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gave Bousfield localization of spectra a more informative Idea-section
added a few basics to the Definition- and the Examples-section of Bousfield localization of spectra from lecture 20 of Chromatic Homotopy Theory
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 $E \wedge (-) : \text{Spectra} \rightarrow \text{Spectra}$ 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…
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 $E$. Since any such $E$ represents a generalized homology theory, this may also be thought of $E$-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.
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.
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.
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 $X \to L_E X$ is $E$-acyclic. (er, why?) But I need to run now.
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.
I have added more details to the Definition-section. For instance I have spelled out why the two equivalent conditions on an $E$-local spectrum are indeed equivalent (here)
In the proof of the existence of localizations of spectra here I am using the notation $A = \underset{\longrightarrow}{\lim}_\sigma c_\sigma$ for exhibiting the cell decomposition of a cell spectrum $A$. That’s maybe a bit too cavalier. I suppose I should instead say that the $\alpha$-stage of the cell decomposition ($\alpha$ 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.
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