## Duality beyond Sober Spaces: Topological Spaces and Observation Frames (1995)

Venue: | and Completion in Semantics |

Citations: | 10 - 4 self |

### BibTeX

@ARTICLE{Bonsangue95dualitybeyond,

author = {Marcello Bonsangue and Bart Jacobs and Joost Kok},

title = {Duality beyond Sober Spaces: Topological Spaces and Observation Frames},

journal = {and Completion in Semantics},

year = {1995},

volume = {151},

pages = {18--20}

}

### OpenURL

### Abstract

We introduce observation frames as an extension of ordinary frames. The aim is to give an abstract representation of a mapping from observable predicates to all predicates of a specific system. A full subcategory of the category of observation frames is shown to be dual to the category of T 0 topological spaces. The notions we use generalize those in the adjunction between frames and topological spaces in the sense that we generalize finite meets to infinite ones. We also give a predicate logic of observation frames with both infinite conjunctions and disjunctions, just like there is a geometric logic for (ordinary) frames with infinite disjunctions but only finite conjunctions. 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 obtain dualities for various subcategories of T 0 spaces. These involve non sober spaces. Contents 1 Introduct...

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Citation Context ... some basic notions and facts on lattices and topological spaces. For more detailed discussions consult [Dug66, Eng89] and [GHK + 80]. We assume some familiarity with basic notions of category theory =-=[Mac71]-=-. A poset L = (L; ) is a set together with a reflexive, transitive and antisymmetric relationson L. A poset which has meets (greatest lower bounds) for every pair of elements is a meet-semilattice. A ... |

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Citation Context ...orphism (notice that for proving F I F = F we use the fact that every open set o 2 O(X) is equal to "a for some a ` K(X)). 2 The power domains on algebraic dcpo's in (i) were first studied by Plo=-=tkin [Plo76]-=- and Smyth [Smy78]. 5.12. Remark. The compact elements of the algebraic dcpo (Fil(O(X)); `) are isomorphic to the principal ideals of the poset (P(K(X)); U ), which are isomorphic, by Lemma 5.9, to th... |

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Citation Context ...ation Lattices 44 1 Introduction A topological duality is a correspondence between two mathematical structures involving points and predicates such that isomorphic structures can be identified. Stone =-=[Sto36]-=- first found such a duality between topology and logic. He considered ordered sets (representing the syntax of some logical system) and constructed from a boolean algebra a set of points using prime f... |

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Citation Context ...at for proving F I F = F we use the fact that every open set o 2 O(X) is equal to "a for some a ` K(X)). 2 The power domains on algebraic dcpo's in (i) were first studied by Plotkin [Plo76] and S=-=myth [Smy78]-=-. 5.12. Remark. The compact elements of the algebraic dcpo (Fil(O(X)); `) are isomorphic to the principal ideals of the poset (P(K(X)); U ), which are isomorphic, by Lemma 5.9, to the M-filters of the... |

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Citation Context ...he set of its directed ideals Idl(P ) ordered by subset inclusion, extends to a functor Idl(\Gamma) : PoSet ! AlgPos which is left adjoint of the forgetful functor U : AlgPos ! PoSet (see for example =-=[Plo81]-=-). 6.9. Lemma. The inclusion functor i : AlgCDLat ,! AlgCDFrm has a left adjoint j that is given by assigning to every algebraic complete lattice L the Alexandrov topology of the set of all prime elem... |

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Citation Context ...by using a topology on a set of points he was able to construct a boolean algebra. For certain topological spaces (later called Stone spaces) these constructions give an isomorphism. In a later paper =-=[Sto37]-=-, he generalized this correspondence from Stone spaces to spectral spaces and from boolean algebra's to distributive lattices. Hofmann and Keimel [HK72] described this duality in a categorical framewo... |

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Citation Context ...n as Scott-open filter theorem) given in Corollary 5.6 below. The latter is about Scott-open sets F ` O(X) of a (sober) space X, which are ordinary filters. This theorem is due to Hofmann and Mislove =-=[HM81]-=-, and can in our present setting be obtained from the following result. It identifies where one uses the Axiom of Choice and the fact that the space is sober. We sketch the proof for reasons of comple... |

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Citation Context ...respondence from Stone spaces to spectral spaces and from boolean algebra's to distributive lattices. Hofmann and Keimel [HK72] described this duality in a categorical framework. Even further, Isbell =-=[Isb72a]-=- gives an adjunction between the category of topological spaces with continuous functions and the opposite category of frames with frame homomorphisms (which yields a duality between sober spaces and ... |

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Citation Context ... and spatial frames). Stone duality in a mathematical context is studied in a book of Johnstone [Joh82], and for the context of domain theory we refer the reader to [GHK + 80]. In his thesis Abramsky =-=[Abr87]-=- applied Stone duality to get logics of domains, as used in denotational semantics. He argues that Stone duality is the bridge between denotational and axiomatic semantics. Smyth [Smy83, Smy92] genera... |

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Citation Context ... are mappings from frames to complete lattices preserving arbitrary joins and finite meets. An example of an observation frame is the embedding of open sets into the power set of points. Also Vickers =-=[Vic89]-=- and Abramsky [Abr87] view frames as collections of observable predicates. Concerning liveness predicates, Abramsky [Abr87] suggests that one has to look for structures more complicated than the simpl... |

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Citation Context ... of ! 1 chains [Plo82]. In [BK93] two of the authors introduced the notion of completely multiplicative predicate transformers. This notion was used in [BK94] for an extension of the results of Smyth =-=[Smy83]-=- from sober to T 0 spaces (considering also lower semi-continuous and Vietoris continuous multifunctions in addition to the upper semi-continuous ones). For this result frames were used whose elements... |

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Citation Context ...eterminism then it seems that one has to go beyond sober spaces: one has to consider a category of ! 0 and ! 1 chain complete partial orders with functions preserving least upper bounds of ! 1 chains =-=[Plo82]-=-. In [BK93] two of the authors introduced the notion of completely multiplicative predicate transformers. This notion was used in [BK94] for an extension of the results of Smyth [Smy83] from sober to ... |

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Citation Context ...e exists a compact set A ` X and an open set o 0 2 O(X) such that x 2 o 0 ` A ` o. Core compact spaces are important because they are exponentiable in Sp [Isb75] (the converse also holds, as shown in =-=[DK70]-=-). Isbell [Isb75] gives an example of a core compact space which is not sober. Denote by cKSp 0 the full sub-category of Sp 0 whose objects are core compact T 0 spaces and let OCFrmM denote the full s... |

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is not always sober
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Citation Context ...cate transformers in the sense of Dijkstra [Dij76, DS90]. There are interesting spaces which are not sober. For example posets taken with the Alexandrov topology are not always sober. Johnstone shows =-=[Joh81]-=- that not every dcpo with the Scott topology is sober. There are also spaces which have an exponent in the category of topological spaces that are not sober [Isb72a]. Also if one wishes to study fairn... |

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Citation Context ...ven, respectively, by ? = W ; and ? = V ;. A complete lattice L is called completely distributive if A2A A = f A2A f(A) j f : A ! [ A & f(A) 2 Ag; for every set of sets A 2 P(P(L)). Equivalently (see =-=[Ran52]-=-), L is completely distributive if A2A A = f A2A f(A) j f : A ! [ A & f(A) 2 Ag for every A 2 P(P(L)). A frame L is a complete lattice which satisfies the following infinite distributivity law, a rest... |

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Citation Context ...ructions give an isomorphism. In a later paper [Sto37], he generalized this correspondence from Stone spaces to spectral spaces and from boolean algebra's to distributive lattices. Hofmann and Keimel =-=[HK72]-=- described this duality in a categorical framework. Even further, Isbell [Isb72a] gives an adjunction between the category of topological spaces with continuous functions and the opposite category of ... |

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Citation Context ...see below) denoted by Idl(B). Let X be a continuous dcpo. The set of finite subsets of X can be made into an abstract basis by putting for a; b 2 P f (X), a OE b , b ` " " a , 8y 2 b: 9x 2 a=-=: xsy see [Hrb87]-=- where this approach stems from (but where it is used for convex power domains). With the same definition of the approximation relation, also the set of all subsets of X can be made in an abstract bas... |

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Citation Context ...en the categories F (A) and G(B) (here F (A) denotes the full subcategory of B whose objects are in the image of F and G(B) denotes the full sub-category of A whose objects are in the image of G). In =-=[Isb72b] it i-=-s shown that an adjunction ! F; G; j; " ?: A ! B is Galois if and only if it restricts to a reflection of A into F (A). By Lemma 3.24 and Remark 3.21 we have that the adjunction of Theorem 3.23 r... |

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Domain theory
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(Show Context)
Citation Context ...x2A " " x. The latter expression is also written as " " A. The way below relationssatisfies a certain interpolation property, which is axiomatized as (INT) below. An abstract basis=-= (according to Jung [JA94]-=-) consists of a set B with a transitive relation OE, such that for finite subsets S ` B and x 2 B, S OE x ) 9y 2 B: S OE y OE x. (INT) A directed ideal in such an abstract basis (B; OE) is a subset I ... |

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(Show Context)
Citation Context ...hen it seems that one has to go beyond sober spaces: one has to consider a category of ! 0 and ! 1 chain complete partial orders with functions preserving least upper bounds of ! 1 chains [Plo82]. In =-=[BK93]-=- two of the authors introduced the notion of completely multiplicative predicate transformers. This notion was used in [BK94] for an extension of the results of Smyth [Smy83] from sober to T 0 spaces ... |

4 | Relating Multifunctions and Predicate Transformers through Closure Operators
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(Show Context)
Citation Context ...ers with functions preserving least upper bounds of ! 1 chains [Plo82]. In [BK93] two of the authors introduced the notion of completely multiplicative predicate transformers. This notion was used in =-=[BK94]-=- for an extension of the results of Smyth [Smy83] from sober to T 0 spaces (considering also lower semi-continuous and Vietoris continuous multifunctions in addition to the upper semi-continuous ones)... |

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4 |
Function spaces and adjoints
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Citation Context ...2 X and open set o 2 O(X) such that x 2 o there exists a compact set A ` X and an open set o 0 2 O(X) such that x 2 o 0 ` A ` o. Core compact spaces are important because they are exponentiable in Sp =-=[Isb75]-=- (the converse also holds, as shown in [DK70]). Isbell [Isb75] gives an example of a core compact space which is not sober. Denote by cKSp 0 the full sub-category of Sp 0 whose objects are core compac... |

4 |
On topological characterization of behavioural properties
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(Show Context)
Citation Context ...ated element) if and only if it satisfies all the observable predicates which constitute such specification. Sometimes specifications of this kind are referred to as liveness predicates (see [Smy83], =-=[Kwi91]-=- and [BK93]). 7 Notice that for an observation frame F.!L, we have that q 2 Q(F.!L) if and only if q = ufa j a 2 F & q v a g. From right to left is clear. For the converse, let q 2 Q(F.!L), say q = uA... |

2 |
Domain theory. To appear
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(Show Context)
Citation Context ... " " x. The latter expression is also written as " " A. The way below relation �� satisfies a certain interpolation property, which is axiomatized as (INT) below. An abstract b=-=asis (according to Jung [JA94]-=-) consists of a set B with a transitive relation OE, such that for finite subsets S ` B and x 2 B, S OE x ) 9y 2 B: S OE y OE x. (INT) A directed ideal in such an abstract basis (B; OE) is a subset I ... |

1 |
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Citation Context ...es has both arbitrary unions and intersections of open sets. We use an abstract framework where open sets are just elements of a frame. A different approach has been followed by Kwiatkowska et al. in =-=[AKM94]-=-. In this approach liveness predicates can be interpreted as greatest fixed points of monotone operators on compact open sets. This interpretation, however, does not coincide with the classical interp... |

1 |
The Art of Pointless Thinking: The Category of Locales
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(Show Context)
Citation Context ...are necessary to describe limits, colimits, monos and epis in this category. A related question is whether the category observation frames is any good for doing some form of pointless topology, as in =-=[Joh91]-=-. Finally, we mention two more points which need to be explored: a representation of general (non sober) directed complete partial orders with Scott topology and the question whether the category of o... |