Extending localization functors

Casacuberta, Carles; Frei, Armin; Tan, Geok Choo

doi:10.1016/0022-4049(94)00099-5

Cited by 13 publications

(8 citation statements)

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“…A general attack on this problem is to be found in [6]. The solution below is arguably more concrete, concentrating attention on groups of a given cardinality, where L P takes on a more recognizable form.…”

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confidence: 99%

The minimal extension of P-localization on groups

Berrick

Tan

1996

Math. Proc. Camb. Phil. Soc.

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Let P be a fixed set of primes, G the category of all groups and grouphomomorphisms, and N the full subcategory of nilpotent groups. In [9], an idempotent functor e: N → N , called P -localization, was defined so as to extend the Z-module-theoretic localization of abelian groups. There are two wellknown extensions of e to G, namely, Bousfield's P -localization [2], [4], denoted by EZ P , and Ribenboim's P -localization [13], usually denoted by ( ) P . Ribenboim's P -localization is the maximal extension among localizations extending e to G in that it maximizes the number of groups in its image [7]. The localized groups obtained after applying Ribenboim's P -localization are precisely the P -local groups, that is, the groups having unique nth-root for every n which is coprime to P , [13]. Being maximal is equivalent to this class of P -localized groups being the saturated class of groups generated by e-equivalences, that is, group homomorphisms between nilpotent groups which become isomorphisms after applying e.This suggests the problem of whether e has a minimal extension to G, a localization L P , say, yielding the minimum number of L P -local groups. A general attack on this problem is to be found in [6]. The solution below is arguably more concrete, concentrating attention on groups of a given cardinality, where L P takes on a more recognizable form. The reason for this strategy is to overcome technical difficulties that arise because the collection of all (P -local nilpotent) groups fails to form a set.In the final section, we discuss the effect of the localization L P on groups whose lower central series stabilizes. We use our localization and Bousfield's P -1

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confidence: 99%

The minimal extension of P-localization on groups

Berrick

Tan

1996

Math. Proc. Camb. Phil. Soc.

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“…The concept of reflections in categories has been investigated by several authors (see for example [3], [4], [5], [6], [9], [11], [15]). This concept serves the purpose of unifying various constructions in mathematics.…”

Section: • (Ii) =⇒ (Iii)mentioning

confidence: 99%

On some properties of $T_0$-ordered reflection

Lazaar

Mhemdi

2014

Appl.Gen.Topol.

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In [12], the authors give an explicit construction of the T0−ordered reflection of an ordered topological space (X, τ, ≤). All ordered topological spaces such that whose T0−ordered reflections are T1−ordered spaces are characterized. In this paper, some properties of the T0−ordered reflection of a given ordered topological space (X, τ, ≤) are studies. The class of morphisms in ORDTOP orthogonal to all T0−ordered topological space is characterized.

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“…We will finish this comparison of locally presentable categories with combinatorial model categories with one example of a statement which was not generalized yet: any orthogonality class generated by a set of objects in a complete and wellpowered category was shown to be reflective in [3]. An analogous statement in topology is known better as a problem of existance of homotopical localization with respect to a cohomology theory.…”

Section: Introductionmentioning

confidence: 96%