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I have expanded the Idea section in localization of model categories as it previously seemed to be a stub. (It said: A localisation for model categories. Doh!) I have given a quote from Hirschhorn’s book.
This article describes left Bousfield localizations. Why is it separate from left Bousfield localization? I suggest deleting this article.
But there is a notion of localization of model categories beyond Bousfield localization. So I think what the article needs is better content, not deletion.
But there is a notion of localization of model categories beyond Bousfield localization.
What do you have in mind, specifically? Do you have a reference? Hirschhorn talks about left Bousfield localizations exclusively.
Hirschhorn’s book starts with a general notion of localization of model categories in section 3.1 and then turns attention to Bousfield localizations from 3.3 on.
From p. 47:
In Section 3.1 we define left and right localizations of model categories […] in Section 3.3 we discuss (left and right) Bousfield localizations of categories, a special case of left and right localizations
I have essentially rewritten the entry to better reflect this.
Re #5: No, this is just Hirschhorn’s idiosyncrasy that is not shared by other authors. Both notions present exactly the same construction of reflective (∞,1)-localizations.
Look at Definition 4.2 in Barwick’s “On left and right…”, he defines left Bousfield localizations exactly as what this article calls “localizations of model category”, i.e., Hirschhorn’s left localization.
Does he also show that the two definitions are equivalent?
Re #7: I would not even call the construction with S-local weak equivalences a definition on its own.
The fundamental property that we want is precisely the same as for reflective localizations: we want the initial object in the category of left adjoint functors that invert S.
In the ordinary case, if the category is locally presentable and S is a set, then the reflective localization exists, and can be computed either as the full subcategory of S-local objects, or as the category of fractions of C with respect to S-local equivalences.
However, we do not say that either of these two constructions is a “new notion of a localization”. These are just computational descriptions of reflective localizations.
Likewise, in the model categorical case, there is only one definition: the initial object in the category of left Quillen functors whose left derived functor inverts S.
If the category is combinatorial and left proper, and S is a set of morphisms, than the localization exists and can be computed either as the full subcategory of S-local objects (if we are willing to work with relative categories, not model categories), or as the original model structure whose weak equivalences are enlarged to S-local weak equivalences.
However, we do not say that either of these two constructions is a “new notion of a localization”. These are just computational descriptions of left Bousfield localizations.
Maybe it’s trivial, just let me know (though it didn’t seem obvious to Hirschhorn, who calls Bousfield loc a “special case”):
Are the two definitions equivalent? I.e. does it follow that every “left localization” fixes the class of cofibrations?
Re #8: I don’t think I’ve ever seen a definition of “reflective localization” that isn’t “has a fully faithful right adjoint”. Is it automatic that the right adjoint to a functor satisfying the “fundamental property” you describe is a fully faithful functor?
Barring an affirmative answer to that question, I would say that “a functor satisfying the fundamental property of #8” is a “new notion of a localization”, one that happens to coincide with “has a fully faithful right adjoint” when the latter exists. (well, I would lean towards saying “generalization”, but you get the point)
My state of knowledge is the same regarding left Bousfield localization; the definitions I’ve encountered are “same underlying category and cofibrations, but more weak equivalences”. And while such a thing satisfies the universal property you allude to… it is by no means obvious to me that such a thing is guaranteed to exist whenever there exists something satisfying said universal property.
Re #9: As far as I know, there is not a single example of a “left localization” in the literature that is not a left Bousfield localization. (Using Hirschhorn’s terminology.)
I am not sure whether having an article about a potential empty set is warranted. And other authors, like Barwick, use the definition of this article as the definition of a left Bousfield localization.
There is also a potential mathematical problem: whereas in the case of left Bousfield localizations we do know that they present reflective localizations of underlying (∞,1)-categories, I am not aware of such a proof for Hirschorn’s “left localizations”. (We would have to show that the strictly initial left Quillen functor is also homotopy initial, which as far as I am aware, nobody ever looked at.) So in fact, we don’t know whether left localizations are localizations!
Re #10:
I don’t think I’ve ever seen a definition of “reflective localization” that isn’t “has a fully faithful right adjoint”. Is it automatic that the right adjoint to a functor satisfying the “fundamental property” you describe is a fully faithful functor?
A priori, one could define two different notions: (1) the initial object in the category of left adjoint functors that invert S; (2) the initial object in the category of left adjoint functors that invert S and whose right adjoint is fully faithful.
In the case when S is a set and the category is locally presentable, (1) and (2) coincide, since any left adjoint functor that inverts S must invert morphisms in the strong saturation of S, and the latter coincides with S-local equivalences when S is a set and the category is locally presentable.
It would be interesting to see examples that distinguish (1) and (2).
For model categories, the situation is the same: if S is a set and the model category is combiantorial (or cellular), then the right derived functor of the right adjoint is automatically homotopically fully faithful.
Thanks for clarifying.
It seems to me only one thing is really clear now: We want to retain that page in order to have a place to discuss all these issues. :-)
Re #13: But then the title of the article is problematic: we do not know that Hirschhorn’s definition gives a localization, until we show that it coincides with the left Bousfield localization. Otherwise, it is just some formal construction for which we have no idea how it relates to reflective localizations of (∞,1)-categories. The only thing we can show is that if is a left localization of C at S in the sense of Hirschhorn, then on the underlying (∞,1)-categories, we have a left adjoint functor that inverts elements of S. However, I see no way to prove that this functor is ∞-initial among all such functors, nor a way to prove that its right adjoint is a fully faithful functor.
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