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nLab > Latest Changes: Bousfield localization of spectra

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2013

    gave Bousfield localization of spectra a more informative Idea-section

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2013

    added a few basics to the Definition- and the Examples-section of Bousfield localization of spectra from lecture 20 of Chromatic Homotopy Theory

    • CommentRowNumber3.
    • CommentAuthorDylan Wilson
    • CommentTimeNov 19th 2013

    The proof of existence of Bousfield localizations in the literature has always left me a bit unsatisfied. It really seems to be a special case of some facts about presentable categories. In particular we are just applying Proposition T.5.5.4.16 to the functor E∧(−):Spectra→SpectraE \wedge (-) : \text{Spectra} \rightarrow \text{Spectra} and the class of equivalences in the codomain to show that E-local equivalences are of small generation, and then localizing with respect to some generating set.

    This covers basically all uses of Bousfield localization that we care about. If we want to be pedantic, Bousfield’s techniques work in “cellular model categories” not all of which are combinatorial. But actually, I kind of suspect that the underlying infinity category of a cellular model category is still presentable? I would need to think about that though…

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2013
    • (edited Nov 19th 2013)

    Hi Dylan, nice to see you here.

    I tend to think of the term “Bousfield localization” as indicating a special case of localization of ∞\infty-categories, namely either specifically the localization of model categories by enlarging the weak equivalences while keeping one of fibrations or cofibrations fixed, or else the specific localization at smashing with a spectrum – and possibly both of these at the same time.

    Just since you brought it up, I have now tried to say this more clearly in the beginning of the (still stubby) nLab entry on Bousfield localization of spectra. It now starts out as such:

    Bousfield localization of spectra refers generally to localizations of the stable (∞,1)-category of spectra at the collection of morphisms which become equivalences under smash product with a given spectrum EE. Since any such EE represents a generalized homology theory, this may also be thought of EE-homology localization.

    More specifically, if the stable (∞,1)-category of spectra is presented by a (stable) model category then the ∞-categorical localization can be presented by the operation of Bousfield localization of model categories. The original article (Bousfield 79) essentially considered localization at the level of homotopy categories.

    If you feel at all motivated to improve that entry a little more by adding to it, please feel invited.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2014

    I have touched Bousfield localization of spectra briefly making edits of the kind we have been discussing here in the context of the entries localization of a space and fracture theorem.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2014

    I have added pointers to a bunch of further little statements from Bousfield79 to Bousfield localization of spectra and to other entries, such as fracture theorem.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2016
    • (edited Jun 10th 2016)

    I have started making a quick note (here) on Bousfield’s original proof of the existence of Bousfield localization of spectra, via the small object argument.

    Still need to say why the homotopy fiber of the resulting X→L EXX \to L_E X is EE-acyclic. (er, why?) But I need to run now.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2016

    I have completed the argument here of what I suppose was Bousfield’s original proof.

    Still a bit telegraphic for the moment. Need to call it quits for the weekend.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2016
    • (edited Jul 15th 2016)

    I have added more details to the Definition-section. For instance I have spelled out why the two equivalent conditions on an EE-local spectrum are indeed equivalent (here)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2016
    • (edited Jul 15th 2016)

    In the proof of the existence of localizations of spectra here I am using the notation A=lim⟶ σc σA = \underset{\longrightarrow}{\lim}_\sigma c_\sigma for exhibiting the cell decomposition of a cell spectrum AA. That’s maybe a bit too cavalier. I suppose I should instead say that the α\alpha-stage of the cell decomposition (α\alpha an ordinal) sits in a homotopy cofiber sequence with the next stage and a wedge of suspensions of spheres (the cells). The remainig argument of the proof would essentially not change, and the present notation is more efficient, but maybe its a bit of an abuse.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2016

    I have finally rewritten the proof here. Have relegate the proof of the relevant small object property to here.

  12. Added a pointer to van der Meer’s thesis.

    Multiplicative Group Scheme

    diff, v40, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2019

    cross-linked with the new page KU-local stable homotopy theory

  14. Changed “Moore” to “Eilenberg-MacLane” after the introduction of E := HZ/pZ

    Anonymous

    diff, v42, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2020

    added full publication data and DOI to:

    diff, v43, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2020

    added publication data to:

    diff, v44, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2021

    added pointer to:

    diff, v46, current

1 to 17 of 17

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