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39
A Categorical Programming Language
, 1987
"... A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Ther ..."
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Cited by 66 (0 self)
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A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Therefore, it becomes important to organize data types systematically so that we can understand the relationship between one data type and another and investigate future directions which lead us to discover exciting new data types. There have been several approaches to systematically organize data types: algebraic specification methods using algebras, domain theory using complete partially ordered sets and type theory using the connection between logics and data types. Here, we use category theory. Category theory has proved to be remarkably good at revealing the nature of mathematical objects, and we use it to understand the true nature of data types in programming.
Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 60 (9 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 26 (9 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Abstract scalars, loops, and free traced and strongly compact closed categories
 PROCEEDINGS OF CALCO 2005, VOLUME 3629 OF SPRINGER LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of ..."
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Cited by 25 (5 self)
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We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in ’ to the free construction.
Generality of proofs and its Brauerian representation
 J. Symbolic Logic
"... The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivati ..."
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Cited by 14 (4 self)
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The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivation. The variables in question are propositional or of another type. A generalization of the derivation consists in diversifying variables without changing the rules of inference. This paper examines in the setting of categorial proof theory the conjecture that two derivations with the same premises and conclusions stand for the same proof if and only if they have the same generality. For that purpose generality is defined within a category of equivalence relations on finite ordinals, where composition is rather complicated. Several examples are given of deductive systems of derivations covering fragments of
Normal Forms and CutFree Proofs as Natural Transformations
 in : Logic From Computer Science, Mathematical Science Research Institute Publications 21
, 1992
"... What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax g ..."
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Cited by 12 (4 self)
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What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cutelimination and asymmetrical interpretations of cutfree proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the KellyLambekMac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...
Selfadjunctions and matrices
 Journal of Pure and Applied Algebra
"... It is shown that the multiplicative monoids of TemperleyLieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a selfadjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronec ..."
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Cited by 10 (4 self)
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It is shown that the multiplicative monoids of TemperleyLieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a selfadjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. Thereby one obtains a representation of braid groups in matrices, which, though different and presumably new, is related to the standard representation of braid groups in TemperleyLieb algebras. Mathematics Subject Classification (2000): 57M99, 20F36, 18A40 1
A Note on Actions of a Monoidal Category
, 2001
"... An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, ..."
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Cited by 8 (0 self)
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An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functorso that, passing to the categories of monoids (also called \algebras") in V and in [A; A], we have an adjunction MonF a MonG between the category MonV of monoids in V and the category Mon[A; A] = MndA of monads on A. We give sucient conditions for the existence of the right adjoint g, which involve the existence of right adjoints for the functors X { and { A, and make A (at least when V is symmetric and closed) into a tensored and cotensored Vcategory A. We give explicit formulae, as large ends, for the right adjoints g and MonG, and also for some related right adjoints, when they exist; as well as another explicit expression for MonG as a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors.
Simple free starautonomous categories and full coherence
, 2005
"... This paper gives a simple presentation of the free starautonomous category over a category, based on EilenbergKellyMacLane graphs and Trimble rewiring, for full coherence. ..."
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Cited by 8 (0 self)
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This paper gives a simple presentation of the free starautonomous category over a category, based on EilenbergKellyMacLane graphs and Trimble rewiring, for full coherence.
Functorial calculus in monoidal bicategories
 Applied Categorial Structures
, 2002
"... The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomp ..."
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Cited by 7 (1 self)
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The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomplete symmetric monoidal