Epimorphisms and cowellpoweredness for separated metrically generated theories

Claes, Veerle; Colebunders, Eva; Gerlo, An

doi:10.1007/s10474-006-0518-6

Cited by 13 publications

(14 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So we can conclude that for X = Lip the uniformly complete T 0 objects form a reflective subcategory with associated firm class consisting of those morphisms that are dense embeddings, where density is expressed in the underlying topology of the codomain. Applying Proposition 4.3.6 in [3] we can conclude that this class coincides with the class of all epimorphic embeddings. It follows that the uniformly complete objects are exactly those Lipschitz spaces which have a complete underlying uniformity.…”

Section: Proposition the Categorymentioning

confidence: 66%

“…Base categories consisting of metric spaces are contained in C ∆,s , the construct of all metric spaces. We require that the base category C satisfies the following supplementary condition which was introduced in [3] and is fulfilled by all concrete examples of base categories just mentioned. C is closed for epimorphic embeddings in C ∆ i.e.…”

Section: Preliminariesmentioning

confidence: 99%

“…We assume that all the expanders appearing in this paper satisfy two more technical conditions taken from [3], namely the condition ξ(D) = D implies that the meter is an ideal and ξ {0} = {0}, where 0 is the zero metric. It should be noted that these conditions are very mild and satisfied by all the examples that are listed below.…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Bicompletion of metrically generated constructs

Colebunders

Vandersmissen

2010

Acta Math Hung

Self Cite

View full text Add to dashboard Cite

We develop a general technique to study completeness in a metrically generated construct X, the objects of which can be isomorphically described as sets endowed with a gauge of quasi metrics, saturated in a certain way. Our approach depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms L(R) is known. This functor will permit to "lift" the notion of completeness from A to X. We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory R F for the construct X0 of T0 objects and that the associated firm class of morphisms can be derived from L(R). Application of this technique to concrete examples results in many interesting completion theories and provides us with a far better understanding of several existing ones.

show abstract

Section: Proposition the Categorymentioning

confidence: 66%

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Bicompletion of metrically generated constructs

Colebunders

Vandersmissen

2010

Acta Math Hung

Self Cite

View full text Add to dashboard Cite

show abstract

“…In a similar way we now dene the symmetrization S A (X) of an approach space X with gauge D and distance δ to be the approach space X, ξ A (D * ) . It was proved in [7] that the distance of S A (X) equals δ ∨ δ ϕ −1 . Here, ϕ denotes the quasi-metrical coreection of X, which is given by ϕ = sup d∈D d. 5.6.…”

Section: Qug-spacementioning

confidence: 99%

“…In [7], it was proved that for a T 0 approach space X, r is given by the topological coreection of S A (X). It is well-known that epimorphisms in Ap 0 are exactly the r-dense maps.…”

Section: Sobrication Of An Approach Space Via Bicompletionmentioning

confidence: 99%

Constructing the sobrification of an approach space via bicompletion

Gerlo

Vandersmissen

2007

Acta Math Hung

Self Cite

View full text Add to dashboard Cite

On every approach space X, we construct a compatible quasiuniform gauge structure which turns out to be at the same time the coarsest functorial structure and the nest compatible totally bounded one. Based on the analogy with the classical CsászárPervin quasi-uniform space, we call this the CsászárPervin quasi-uniform gauge space. By means of the bicompletion of this CsászárPervin quasi-uniform gauge space of a T0 approach space X, we succeed in constructing the sobrication of X.

show abstract

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

Gerlo¹,

Vandersmissen²,

Olmen³

2006

Appl Categor Struct

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Epimorphisms and cowellpoweredness for separated metrically generated theories

Cited by 13 publications

References 13 publications

Bicompletion of metrically generated constructs

Bicompletion of metrically generated constructs

Constructing the sobrification of an approach space via bicompletion

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

Contact Info

Product

Resources

About