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Complete Cuboidal Sets in Axiomatic Domain Theory (Extended Abstract)
 In Proceedings of 12th Annual Symposium on Logic in Computer Science
, 1997
"... ) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichme ..."
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) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, viz. that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domaintheoretic structures. Introduction The aim of Axiomatic Domain Theory (ADT) is to provide a conceptual understanding of why domains are adequate as mathematical models of computation. (For a discussion see [12, x Axiomatic Domain Theory ].) The approach taken is to axiomatise the structure needed on a category so that its objects can be considered as domains, and its maps as continuous...
An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory
 In Proceedings of CSL 96
, 1997
"... . We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of ..."
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. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !inductive fixedpoint object (Definition 1...
Linearlyused state in models of callbyvalue
"... Abstract. We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fullycomplete statepassing translation from an impure callbyvalue language to a new linear type theory: the enriched callbyvalue calculus. The results are not specific to store, but can b ..."
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Abstract. We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fullycomplete statepassing translation from an impure callbyvalue language to a new linear type theory: the enriched callbyvalue calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not usually thought of as stateful. There is a bijective correspondence between generic effects in the source language and state access operations in the enriched callbyvalue calculus. From the perspective of categorical models, the enriched callbyvalue calculus suggests a refinement of the traditional Kleisli models of effectful callbyvalue languages. The new models can be understood as enriched adjunctions. 1
REFLECTIVE KLEISLI SUBCATEGORIES OF THE CATEGORY OF EILENBERGMOORE ALGEBRAS FOR FACTORIZATION MONADS
"... It is well known that for any monad, the associated Kleisli category is embedded in the category of EilenbergMoore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left ..."
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It is well known that for any monad, the associated Kleisli category is embedded in the category of EilenbergMoore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left
HigherOrder Containers
"... Abstract. Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category ..."
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Abstract. Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category of containers is cartesian closed, giving rise to a full cartesian closed subcategory of endofunctors. The result has interesting applications syntax. We also show that the category of containers has finite limits, but it is not locally cartesian closed. 1
The King's Buildings Edinburgh EH9 3JZ, Scotland
"... We provide an internal characterization of the category!Cpo of!complete posets and!continuous functions within the model H of SDT recently introduced by the authors. It follows that!cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. ..."
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We provide an internal characterization of the category!Cpo of!complete posets and!continuous functions within the model H of SDT recently introduced by the authors. It follows that!cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness.
Domains in H
"... We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Int ..."
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We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domainlike sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a settheoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...