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Building Metric Structures with the Meas-Functor - den Hartog, de Vink (Correct)
....Finally, section 4 illustrates the use of the functor for the construction of a probabilistic semantical domain. 2 Mathematical preliminaries We assume the reader to be familiar with the basic notions from metric topology, measure theory and category theory. See, e.g. 9, 30] 18, 26] and [22, 11], respectively. De nition 2.1 A function d: M M [0; 1] is a 1 bounded ultrametric on the nonempty set M if the following conditions are met: i) d(x; y) d(y; x) ii) d(x; y) 0 ( x = y (iii) d(x; z) maxf d(x; y) d(y; z) g 3 for all x; y; z 2 M . Condition (iii) of De nition 2.1 ....
M. Barr and C. Wells. Category Theory for Computing Science. Prentice Hall, 1990.
Comparison of Categorical Foundations - Of Object-Oriented Database (1997) (Correct)
....in [5] 7] 17] 18] 19] and [21] In these works, classes and database objects are basically represented as categorical objects, operations as arrows, generalization and specialization constructions as limits. For our purpose , we will use just the basic notions of CT as presented in [1]. The paper is organized as follows: The next section reviews the current research in the area of the object oriented modeling based on CT. We discuss advantages and disadvantages of the existing models. Section 3 compares these models with a presentation of limit data model. Section 4 describes, ....
....to the aggregation with the exception that we have to solve the problem of attribute access for the derived classes. The attributes brought from the superclasses can be accessed again by using the composition of projections. The arrow composition is an essential property of CT and according to [1] we use the notation ffi (a) for a composed projection ( a) In addition to the aggregation, we will introduce a specific labeling assigning meaningful names to the (composed) attribute projections. The labeling is essential to model the inheritance and will describe the flat naming ....
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BARR, M., WELLS, C.: Category Theory for Computing Science. Prentice Hall, 1995.
Towards an Algebraic Semantics for the Object Paradigm - Goguen, Diaconescu (1994) (43 citations) (Correct)
....familiarity with some basics of term rewriting, including con uence, termination and narrowing; these are explained, for example, in [7] and [16] We sometimes use basic notions from category theory, including category, functor, and initial object. For an introduction to category theory, see [3] or [27] We use the bbold font to denote categories, e.g. C. Given morphisms f : A B and g : B C, we let f ; g denote their composition, a morphism A C; also, we let 1 A denote the identity morphism at an object A. Sections 4.1 and 7 use colimits, and Section 7 also uses universal ....
Michael Barr and Charles Wells. Category Theory for Computing Science. Prentice-Hall, 1990.
Categorical and Graphical Models of Programming Languages - Schweimeier (2001) (1 citation) (Correct)
....some of the mathematical structures which are used in this thesis. We presuppose some fundamental definitions of category theory such as category, functor, natural transformation, co )limit, adjunction, and cartesian closed category, which can be found in any textbook on category theory, such as [ML71, Pie91, BW95, Bor94a]. 2.1 Monoidal Categories Definition 2.1. A monoidal category is a category C with a functor : C C C , called the tensor product, an object I in C , called the tensor unit, and natural isomorphisms a, l and r with components B) B I A satisfying the coherence conditions given ....
Michael Barr and Charles Wells. Category Theory for Computing Science. Prentice Hall, 2nd edition, 1995.
Composition and Refinement of Behavioral Specifications - Pavlovic, Smith (2001) (1 citation) (Correct)
....The paper is structured straightforwardly. We rst discuss how to extend logical speci cations to model behavior, and then de ne especs and how to re ne and compose them formally. These concepts are illustrated by simple examples. This paper presumes some knowledge of basic category theory (see [2, 11] for relevant background) More details about especs may be found in [8] Related approaches to providing categorical foundations for specifying, composing and re ning behaviors may be found in [3, 5] 2 From Logical Theories to State Machines Behaviors EPOXI is made of two basic building ....
Barr, M., and Wells, C. Category Theory for Computing Science. Prentice-Hall, Englewood Cli s, NJ, 1990.
New Semantics for the Simply Typed lambda-calculus - Silterra (2003) (Correct)
....related to free models very easily become undecidable. However, as an immediate corollary, we have t 1 = t 2 i for all posets P and models [ T Sets we have [ t 1 ] t 2 ] Proof of Soundness and Completeness. The proof is an extension of results from [2] First, recall from [3] or [10] that for any theory T, there is a complete CCC category C (T) and interpretation [ C : T C (T) By Yoneda, there is a full faithful and CCC functor y : C (T) Sets . Next, by [2] and [11] there is a topological space X and two full, faithful and CCC functors, Sets ....
....category theory. 7.1 Cartesian Closed Categories and the calc A cartesian closed category is a category with all nite products and exponentials. In general, one can interpret STT in a cartesian closed category (CCC) An interpretation [ of a theory T in a CCC C is sketched here, see [3] or [10] for more details. Let ( 1 ; 2 ; be the canonical operations in C of pairing, projections, transposition and evaluation. Basic types are interpreted as objects in C and then extended inductively to all types by [ and [ Also, we ....
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Barr, M., Wells, C., \Category Theory For Computing Science", Montreal: Les Publications CRM, 1999.
Categorical Product As A Generic Graph Rewriting Mechanism - Bauderon, Jacquet (1996) (1 citation) (Correct)
....all those kinds of nodes. This graph will be called the structure graph and will often be denoted by S. Fortunately, structured graphs have the nice idea of forming a well behaved category, since they are a slice category of the standard category of graphs which is well known to be a topos [2]. To have a good understanding of our approach, is must be clear that, from our point of view, only nodes are active items in a graph, edges being only used to express limitations in the possible neighbourhood of a node. If we want to promote an edge to an active role, we shall replace it by a ....
....try to provide the reader with an intuitive description of all the objets we build. But we shall not recall the standard definition of basic constructions like product, pullback or slice category. For any such definition or further details we refer the reader to standard textbooks such as [1] [2] or [27] 2 A generic framework 2.1 Graphs In this paper, we shall let ZZ denote the set of integers, or IN the set of non negative integers and IN that of strictly positive ones. We let #E be the number of elements of a set E. As usual ; denotes the empty set. As a starting point, let us ....
M. Barr, C.F. Wells, Category theory for computing science, Prentice-Hall, 1990.
Three Approaches to Partiality in the Sketch Data Model - Johnson, Rosebrugh (2002) (Correct)
.... [10] 37] and [12] The extra constraint information has proved valuable theoretically too, leading to a new treatment of the view update problem [11] and to new techniques for database interoperability [29] and [9] For background material on the theory of sketches the reader can consult [3] or [1]. Definition 2.1 A cone C = v C ; I C : B C ## G;hp b i b2(B C ) 0 ) in a graph G consists of a node v C of G (the vertex of C) a graph morphism I C : B C ## G (the base diagram of C) and, for each node b in B C , an edge p b : v C ## I C b. Cocones are dual (that is we reverse all the edges ....
M. Barr and C. Wells. Category theory for computing science. Prentice-Hall, second edition, 1995.
Semantics for Interoperability: relating ontologies and.. - Bench-Capon, Malcolm.. (2002) (Correct)
....a concrete syntax; the motivating examples given below are in an ad hoc syntax. The presentation of ontologies below follows [3] and is rather dense, assuming some familiarity with algebraic specifications (introductions can be found in [33, 12] and also with basic category theory (see [27, 2] for introductions) Our ontologies specifiy classes of entities with attributes. These attributes take values in data types such as numbers, booleans, lists and so on. Sometimes it is convenient to make changes in the types of attributes, for example an ontology may be refined by refining the ....
Michael Barr and Charles Wells. Category Theory for Computing Science. Prentice Hall, 1990.
Component-Based Coalgebraic Specification and Verification in.. - Meng, Aichernig (2002) (3 citations) (Correct)
....functors, coalgebras for polynomial functors together with the associated definitions of bisimulation, invariant and finality. Familiarity with basic category theory is required when reading this paper. Readers can see e.g. JR97, Rut00] for more detailed information about coalgebras and [AL91, BW99] as references to category theory. The concept of functor comes from category theory. In this paper, we use polynomial functors to describe coalgebraic signatures. All the functors being used are the endofunctors on the category Set of sets and Coalgebras, Bisimulation, Invariant and Finality 3 ....
Michael Barr and Charles Wells. Category Theory for Computing Science, Third Edition. Les Publications CRM, 1999.
Expressive Logics for Coalgebras via Terminal Sequence Induction - Pattinson (2002) (2 citations) (Correct)
....of relators in a coalgebraic setting. The extension of endofunctors to relations is an integral part of Moss de nition of the semantics of coalgebraic logic, which is given in terms of initial algebras. For the basic notions regarding (initial) algebras we refer to the book by Barr and Wells [5]. De nition 5.3. Suppose T is accessible and put L = P T . The language L(T ) of coalgebraic logic associated to T is the carrier of the initial L algebra (L; If (C; 2 CoAlg(T ) put d : P P(C) P(C) d(x) x and e : TP(C) P(C) e(x) fc 2 C j ( c) t) 2 T ( C )g, where ....
M. Barr and C. Wells. Category Theory for Computing Science. Prentice Hall, 1989.
Sketch Data Models, Relational Schema and Data Specifications - Johnson, Rosebrugh (2001) (Correct)
....(iii) denoting G (s; t; k) s ; t ; k ) s (s) t (t) and k =G(s)k (compatibility for foreign keys) In the sequel, when the component of G is obvious, we will omit the superscript. Evidently, morphisms of database schemes form a category which we denote DbSch. A sketch [2] is a quadruple consisting of a graph, a set of diagrams in the graph, a set of cones in the graph, and a set of cocones in the graph. Let P denote the function taking each cone to its base, and let S denote the function taking each cocone to its base. We recall from [19] the definition of EA ....
M. Barr and C. Wells. Category theory for computing science. Prentice-Hall, second edition, 1995.
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M. Barr and C. Wells. Category Theory for Computing Science. London: PrenticeHall, 1990.
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Combining Computational Effects: Commutativity and Sum - Hyland, Plotkin, Power (2002) (1 citation) (Correct)
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Barr, M., and Wells, C. 1990. Category Theory for Computing Science. Prentice-Hall.
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M. Barr and C. Wells. Category Theory for Computing Science. Prentice Hall, 1990.
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