Reflective subcategories, localizations and factorization systems: Corrigenda

Cassidy, Charles; Hébert, Michel; Kelly, G. M.

doi:10.1017/s1446788700033693

Cited by 56 publications

(151 citation statements)

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“…which is also a special case of Corollary 3.4 [8] and Proposition 10 [11], although the proof given here holds for a more general setting. Here, we have applied the technique of Bousfield [3] in proving this theorem.…”

Section: Appendixmentioning

confidence: 76%

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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Section: Appendixmentioning

confidence: 76%

The minimal extension of P-localization on groups

Berrick

Tan

1996

Math. Proc. Camb. Phil. Soc.

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“…b) By Lemma 2.2, Pro(DQ,C) is also oomplete and well-powered. This implies in particular that it is finitely well-complete, and we can apply ¡Corollary 3.4 of [3].…”

Section: Mhébertmentioning

confidence: 92%

“…In [3], a bisection has been constructed, in sufficiently nice categories, between reflective subcategories and factorization systems for morphisms verifying a certain property of maximality. Prom [4], we also obtain a correspondence between epirefleotive subcategories and the factorization systems * (B,M) for morphisms verifying a property of minimality and such that B is a class of epimorphisms (this result being equivalent to one obtained in [6] involving the so-called dispersed factorization systems for sources).…”

Section: Introductionmentioning

confidence: 99%

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Multireflections and Multifactorization Systems

Hébert¹

1987

Demonstratio Mathematica

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“…Definition 3.1. A factorisation system (E, M) is a reflective factorisation system [8] if C has a terminal object 1 and the maps in E satisfy 2 from 3.…”

Section: Factorisation Systems On 2-catmentioning

confidence: 99%

The Gray Tensor Product Via Factorisation

Bourke

Gurski

2016

Appl Categor Struct

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Abstract. We discuss the folklore construction of the Gray tensor product of 2-categories as obtained by factoring the map from the funny tensor product to the cartesian product. We show that this factorisation can be obtained without using a concrete presentation of the Gray tensor product, but merely its defining universal property, and use it to give another proof that the Gray tensor product forms part of a symmetric monoidal structure. The main technical tool is a method of producing new algebra structures over Lawvere 2-theories from old ones via a factorisation system.

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Reflective subcategories, localizations and factorization systems: Corrigenda

Abstract: Reflective Subcategories, Localizations and Factorization Systems: Corrigenda C. Cassidy, M. Hébert and G. M. Kelly, 1980 Mathematics subject classification (Amer. Math. Soc.): 18 A 20.

Cited by 56 publications

References 1 publication

The minimal extension of P-localization on groups

The minimal extension of P-localization on groups

Multireflections and Multifactorization Systems

The Gray Tensor Product Via Factorisation

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