Localizations as idempotent approximations to completions

Casacuberta, Carles; Frei, Armin

doi:10.1016/s0022-4049(98)00074-7

Cited by 5 publications

(5 citation statements)

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“…They also observe that the first few terms of this tower are familiar constructions in terms of the R-completion of X in the sense of Bousfield-Kan (see Example 3.20). These ideas were further studied in [CF99].…”

Section: Localization and Completionmentioning

confidence: 99%

The Monadic Tower for $\infty$-Categories

Yanovski¹

2021

Preprint

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Every right adjoint functor between presentable ∞-categories is shown to decompose canonically as a coreflection, followed by, possibly transfinitely many, monadic functors. Furthermore, the coreflection part is given a presentation in terms of a functorial iterated colimit. Background material, examples, and the relation to homology localization and completion are discussed as well.Weltchronik in Versen, Szene: Der Turmbau zu Babel (c. 1370s). Depiction of the construction of the tower of Babel.

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Section: Localization and Completionmentioning

confidence: 99%

The Monadic Tower for $\infty$-Categories

Yanovski¹

2021

Preprint

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“…An idempotent approximation to a monad M on C is an idempotent monad M on C together with a map of monads M → M which is terminal among all maps from idempotent monads to M . Casacuberta and Frei [CF99] give a convenient characterization of idempotent approximations, without any conditions on the underlying category C . Moreover, their perspective allows to easily deduce some basic properties of idempotent approximations.…”

Section: Idempotent Approximationmentioning

confidence: 99%

Chromatic completion

Barthel

2015

Proc. Amer. Math. Soc.

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“…In the same year they were studied in [10,Section 2] where it is shown that their Kleisli categories are isomorphic to the category of fractions (of invertible morphisms). Extending these ideas, idempotent approximations to any monad are the topic of [5].…”

Section: Remarksmentioning

confidence: 99%

Idempotent monads and ⋆-functors

Clark¹,

Wisbauer²

2011

Journal of Pure and Applied Algebra

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a b s t r a c tFor an associative ring R, let P be an R-module with S = End R (P). C. Menini and A. Orsatti posed the question of when the related functor Hom R (P, −) (with left adjoint P ⊗ S −) induces an equivalence between a subcategory of R M closed under factor modules and a subcategory of S M closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphism and the counit is a monomorphism. A module P inducing these properties is called a -module.The purpose of this paper is to consider the corresponding question for a functor G : B → A between arbitrary categories. We call G a -functor if it has a left adjoint F : A → B such that the unit of the adjunction is an extremal epimorphism and the counit is an extremal monomorphism. In this case (F , G) is an idempotent pair of functors and induces an equivalence between the category A GF of modules for the monad GF and the category B FG of comodules for the comonad FG. Moreover, B FG = Fix(FG) is closed under factor objects in B, A GF = Fix(GF ) is closed under subobjects in A.

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Localizations as idempotent approximations to completions

Cited by 5 publications

References 12 publications

The Monadic Tower for $\infty$-Categories

The Monadic Tower for $\infty$-Categories

Chromatic completion

Idempotent monads and ⋆-functors

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