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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJun 23rd 2014

    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 (,1)(\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 SEModS \rightleftarrows E Mod for some ring spectrum EE (with SS either spaces or spectra), and we want to factor it through the inclusion of a reflective subcategory of SS consisting of the objects that are orthogonal to the morphisms inverted by the left adjoint SEModS\to E Mod (the EE-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 (,1)(\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 EModE 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?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2014

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

    (I am not at the institute these days…)

    • CommentRowNumber3.
    • CommentAuthorThomas Holder
    • CommentTimeJun 24th 2014

    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.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 25th 2014

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

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2014
    • (edited Jul 11th 2014)

    I have added to idempotent monad:

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2014

    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

    • CommentRowNumber7.
    • CommentAuthorThomas Holder
    • CommentTimeJul 11th 2014

    Added a link to the Fakir paper.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2014

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

    • CommentRowNumber9.
    • CommentAuthorThomas Holder
    • CommentTimeAug 8th 2014
    • (edited Aug 8th 2014)

    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 CC.