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The largest Cartesian closed category of domains, considered constructively

2005
*
Mathematical Structures in Computer Science
*

Theorem

doi:10.1017/s0960129504004591
fatcat:vp4xagskqzhqzhemvfwih46msy
*The**category**of**constructive*SFP*domains*is*the**largest**constructively**Cartesian**closed*weakly indexed full subcategory*of**the**category**of**constructive**domains*that have a computable completeness ... Corollary*The**category**of*effective SFP objects and continuous maps is*the**largest**Cartesian**closed*full subcategory*of**the**category**of*ω-algebraic cpo's that have a computable completeness test. ... It is shown that*the**category**of**constructive*SFP*domains*is*the**largest**constructively**Cartesian**closed*weakly indexed full subcategory*of**the**category**of**constructive**domains*having a computable completeness ...##
###
On the Largest Cartesian Closed Category of Stable Domains

2014
*
Electronical Notes in Theoretical Computer Science
*

Zhang showed that

doi:10.1016/j.entcs.2014.01.011
fatcat:27gq52katrg2jjcuunrgvnvjjq
*the**category**of*dI-*domains*is*the**largest**cartesian**closed*subcategory*of*ω-SABC and ω-SABC, with*the*exponential being*the*stable function space, where ω-SABC and ω-SABC are full subcategories ...*cartesian**closed*subcategory*of*SABC;*The*compact elements*of*function spaces in*the**category*SABC are also studied. ... Theorem 4.1*The**category**of*dI-*domains*is*the**largest**cartesian**closed*subcategory*of*ω-SABC. Next we show that*the**category*SDABC is*the**largest**cartesian**closed*subcategory*of*SABC. ...##
###
The largest cartesian closed category of domains

1983
*
Theoretical Computer Science
*

*The*importance

*of*this is that it enables us to conclude that SFP is

*the*

*largest*

*category*

*of*

*domains*that iq

*closed*under

*the*

*constructions*

*of*interest in semantics. ... In

*the*final section we briefly

*consider*how our results may need to be extended if our, admittedly rather restrictive, notion

*of*'

*category*

*of*

*domains*' is relaxed in certain ways. ... Acknowledgment

*The*author is grateful tcl Gordon Tlotkin, Dana Scott. Flynn Winskel and an anonymous referee for suggesting improvements and corrections to

*the*wiginal version

*of*this paper. ...

##
###
Weakly Distributive Domains
[chapter]

2007
*
Lecture Notes in Computer Science
*

This accomplishes

doi:10.1007/978-3-540-73228-0_15
fatcat:xvl4t2mkbrff3dz7uqk5s5xne4
*the*first*of*a possible, two-step process in solving*the*problem raised in [1, 2] : whether*the**category**of*stable bifinite*domains**of*Amadio-Droste-Göbel [1, 6] is*the**largest**cartesian*... We introduce a new class*of**domains*called weakly distributive*domains*and show that for these*domains*to be in a*cartesian**closed**category*using ω-algebraic meet-cpos, property MI ∞ must not be violated ...*The*existence*of*a variety*of**cartesian**closed**categories**of**domains*motivated a systematic investigation*of**the*question*of*"*largest**cartesian**closed**categories**of**domains*", starting with*the*work*of*...##
###
Book review

1999
*
Science of Computer Programming
*

*the*

*largest*

*cartesian*

*closed*subcategory

*of*

*the*

*category*

*of*algebraic dcpo's (with a countable basis) and some

*of*its variants. ...

*of*

*the*theory developed in Chapter 2), and give rise to several

*cartesian*

*closed*

*categories*

*of*

*domains*whose morphisms restrict Scott-continuous functions to match better

*the*operational behavior

*of*programs ...

##
###
On exponentiating exponentiation

1981
*
Journal of Pure and Applied Algebra
*

One

doi:10.1016/0022-4049(81)90050-5
fatcat:a4vam3fykngtna22sv5ifcx74a
*of**the*simplest examples*of*a*Cartesian**closed**category*which is not complete is*the**category*FEns*of*finite sets;*the*last section investigates*the**Cartesian**closedness**of**categories*FEnsJ. ... each complete,*Cartesian**closed**category*K,*the**category**of*all K-valued sheaves on any topological space is also*Cartesian**closed*. ...##
###
All cartesian closed categories of quasicontinuous domains consist of domains

2015
*
Theoretical Computer Science
*

In this paper we ask which

doi:10.1016/j.tcs.2015.05.014
fatcat:4q4huwdaafbepbdhmq4b3gmuk4
*cartesian**closed*full subcategories exist in qCONT,*the**category**of*all quasicontinuous*domains*and Scottcontinuous functions. ... Quasicontinuity is a generalisation*of*Scott's notion*of*continuous*domain*, introduced in*the*early 80s by Gierz, Lawson and Stralka. ... Acknowledgements*The*first author acknowledges support by*the*University*of*Birmingham ...##
###
A Convenient Category of Domains

2007
*
Electronical Notes in Theoretical Computer Science
*

Our

doi:10.1016/j.entcs.2007.02.004
fatcat:tfvmwjta3bghfip63xew2onqvi
*category*supports all*the*standard*constructions**of**domain*theory, including*the*solution*of*recursive*domain*equations. ... We motivate and define a*category**of*topological*domains*, whose objects are certain topological spaces, generalising*the*usual ω-continuous dcppos*of**domain*theory. ... Acknowledgements We are happy to acknowledge*the*enormous influence Gordon Plotkin has had on*the*development*of*this research. ...##
###
Coherence and consistency in domains

1990
*
Journal of Pure and Applied Algebra
*

These form a

doi:10.1016/0022-4049(90)90055-m
fatcat:j5f7xcez3vgpvl2wd4ajy3wn5q
*Cartesian**closed**category*which has fixed points for*domain*equations. It is shown that a 'universal*domain*' exists. ... Since*the**construction**of*this*domain*seems to be*of*general significance, a categorical treatment is provided and applied to other classes*of**domains*. ...*Cartesian**closed*subcategories:*the**category**of*Ldomains and*the**category**of*bifinite*domains*, which we now proceed to define. ...##
###
The versatile continuous order
[chapter]

1988
*
Lecture Notes in Computer Science
*

*The*earlier sections concentrate on

*the*order-theoretic aspects

*of*continuously ordered sets and then specifically

*of*

*domains*. ...

*The*ideal completion can be characterized alternately as arising from

*the*adjoint functor to

*the*forgetful functor from

*the*

*category*

*of*CPO's and continuous morphisms to

*the*

*category*

*of*partially ordered ... Do

*the*finitely continuous CPO's form

*the*

*largest*

*cartesian*

*closed*full subcategory contained in

*the*

*category*

*of*continuous CPO's? Problem. ...

##
###
Compactly generated domain theory

2006
*
Mathematical Structures in Computer Science
*

*The*

*category*

*of*such spaces enjoys

*the*usual properties

*of*

*categories*

*of*"predomains" in denotational semantics. ... We compare

*the*standard

*domain*-theoretic

*constructions*

*of*products and function spaces on dcpos with their compactly generated counterparts, showing that these agree in important cases though not in general ... We also thank

*the*editors

*of*this volume for their patience and encouragement. ...

##
###
Comparing categories of domains
[chapter]

1986
*
Lecture Notes in Computer Science
*

We discuss some

doi:10.1007/3-540-16816-8_27
fatcat:uz3bqcsf3ndvrcoswzle4y6o2e
*of**the*reasons for*the*proliferation*of**categories**of**domains*suggested for*the*mathematical foundations*of**the*Scott-Strachey theory*of*programming semantics. ... Five general conditions are presented which such a*category*should satisfy and they arc used to motivate a number*of*examples. ... I would like to thank*the*following people for suggestions and encouragement: Achim Jung, Gordon Plotkin, Pino Itosolini, Dana Scott, Rick Statman, Adrian Tang. ...##
###
A uniform approach to domain theory in realizability models

1997
*
Mathematical Structures in Computer Science
*

One possibility is to cut down

doi:10.1017/s0960129597002387
fatcat:j6roaodr2fgbzjrth4emkq5ctu
*the**category**of*topological spaces to a full subcategory that is*cartesian**closed*. ... spaces) (Hyland 1979b); or*the*even larger*category**of*quotients*of*exponentiable spaces*considered*in Day (1972) . ... We are especially grateful to Matthias Schröder for permission to include his proof*of*Theorem 3. We also acknowledge*the*use*of*Paul Taylor's diagram macros. ...##
###
Topological and limit-space subcategories of countably-based equilogical spaces

2002
*
Mathematical Structures in Computer Science
*

Under one approach, one restricts to a full subcategory

doi:10.1017/s0960129502003699
fatcat:tg66vh7on5amneyxftvenedxlu
*of*topological spaces that happens to be*cartesian**closed*-for example,*the**category**of*sequential spaces. ... There are two main approaches to obtaining "topological"*cartesian*-*closed**categories*. ... We are especially grateful to Matthias Schröder for permission to include his proof*of*Theorem 3. We also acknowledge*the*use*of*Paul Taylor's diagram macros. ...##
###
A characterisation of the least-fixed-point operator by dinaturality

1993
*
Theoretical Computer Science
*

., A characterisation

doi:10.1016/0304-3975(93)90112-7
fatcat:whulglkzcvaoporhup5ptquvza
*of**the*least-fixed-point operator by dinaturality, Theoretical Computer Science 118 (1993) 301-314. ... (*The**category**of*countably based algebraic L-*domains*is not*Cartesian*-*closed*.201, and so*the**category*ContL,*of*continuous L-*domains*, is*Cartesian*-*closed*. Theorem 5.1. ... D is an L-*domain*if, for every XED,*the*set Jx is a complete lattice under*the*induced ordering. AlgL,*the**category**of*algebraic L-*domains*, is*Cartesian*-*closed*[S]. ...
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