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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJun 23rd 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI just noticed that the notion of [[Bousfield localization of spectra]], and also the similar sort of localization of spaces with respect to homology, is an $(\infty,1)$-categorical version of the "adjoint-functor factorization" of Applegate-Tierney-Day (e.g. Applegate-Tierney "Iterated Cotriples" or Day "On adjoint-functor factorization"). We have an adjunction $S \rightleftarrows E Mod$ for some ring spectrum $E$ (with $S$ either spaces or spectra), and we want to factor it through the inclusion of a reflective subcategory of $S$ consisting of the objects that are orthogonal to the morphisms inverted by the left adjoint $S\to E Mod$ (the $E$-homology equivalences). Is this analogy known? Can the specific constructions be related as well? E.g. Applegate-Tierney construct the factorization by a transfinite tower of adjunctions, while Day does it by factoring the adjunction units and then applying an adjoint functor theorem; do either of those strategies generalize to the $(\infty,1)$-context? Conversely, the Bousfield-Kan construction of localization (as the totalization of a cosimplicial object) seems to be the composite of the comparison functor for the induced comonad on $E Mod$ and the functor that would be its inverse if the adjunction were comonadic; are there general conditions under which this yields the desired sort of reflection?

   I just noticed that the notion of Bousfield localization of spectra, and also the similar sort of localization of spaces with respect to homology, is an $(\infty,1)$-categorical version of the “adjoint-functor factorization” of Applegate-Tierney-Day (e.g. Applegate-Tierney “Iterated Cotriples” or Day “On adjoint-functor factorization”). We have an adjunction $S \rightleftarrows E Mod$ for some ring spectrum $E$ (with $S$ either spaces or spectra), and we want to factor it through the inclusion of a reflective subcategory of $S$ consisting of the objects that are orthogonal to the morphisms inverted by the left adjoint $S\to E Mod$ (the $E$-homology equivalences).

   Is this analogy known? Can the specific constructions be related as well? E.g. Applegate-Tierney construct the factorization by a transfinite tower of adjunctions, while Day does it by factoring the adjunction units and then applying an adjoint functor theorem; do either of those strategies generalize to the $(\infty,1)$-context? Conversely, the Bousfield-Kan construction of localization (as the totalization of a cosimplicial object) seems to be the composite of the comparison functor for the induced comonad on $E Mod$ and the functor that would be its inverse if the adjunction were comonadic; are there general conditions under which this yields the desired sort of reflection?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 24th 2014
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   Author: Urs
   Format: MarkdownItexIs there available an electronic copy with details on the Applegate-Tierney-Day story? (I am not at the institute these days...)

   Is there available an electronic copy with details on the Applegate-Tierney-Day story?

   (I am not at the institute these days…)

3. • CommentRowNumber3.
   • CommentAuthorThomas Holder
   • CommentTimeJun 24th 2014
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   Author: Thomas Holder
   Format: MarkdownItexThe papers of Applegate-Tierney and Day have appeared in LNM volumes. In a recent paper [1406.2361](http://arxiv.org/abs/1406.2361) on the arXiv Lucyshyn-Wright derives some of their results. In the 90s Casacuberta and Frei did some work on localizations and idempotent approximations. It might be worthwhile to have a look at 'extending localizing functors' and 'localizations as idempotent approximations to completions' on Casacuberta's [homepage](http://atlas.mat.ub.es/personals/casac/publicacions.html).

   The papers of Applegate-Tierney and Day have appeared in LNM volumes. In a recent paper 1406.2361 on the arXiv Lucyshyn-Wright derives some of their results.

   In the 90s Casacuberta and Frei did some work on localizations and idempotent approximations. It might be worthwhile to have a look at ’extending localizing functors’ and ’localizations as idempotent approximations to completions’ on Casacuberta’s homepage.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJun 25th 2014
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   Author: Mike Shulman
   Format: MarkdownItex@Thomas, thanks! Lots of good stuff to read there.

   @Thomas, thanks! Lots of good stuff to read there.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2014
   • (edited Jul 11th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[idempotent monad]]_: * the above [references](http://ncatlab.org/nlab/show/idempotent+monad#References); * a lead-in paragraph at the beginning of _[The associated idempotent monad of a monad](http://ncatlab.org/nlab/show/idempotent+monad#AssociatedIdemopotentMonad)_; * a minimum of general text in the [Idea-section](http://ncatlab.org/nlab/show/idempotent+monad#Idea).

   I have added to idempotent monad:

   • the above references;

   • a lead-in paragraph at the beginning of The associated idempotent monad of a monad;

   • a minimum of general text in the Idea-section.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added statement of the theorem that the idempotent completion/core of a monad is that induced from a reflection which factors any adjunction that gives the original monad through a conservative left adjoint. [here](http://ncatlab.org/nlab/show/idempotent+monad#FactorizationOfIdempotentCore)

   I have added statement of the theorem that the idempotent completion/core of a monad is that induced from a reflection which factors any adjunction that gives the original monad through a conservative left adjoint. here

7. • CommentRowNumber7.
   • CommentAuthorThomas Holder
   • CommentTimeJul 11th 2014
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexAdded a link to the Fakir paper.

   Added a link to the Fakir paper.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the [remark](http://ncatlab.org/nlab/show/idempotent+monad#IdempotentFactorizationAndBousfieldLocalization) that on model categories the idempotent factorization is Bousfield localization.

   I have added the remark that on model categories the idempotent factorization is Bousfield localization.

9. • CommentRowNumber9.
   • CommentAuthorThomas Holder
   • CommentTimeAug 8th 2014
   • (edited Aug 8th 2014)
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexAs I lately try to figure out how the nucleus of an adjunction is related to what Lawvere (TAC 2008) calls the core (variety) and this in turn to Lucyshyn-Wright's above idempotent core (following a terminological suggestions by Lawvere) I came across this [squib](http://arxiv.org/pdf/0909.5010v1) by Brian Day. Unless I am not completely mistaken this construction by Day actually answers the following MO [question](http://mathoverflow.net/questions/59291/completion-of-a-category) because as I understand him using the above adjoint functor factorization for what Todd calls L yields the desired bicompletion for small $C$.

   As I lately try to figure out how the nucleus of an adjunction is related to what Lawvere (TAC 2008) calls the core (variety) and this in turn to Lucyshyn-Wright’s above idempotent core (following a terminological suggestions by Lawvere) I came across this squib by Brian Day. Unless I am not completely mistaken this construction by Day actually answers the following MO question because as I understand him using the above adjoint functor factorization for what Todd calls L yields the desired bicompletion for small $C$.