Relative group completions
type: Item
instance of: scholarly article
title: Relative group completions
zbMATH Open document ID: 1068.55011
author: Carles Casacuberta
author: An Descheemaeker
published in: Journal of Algebra
publication date: 2005-04-18T00:00:00Z
review text: Let \(\mathcal{C}\) be a category. A functor \(L: \mathcal{C} \rightarrow \mathcal{C}\) is called a pointed endofunctor if there exists a natural transformation \(l: Id_{\mathcal{C}} \rightsquigarrow L\). The first result of this paper shows that a pointed endofunctor on the category of groups, \(\mathcal{G}\), which is compatible with conjugation, induces a pointed endofunctor on the category of pairs of groups. \noindent The second result shows that if a pointed endofunctor on \(\mathcal{G}\) is a completion, then it is compatible with conjugation, so it admits a relative version. The authors remark that, since localizations are special cases of completions, then they admit a relative version. The authors explicit the universal property of relative group localizations and give some applications of their results, namely the study of the effect of plus-constructions on homotopy fibre sequences.
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 20J15
Mathematics Subject Classification ID: 19D06
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 2157742
zbMATH Keywords: relative localizations
zbMATH Keywords: relative completions
zbMATH Keywords: category of groups
reviewed by: Saïd Zarati
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.10.023
OpenAlex ID: W1978121582
cites work: THE PLUS-CONSTRUCTION AND FIBRATIONS
cites work: Q3314044
cites work: A universal space for plus-constructions
cites work: Fibration and nullifications
cites work: A push-out construction for groups with operators
cites work: Localization and Periodicity in Unstable Homotopy Theory
cites work: Homotopy limits, completions and localizations
cites work: Q4762696
cites work: Localization methods in the study of the homology of virtually nilpotent groups
cites work: On Towers Approximating Homological Localizations
cites work: Q4240517
cites work: \(P\)-localization of relative groups
cites work: Cellular spaces, null spaces and homotopy localization
cites work: Q3950728
cites work: Q3316214
cites work: Localization of fibrations with nilpotent fibre
cites work: Q4093629
cites work: Q4215784
cites work: Fibrewise Localization and Completion
cites work: HOMOTOPY GROUPS OF ACYCLIC SPACES
cites work: Varieties of groups
cites work: Q5631287
cites work: Universal epimorphic equivalences for group localizations
cites work: Q4156898
cites work: Genetics of homotopy theory and the Adams conjecture
DOI: 10.1016/J.JALGEBRA.2004.10.023
description: scientific article; zbMATH DE number 2157742