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Categorical and Graphical Models of Programming Languages - Schweimeier (2001) (1 citation) (Correct)
.... f : n 1 ]s 1 ; n k ]s k ) s 0 , a i 2 (d n i [ s i ] A ) X) s i ] A (X 1 n i ; X 2 ; X n ) we have h s 0 ( f ] A (a 1 ; a n ) f ] B (d (h s 1 ) a 1 ) d (h s n ) a n ) Categorically, a S algebra is an algebra (see e.g. ML71] [BW85]) for a functor F = F 1 ; F n ) Set (Set F i (X 1 ; X n ) 1 X i 1 d 1 X i k ) and for g : X 1 Y 1 ; g n : X n Y n , F i (g 1 ; g n ) 1 g i 1 d 1 g i k ) and a S homomorphism is an algebra homomorphism ....
Michael Barr and Charles Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.
Similarities Between Powersets of Terms - Eklund, Galan, Medina.. (Correct)
....in a sense, an abstraction of universal algebra. It is interesting to note that monads are useful not only in universal algebra, but they are also an important tool in topology when handling regularity, iteratedness and compactifications, and also in the study of toposes and related topics. See [1,3] for category theoretic notions. As remarked in [3] the naming and identification of monads, in particular as associated with adjoints, can be seen as initiated around 1958. Godement was at that time one of the very first authors to use monads, even if then only named standard constructions . ....
....is interesting to note that monads are useful not only in universal algebra, but they are also an important tool in topology when handling regularity, iteratedness and compactifications, and also in the study of toposes and related topics. See [1,3] for category theoretic notions. As remarked in [3], the naming and identification of monads, in particular as associated with adjoints, can be seen as initiated around 1958. Godement was at that time one of the very first authors to use monads, even if then only named standard constructions . Huber in 1961 showed that adjoint pairs give rise to ....
M. Barr, C. Wells, Toposes, Triples and Theories, Springer-Verlag, 1985.
Monad-independent dynamic logic in HasCasl - Schröder, Mossakowski (2003) (Correct)
.... monads make sense in an arbitrary topos with natural numbers object (this is a suciently strong setting for recursive datatypes [10] i.e. essentially in an ambient intuitionistic higher order type theory; cf. 7] In a topos, the required partial function types do indeed exist; see for example [1]. Figure 1 shows a speci cation of monads in HasCasl. As an example of an instance for this type class, a speci cation of the state monad is shown in Figure 2. Since the operations of the monad are functions in the model, the monads thus speci ed are automatically strong, strength being ....
M. Barr and C. Wells. Toposes, Triples and Theories. Springer, 1984.
Set Functors and Generalised Terms - Eklund, Galan, Ojeda-Aciego.. (2000) (Correct)
....category associated with the term monad [13] # Corresponding author. Supported by the Swedish Research Council for Engineering Sciences. # Partially supported by the Spanish CICYT project TIC97 0579 C02 02. For notations and results within category theory and universal algebra, we refer to [1, 2, 10, 11]. For a more detailed treatment of set functors used in this paper, also including many valued sets, we refer to [4, 5, 6] For a survey of many valued logic, see e.g. 8] 2 Monads and Kleisli categories A monad can be seen as the abstraction of the concept of adjoint functors and in a sense an ....
M Barr, C Wells, Toposes, Triples and Theories, Springer, 1985.
AFramework for Unification using Powersets of Terms - Eklund, Galan, Medina.. (2002) (Correct)
....C is written as # = #,#, where # : C C is a (covariant) functor, and # : id # and :## # # are natural transformations for which # = and ## = ##=id# hold. The Kleisli category C# for # over C consists of objects in C, and the morphisms are given by hom C (X,#Y ) See [1, 3] for category theoretic notions. Let 2 be the usual covariant powerset monad (2,#, where 2X is the set of subsets of X, # X (x) x and X (B) B. The term functor T#,orT for short, with TX being the set of terms over the operator domain # and the variable set X,isextended to a monad in the ....
M. Barr, C. Wells, Toposes, Triples and Theories, Springer-Verlag, 1985.
Comprehension for Coalgebras - Jacobs (2002) (Correct)
....of sinks (collections of maps with a common codomain) to construct limits of coalgebras. Implicitly, this factorisation of sinks amounts to a greatest subcoalgebra construction. 5 Colimits for algebras Colimits of algebras are well studied topic in the categorical literature, see for instance [17,1,2]. That work concentrates on algebras for monads. This section obtains existence results for colimits of algebras of functors, by dualising the approach of the previous sections. Especially, it introduces a notion of least quotient algebra , as dual to greatest subcoalgebra . It also gives ....
M. Barr and Ch. Wells. Toposes, Triples and Theories. Springer, Berlin, 1985.
A Categorical Axiomatics for Bisimulation - Cattani, Power, Winskel (1998) (7 citations) (Correct)
....congruence properties with respect to bisimulation from open maps. Let R be a dense KZ monad on Cat and let S be another 2 monad on Cat (not necessarily KZ) By a distributive law of S over R one mean a natural transformation : SR ) RS that preserves multiplications ad units of the two 2 monads [2]. If such a distributive law is given, then a 2 monad 17 structure is induced on the composite functor T = RS. So we have (cf. Corollary 5.4) Kl(R) TC ; TD ] Kl(T ) T SC ; TD ] In particular, one obtains a functor in Kl(R) from a functor F : TSC TD in Kl(T ) by precomposing F with R ....
M. Barr, C. Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.
Entity-Relationship-Attribute Designs And Sketches - Johnson, Rosebrugh, Wood (2002) (Correct)
....idea, applied to a fixed EA sketch, can be used to capture not only the category of states or statics but also both the category of queries on the database design and the updates (or dynamics ) of database states. Certain sketches, in particular our EA sketches, have what has been called in [2] an associated theory which is, in a sense, an enlargement of the sketch to include all possible derived operations and specifications. There are a number of special cases of such theories that may come to mind but an early observation of ours was that the theory expresses the query language of ....
....a graph homomorphism M : G # C which sends diagrams belonging to D to commutative diagrams, cones belonging to L to limit cones and cocones belonging to R to colimit cocones. A homomorphism of models h : M # N is a natural transformation. For a fuller treatment we refer the reader to [1] or [2]. Models and their homomorphisms determine a category that we denote by Mod(S, C) If we write C for the category generated by G subject to the relations D then a graph homomorphism M : G # C that sends diagrams in D to commutative diagrams is the same thing as a functor M : C # C. A ....
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M. Barr and C. Wells. Toposes, Triples and Theories. Grundlehren Math. Wiss. 278, Springer Verlag, 1985.
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Generic Exception Handling and the Java Monad - Schröder, Mossakowski (Correct)
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M. Barr and C. Wells. Toposes, Triples and Theories. Springer, 1985.
Monad-independent dynamic logic in HasCasl - Schröder, Mossakowski (2003) (Correct)
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Basic Category Theory for Models of Syntax - Crole (2002) (2 citations) (Correct)
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What is the Coalgebraic Analogue of Birkhoff's Variety Theorem? - Goldblatt (2000) (2 citations) (Correct)
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Michael Barr and Charles Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.
Duality for Some Categories of Coalgebras - Goldblatt (2001) (2 citations) (Correct)
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Michael Barr and Charles Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.
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