Localizations as idempotent approximations to completions
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.
scientific article; zbMATH DE number 1355520