Localizations as idempotent approximations to completions Original Research Article

Given a monad (also called a triple) T on an arbitrary category, an idempotent approximation to T is defined as an idempotent monad image rendering invertible precisely the same class of morphisms which are rendered invertible by T. One basic example is homological localization with coefficients in a ring R, which is an idempotent approximation to R-completion in the homotopy category of CW-complexes. We give general properties of idempotent approximations to monads using the machinery of orthogonal pairs, aiming to a better understanding of the relationship between localizations and completions.