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Duality beyond sober spaces: Topological spaces and observation frames

Marcello M. Bonsangue, Bart Jacobs, Joost N. Kok
1995 Theoretical Computer Science  
This theory is then applied to two situations: firstly to upper power spaces, and secondly we restrict the adjunction between the categories of topological spaces and of observation frames in order to  ...  A full subcategory of the category of observation frames is shown to be dual to the category of F! topological spaces.  ...  We finally wish to thank the two anonymous referees and Paul Johnson for their careful reading of the preliminary version of the paper and for their valuable suggestions to improve and correct some of  ... 
doi:10.1016/0304-3975(95)00048-2 fatcat:6b4iewfv2zgybfjlryrpfp4izy

Computably Based Locally Compact Spaces

Paul Taylor, Valeria de Paiva
2006 Logical Methods in Computer Science  
ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the  ...  In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones.  ...  Graham White has given continuing encouragement throughout the abstract Stone duality project, besides being an inexhaustible source of mathematical ideas.  ... 
doi:10.2168/lmcs-2(1:1)2006 fatcat:flaa7lepr5bglaansvgindbiia

Domain-complete and LCS-complete spaces [article]

Matthew de Brecht, Jean Goubault-Larrecq, Xiaodong Jia, Zhenchao Lyu
2019 arXiv   pre-print
Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense of Čech, and all quasi-Polish spaces-in particular, all Polish spaces  ...  We study G_δ subspaces of continuous dcpos, which we call domain-complete spaces, and G_δ subspaces of locally compact sober spaces, which we call LCS-complete spaces.  ...  The coproduct of arbitrarily many sober spaces is sober, too [13, Lemma 8.4.2] . 2 In order to show that coequalizers fail to exist, we make the following observation.  ... 
arXiv:1902.11142v1 fatcat:fzas6lzl6vet3j4tpkanwse3cu

Domain-complete and LCS-complete Spaces

Matthew de Brecht, Jean Goubault-Larrecq, Xiaodong Jia, Zhenchao Lyu
2019 Electronical Notes in Theoretical Computer Science  
Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense of Čech, and all quasi-Polish spaces-in particular, all Polish spaces  ...  We study G δ subspaces of continuous dcpos, which we call domain-complete spaces, and G δ subspaces of locally compact sober spaces, which we call LCS-complete spaces.  ...  Stone duality for domain-complete and LCScomplete spaces There This adjunction restricts to an adjoint equivalence between the full subcategories of sober spaces and spatial locales, between the category  ... 
doi:10.1016/j.entcs.2019.07.014 fatcat:igaa3hkbkfbcvolfpb2rgd3lfa

Canonical Extensions, Esakia Spaces, and Universal Models [chapter]

Mai Gehrke
2014 Leo Esakia on Duality in Modal and Intuitionistic Logics  
We describe Stone duality as derived from canonical extension and derive Priestley and Esakia duality from Stone duality for maps.  ...  Finally we relate the N -universal model of intuitionistic logic to the Esakia space of the corresponding Heyting algebra via bicompletion of quasi-uniform spaces.  ...  are continuous both in the Boolean topology of the spaces on the top and in the spectral topology of the spaces on the bottom.  ... 
doi:10.1007/978-94-017-8860-1_2 fatcat:epjnkd36prfw3lazfms2zoqmgq

Semi-metrics, closure spaces and digital topology

M.B. Smyth
1995 Theoretical Computer Science  
Within an appropriate category of these structures, ordinary spaces arise as inverse limits of digital spaces.  ...  geometry) with "ordinary" topology.  ...  The author is grateful to the referee for some corrections and improvements.  ... 
doi:10.1016/0304-3975(95)00053-y fatcat:olklenlncbe7zbwl4tmyepn3sy

Infinitary domain logic for finitary transition systems [chapter]

Marcello M. Bonsangue, Joost N. Kok
1997 Lecture Notes in Computer Science  
As a consequence soundness and completeness of the in nitary logic is obtained for the class of nitary transition systems.  ...  We thank also the anonymous referees and Achim Jung for their constructive comments.  ...  Acknowledgements: The authors are grateful to Jaco de Bakker, Bart Jacobs, Michael Mislove and Yde Venema for several fruitful discussions on the contents of this paper.  ... 
doi:10.1007/bfb0014553 fatcat:bmwkg5ofmvc5niog3lowwmzv6e

Page 3446 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
(NL-UTRE-C; Utrecht) Duality beyond sober spaces: topological spaces and observation frames. (English summary) Topology and completion in semantics (Chartres, 1993). Theoret. Comput.  ...  obtain adjunc- tions and dualities between categories of topological spaces (or,  ... 

Toward an Infinitary Logic of Domains: Abramsky Logic for Transition Systems

Marcello M. Bonsangue, Joost N. Kok
1999 Information and Computation  
We give a new characterization of sober spaces in terms of their completely distributive lattice of saturated sets.  ...  As a consequence soundness and completeness of the in nitary logic is obtained for a class of transition systems that is computational interesting.  ...  We also thank Achim Jung and the anonymous referees for their constructive comments.  ... 
doi:10.1006/inco.1999.2827 fatcat:rjqzcyirh5dzdhtkuf2rgrxcjy

Modal Logic and the Vietoris Functor [chapter]

Yde Venema, Jacob Vosmaer
2014 Leo Esakia on Duality in Modal and Intuitionistic Logics  
In [17] , Esakia uses the Vietoris topology to give a coalgebra-flavored definition of topological Kripke frames, thus relating the Vietoris topology, modal logic and coalgebra.  ...  Specifically, we look at Stone duality for the Vietoris hyperspace and the Vietoris powerlocale, and at recent work combining coalgebraic modal logic and the Vietoris functor.  ...  Additionally, we would like to thank the anonymous referee, whose thorough criticism contributed significantly to the quality of this paper, and in particular to the presentation and choice of contents  ... 
doi:10.1007/978-94-017-8860-1_6 fatcat:awd6iv5xhbfjtk2hoh2h746kxm

A topos-theoretic approach to Stone-type dualities [article]

Olivia Caramello
2011 arXiv   pre-print
In fact, infinitely many new dualities between preordered structures and locales or topological spaces can be generated through our topos-theoretic machinery in a uniform way.  ...  We then apply our topos-theoretic interpetation to obtain results connecting properties of preorders and properties of the corresponding locales or topological spaces, and we establish adjunctions between  ...  preorders and Alexandrov spaces, the Lindenbaum-Tarski duality, the duality between spatial frames and sober spaces, and we establish new ones, including localic and topological dualities for meet-semilattices  ... 
arXiv:1103.3493v1 fatcat:dtsmsjz4lremrcgvrnlmpglsgq

Priestley-type dualities for partially ordered structures [article]

Olivia Caramello
2012 arXiv   pre-print
We introduce a general framework for generating dualities between categories of partial orders and categories of ordered Stone spaces; we recover in particular the classical Priestley duality for distributive  ...  lattices and establish several other dualities for different kinds of partially ordered structures.  ...  Note that under this duality, finite posets correspond precisely to the finite and sober (equivalently, finite and T 0 ) topological spaces. Let P be a poset.  ... 
arXiv:1203.2800v1 fatcat:75lnxi74wjdxdf6ruiulg57ixm

Bibliography

2001 Electronical Notes in Theoretical Computer Science  
Jacobs, and J.N. Kok. Duality beyond sober spaces: topological spaces and observation frames. Theoretical Computer Science, 151(1):79 124, 1995. 37] M.M. Bonsangue and J.N. Kok.  ...  Topological Spaces -including a treatment of multi-valued functions, vector spaces and convexity. Oliver & Boyd, 1963.  ... 
doi:10.1016/s1571-0661(05)82567-8 fatcat:x7qb7rygq5bafjjdmi24lbdlse

Topological duality for distributive lattices, and applications [article]

Mai Gehrke, Sam van Gool
2022 arXiv   pre-print
This book is a course in Stone-Priestley duality theory, with applications to logic and theoretical computer science.  ...  Our aim is to get in a fairly full palette of duality tools as directly and quickly as possible, then to illustrate and further elaborate these tools within the setting of three emblematic applications  ...  The Ω-Pt adjunction cuts down to a duality between the category of spatial frames with frame homomorphisms and the category of sober topological spaces with continuous maps.  ... 
arXiv:2203.03286v2 fatcat:bs6pfodmkfgnhpegiuvcymb3fy

The spectrum of a well-generated tensor triangulated category [article]

Henning Krause, Janina C. Letz
2022 arXiv   pre-print
For a well-generated category and its frame of localizing tensor ideals we provide conditions such that the associated space is obtained by refining the topology of the corresponding space for the triangulated  ...  For a tensor triangulated category and any regular cardinal α we study the frame of α-localizing tensor ideals and its associated space of points.  ...  Stone duality We recall the definition of a frame and review the correspondence between frames and topological spaces, which is known as Stone duality.  ... 
arXiv:2203.03249v2 fatcat:q2kjzudxifhfrnlohzvnoo4jk4
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