Localizations as idempotent approximations to completions
type: Item
instance of: scholarly article
title: Localizations as idempotent approximations to completions
zbMATH Open document ID: 0931.18005
DOI: 10.1016/S0022-4049(98)00074-7
author: Carles Casacuberta
author: Armin Frei
published in: Journal of Pure and Applied Algebra
publication date: 2000-02-29T00:00:00Z
review text: The authors observe that, given a monad \(T\) on a category \({\mathcal C}\), an idempotent monad \(\widehat T\) on \({\mathcal C}\) rendering invertible the same class of morphisms as \(T\) is an idempotent approximation to \(T\) in the sense that there exists a natural transformation \(\lambda:\widehat T\to T\) which is universally attractive. They take the first property as the definition of an idempotent approximation and derive some general properties which are to lead to a better understanding of the relationship between localization and completion. The paradigmatic example is given by the category of pointed \(CW\)-complexes and homotopy classes of maps. There, for a class of commutative rings \(R\), the Bousfield-Kan \(R\)-completion functor \(R_\infty\) is part of a monad \(T\), which however is not idempotent. On the other hand, the class of maps \(f\) such that \(R_\infty f\) is a homotopy equivalence is the class of homology equivalences with coefficients in \(R\). Hence the \(R\)-homology localization functor \(E_R\), which is part of an idempotent monad \(\widehat T\), is the idempotent approximation of the \(R\)-completion.
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18A35
zbMATH DE Number: 1355520
zbMATH Keywords: pointed \(CW\)-complexes
zbMATH Keywords: Bousfield-Kan completion functor
zbMATH Keywords: idempotent monad
zbMATH Keywords: idempotent approximation
zbMATH Keywords: localization
zbMATH Keywords: completion
zbMATH Keywords: homotopes classes of maps
reviewed by: Heinrich Kleisli
MaRDI profile type: Publication
Wikidata QID: Q127956981
full work available at URL: s0022-4049(98)00074-7
OpenAlex ID: W2038487472
description: scientific article; zbMATH DE number 1355520