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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?
Is there available an electronic copy with details on the Applegate-Tierney-Day story?
(I am not at the institute these days…)
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.
@Thomas, thanks! Lots of good stuff to read there.
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.
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
Added a link to the Fakir paper.
I have added the remark that on model categories the idempotent factorization is Bousfield localization.
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 $C$.
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