Zariski closure, completeness and compactness

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For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X op . Its full subcategory of M-extensive K-Chu spaces is topological over X when X is M-complete, for any morphism class M. Often this subcategory may be presented as a full coreflective subcategory of Diers' category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and their connections with linear logic (Seely), the Dialectica categories (Hyland, de Paiva), and with the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to describe the objects of the * -autonomous category of M-extensive and M-coextensive K-Chu spaces in terms of Zariski separation and to identify its important subcategory of complete objects.

Section: Definition 54 Recall That For An M-subobjectmentioning

confidence: 99%

Section: Definition 54 Recall That For An M-subobjectmentioning

confidence: 99%

mentioning

confidence: 99%

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A Topologist’s View of Chu Spaces

Giuli

Tholen

2007

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“…Note that in [6], E. Giuli obtains analogous results, using different techniques involving the Zariski closure. In particular, he proved that the subconstruct of absolutely Emb(Ap 0 )-closed objects of Ap 0 coincides with the epireflective hull of P in Ap 0 and that this construct is firmly U -reflective in Ap 0 .…”

Section: U -Firmness Of the Sober Approach Spaces In Apmentioning

confidence: 75%

Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

Gerlo¹,

Vandersmissen²,

Olmen³

2006

In [2] the subconstruct Sob of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category Ap 0 of T 0 approach spaces. The main result of this paper states that moreover Sob is firmly U-reflective in Ap 0 for the class U of epimorphic embeddings. 'Firm U-reflective' is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category Unif 0 of Hausdorff uniform spaces with uniformly continuous maps. It means that Sob is U-reflective in Ap 0 and that the reflector is such that f : X → Y belongs to U if and only if ( f ) is an isomorphism. Firm U-reflectiveness implies uniqueness of completion in the sense that whenever f : X → Y is a map with f ∈ U and Y sober, the associated f * : (X ) → Y is an isomorphism. Our result generalizes the fact that in the category Top 0 the subconstruct of sober topological spaces is firmly reflective for the class U b of b-dense embeddings in Top 0 . Also firmness in some other subconstructs of Ap 0 will be easily obtained.

“…Of course, one can now also take a look at the composition P * of the functor D * with the coreflector T A : Prap → Prtop to obtain the functor P * : Ap → Prtop : (X, λ) → (X, T λ ∨ T ϕ −1 ). So P * sends an approach space to its Zariski closure [4,16]. Note that P * (…”

Section: Examplesmentioning

confidence: 99%

Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

Gerlo

2007

In a topological construct X endowed with a proper (E, M)-factorization system and a concrete functor : X → Prap, we study F -compactness and FHausdorff separation, where F is a class of "closed morphisms" in the sense of Clementino et al. (A functional approach to general topology. In: Categorical by . In particular, we point out under which conditions on , the notion of F -compactness of an object X of X coincides with 0-compactness of the image (X) in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b -compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of bcompactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability.