra homomorphism (Tmorphism), if ffi ffi f = T f ffi fl, that is, if the diagram C fl fflffl f // D
Under consideration for publication in Math. Struct. in Comp. Science A Comonadic Account of Behavioural
, 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 coalgebras ..."
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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 significant property. In any logical system suitable for specifying properties of statetransition systems in the HennessyMilner style, each formula will define a class of models that is a behavioural covariety. Assume that the forgetful functor on Tcoalgebras has a right adjoint, providing for the construction of cofree coalgebras, and let G T be the comonad arising from this adjunction. Then we show that behavioural covarieties K are (isomorphic to) the EilenbergMoore categories of coalgebras for certain comonads G K naturally associated with G T. These are called pure subcomonads of G T, and a categorical characterization of them is given, involving a pullback condition on the naturality squares of a transformation from G K to G T. We show that there is a bijective correspondence between behavioural covarieties of Tcoalgebras and isomorphism classes of pure subcomonads of G T.
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"... Abstract. In the theory of coalgebras C over a ring R, the rational functor relates the ..."
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Abstract. In the theory of coalgebras C over a ring R, the rational functor relates the