Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding

Bonsangue, Marcello M.; Breugel, Franck van; Rütten, Jan

doi:10.1016/s0304-3975(97)00042-x

Cited by 103 publications

(144 citation statements)

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“…In some occasions we will need that the (Set-based) natural transformation m : T T → T has (BC) (that is, each naturality square is a weak pullback); this guarantees that m is also a (strict) natural transformation for the extension of T to V-Mat described above. 4 The conditions for our extension are stronger than Seal's in [24]. 5 Recall that a lattice Y is (ccd) if W : 2 Y op → Y has a left adjoint; for more details see [27].…”

Section: ì and Its Extensionmentioning

confidence: 99%

Lawvere Completeness in Topology

Clementino

Hofmann

2008

Appl Categor Struct

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It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical (Ì, V)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (Ì, V)-categories. (2000): 18A05, 18D15, 18D20, 18B35, 18C15, 54E15, 54E50. Mathematics Subject ClassificationKey words: V-category, bimodule, monad, (Ì, V)-category, completeness. IntroductionLawvere in his 1973 paper Metric spaces, generalized logic, and closed categories formulates a notion of complete V-category and shows that for (generalised) metric spaces it means Cauchy completeness. This notion of completeness deserved the attention of the categorical community, and the notion of Cauchy-complete category, or Freyd-Karoubi complete category is well-known, mostly in the context of Ab-enriched categories. However, it never got the attention of the topological community. In this paper we interpret Lawvere's completeness in topological settings. We extend Lawvere's notion of complete V-category to the (topological) setting of (Ì, V)-categories (for a symmetric and unital quantale V), and show that it encompasses well-known notions in topological categories, meaning weakly sober space in the category of topological spaces and continuous maps, weakly sober approach space in the category of approach spaces and non-expansive maps, and Cauchy-completeness in the category of quasi-uniform spaces and uniformly continuous maps.We present also a first step towards a possible construction of completion. Indeed, in the setting of V-categories, it is well-known that the completion of a V-category may be built out of the Yoneda embedding X → V X op . In the (Ì, V)-setting, we could prove that V has a canonical (Ì, V)-categorical structure and that every (Ì, V)-category X has a canonical dual * The authors acknowledge partial financial assistance by Centro de Matemática da Universidade de Coimbra/FCT and Unidade de Investigação e Desenvolvimento Matemática e Aplicações da Universidade de Aveiro/FCT. 1 X op . Using this structure and the free Eilenberg-Moore algebra structure |X| on T X, we get two "Yoneda-like" morphismsFor the latter one we prove a Yoneda Lemma (see 4.2). Furthermore, we show that, under suitable conditions, V is a Lawvere-complete (Ì, V)-category, a first step towards a completion construction which will be the subject of a forthcoming paper.In order to make the presentation of this paper smoother, in Section 1 we recall the notions and properties of V-categories we will generalize throughout. First we introduce V-categories and V-bimodules, and de...

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Section: ì and Its Extensionmentioning

confidence: 99%

Lawvere Completeness in Topology

Clementino

Hofmann

2008

Appl Categor Struct

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“…Well known are completions of metric spaces, uniform spaces, normed linear spaces, lattices, the algebraic completion of fields, the ideal completion of posets, and so forth. In [6] Bonsangue, van Breugel, and Rutten investigated the completion of generalized metric spaces. This yields a generalization both of the chain completion of (pre)ordered sets and of the metric Cauchy completion.…”

Section: Completion Of Posets With Projectionsmentioning

confidence: 99%

“…This shows (5). (6) results from Proposition 5.3 (the proof that ψ be Scott-continuous is deferred to Proposition 5.8 below).…”

Section: C(d)mentioning

confidence: 99%

Partially ordered sets with projections and their topology

Kummetz

2004

Dissertationes Math.

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The family F gives rise to a uniformity on D, which we call the F-uniformity. The uniform topology is the F-topology. We characterize all uniformities on a poset arising as F-uniformities and investigate basic properties of F-uniformity and F-topology. As we have already emphasized, the mappings in F will be used for approximation purposes. We therefore say that an F-poset (D, ≤, F) is approximating if sup f ∈F f (d) = d for all d ∈ D. It turns out that this order-theoretic condition is equivalent to saying that the poset is a partially ordered space in the F-topology. We shall mainly focus on approximating F-posets. Then a close relationship between the partial order and the F-topology can be established. We are interested in the question when suprema and infima of subsets A ⊆ D exist and, moreover, when they are accumulation points of A with respect to the F-topology. Here continuous domains come into play. We show that each monotone net in D converges in the F-topology if and only if (D, ≤) is a continuous domain such that f (d) is "essentially" below d for all f ∈ F and all d ∈ D. In addition, a similar result holds concerning the convergence of monotone nets which are bounded. We give a domain-theoretic characterization of approximating F-posets that are compact in their F-topology. In this case, the F-topology coincides with the Lawson topology of the poset. Moreover, provided that they have a least element, these F-posets arise exactly from FS-domains. Part of these results can also be found in [33].Chapter 3 introduces a special sort of F-posets involving projections. Therefore, we first collect some basic facts on projections. After that, we define a partially ordered set with projections (or pop for short) to be an F-poset (D, ≤, P) where P consists of projections only. We investigate the pop uniformity and the pop topology, i.e. the Funiformity and the F-topology of a pop, and give a list of examples. For instance, Flagg and Kopperman's [19] closed ball model carries such a pop structure. Moreover, we show how several domains of traces can be made into pop's. This enables us to apply our results in order to obtain a uniform proof for topological properties shared by all these trace models.Coming back to the general theory, we show that approximating pop's are complete in their pop uniformity if and only if they can be represented as an inverse limit built up by the images of their projections. We resume the problem when the supremum or the infimum of a subset A exists in the closure of A. This leads to algebraic domains. Using the results of Chapter 2, we prove that each monotone net of an approximating pop (D, ≤, P) converges in the pop topology if and only if (D, ≤) is an algebraic domain with its compact elements being exactly the images of all projections of P. Furthermore, we raise the question under which (order-theoretic) conditions an algebraic domain (D, ≤) admits such a set P. We then call (D, ≤) a P-domain. Here posets satisfying the ascending chain condition (ACC) come into play. As we s...

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“…For instance, see quasi metric spaces [1], generalized quasi metric spaces [2], pseudometric spaces ( [3], Chapter 2), approach spaces [4], -spaces [5], inframetric spaces [6], and -metric spaces [7]. Sometimes, as in [8,9], even the very notion of generalized metric spaces (or even gms [10]) is used, but it has a different meaning. In 2000, Branciari [11] introduced the notion of generalized metric space where the triangle inequality of a metric space is replaced by a rectangular inequality involving four terms instead of three.…”

Section: Introductionmentioning

confidence: 99%