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So I was looking at the nLab page on localization, and I saw that Zoran noted that all localizations do not have adjoints (which is what I thought), but apparently later on, it says that the localization is just the left adjoint of a fully faithful embedding (when it exists). How can we inject Ho(M) back into M fully and faithfully for a model category M?
If this is not the case as I suspect (and Zoran agrees), what are the conditions on a localization for it to be the adjoint of a full and faithful embedding?
The localizations which do have a left adjoint are specially nice ones. We did discuss this at various points, but maybe the discussion is lost here on the nForum instead of archived on the nLab.
[…] what are the conditions on a localization for it to be the adjoint of a full and faithful embedding?
There is a certain saturation condition on the set of morphisms that is being inverted that characterizes reflective localizations.
See the lemma called “localization lemma” at reflective sub-(infinity,1)-category.
@Urs: I think you should split that article up or at least link some of the material from pages like “localization”, since it is extremely relevant, but often hard to find (I literally never would have thought to look at reflective-sub-oo-category). Also, if it happens to be the case that the article for the oo,1 case is written up, but it is not covered for the ordinary case, it seems like it would be better just to leave the oo,1 material on the original page until someone finishes the page for the ordinary case (that is, in this case, merge reflective subcategory and reflective sub-(infinty,1)-category or at least add in links from reflective subcategory to the (oo,1) case and let the reader know that there is information present on the (oo,1) page that isn’t covered in the ordinary case).
I think you should split that article up or at least link some of the material from pages like “localization”
If you find it useful, you can help me do it. I am extremely busy. And about to go to bed.
To be more explicit: please go ahead and copy, re-link, re-organize the materal as need be. I certainly do see and agree of the need of doing so. But there are only so many man hours in the day of one man. So help is needed.
I will do it for this article, but I mean that in the future when you write a long article like the one you linked, it would be really helpful to link the non-higher-version to it if the non-higher-version does not have all of the information in the (oo,1)-version is all I mean.
Or at least, if you can’t do that, could you comment on the nForum (preferably in the title of the post) that others should add links to the ordinary and related pages?
Harry, considering the scores and sometimes hundreds of things Urs does in the nLab every day, concentrating hard on his research needs and not on the needs of others, I’m not sure we should be bugging him about this sort of thing. Let him blaze forth in whatever way works best for him, and if you see a way to improve things, just roll up your sleeves and do it, and then let others know what you have done. He and everyone else would appreciate it!
First an English remark here Todd or Toby can judge. Urs often writes “in that” for “in the sense that”. To me this syntax does not parse, and I can not comprehend the sentence unless I pause and read it 3 times and get it by reverse engineering. In German it parses. But I am not native English speaker so I do not know if such an elliptic shortening parses smoothly to native English speakers. From the reflective (infinity,1)-subcategory entry:
“Recall that the reflective subcategory … is exact if L is a left exact functor in that it preserves finite limits.”
Now about localization. Localization is the inverse image functor. Urs in 2 says something about the left adjoint of localization. In fact he probably meant about the right adjoint, as the inclusion is the right adjoint of the localization. Or maybe Urs talks about a different thing: Harry asked about when a localization has a right adjoint (which is then an inclusion), and Urs talks about the case when an inclusion has a left adjoint (which is then equivalent to a localization).
Now the problem is also about the terminology, what we call localization. Unlike in algebra, in topos theory people use word localization for a left exact functor having right adjoint which is fully faithful. This is taken also by many pure category theorists. Almost all people in algebra and many in algebraic topology by a localization mean the universal functor inverting some family of arrows, that is the canonical functor into a category of fractions. That is the setup of the ur-reference about localization, the Gabriel-Zisman book. Such a localization does not have to have a right adjoint. If it has it must be fully faithful for such a localization. Next, there is a more special case of a localization when it admits a left calculus of fractions. Such one does not need to have a right adjoint either. The condition when it has is GZ Proposition 1.4.1.
Conversely, if a functor has a fully faithful right adjoint, then the class of morphisms which are inverted under it form always a left category of fractions, and the original functor can be factored into a canonical functor into the category of fractions and an equivalence. Note that the localization in narrow sense of canonical functor into a category of fractions satisfies a strong form of universality property (functor inverting all arrows in the distinguished class factorizes uniquely through localization), while the equivalent functors in the above sense satisfy a weak form of the universality, namely there is a functor which completes the factorization only to a unique isomorphism. In algebra, one usually indeed works with strong universality property (the affiness in the algebraic geometry sense fixes a choice in the Morita equivalence class) which people in this community would possibly hastily call “evil” (but it is not, as one has distinguished generators by choosing the structure sheaf as an object in the category to localize) while the nlab page (and if recall right Kashiwara-Schapira) consider the weak universality property. Gabriel-Zisman has a strong universality property and carefully says that the reflection is (only) equivalent to the localization.
Lurie says in his chapter on localization that he means the case with right adjoint notes that many consider the general case of universal functor when the right adjoint does not exist, but this case is less important to him.
My vote for the terminology is this. Let us call localization the universal functor inverting some family of arrows, that is a canonical functor into a category of fractions in the sense of Gabriel-Zisman. No need for asking to admit a calculus of fractions, left or right for that. Let us call the case when the localization functor is flat simply flat localization. All this is standard in algebra, at least. In a context we can restrict to localizations having right adjoint by saying this at least at the beginning of a treatment; or we may emphasize if we mean the universality in strong or weak sense. This would be a good practice in the exposition. In any case, we can not assume any of these specifics for a practical mathematician in general mathematics.
As for the English, “in that” is certainly accepted and used.
I agree (being English!) but I would expand it also as ’ in as much as’. Many English language expressions are a mix of Old English (which was very German/Dutch/Flemish in flavour) and Old Norse, so I don’t know who to blame for that one.
Now the problem is also about the terminology, what we call localization. Unlike in algebra, in topos theory people use word localization for a left exact functor having right adjoint which is fully faithful. This is taken also by many pure category theorists. Almost all people in algebra and many in algebraic topology by a localization mean the universal functor inverting some family of arrows,
Every localization in the sense of a fully faithful functor with a left adjoint does come from universally inverting a collection of morphisms! So it is a special case of the more general notion of localization.
@Tim: I had begun to mention “in as much as” (“inasmuch as”) as well, and then redacted it for some reason. There is also “in so far as” or “insofar as”.
Urs, please read carefully the comment which you responded to this time. I made very careful series of distinctions, including more than once stating the very content of your remark, but more carefully.
To repeat with an emphasis, a fully faithful functor having a right adjoint is not a localization in strictest Gabriel-Zisman sense of a canonical functor into the category of fractions but only such up to an equivalence (not isomorphism!); moreover it does not satisfy the universal property of a localization in its strict Gabriel-Zisman sense. That is is a localization in at in the strict GZ sense if for every functor inverting all morphisms in , there exists a unique such that (strict equality, very useful in noncommutative algebra!). This universality is not invariant under composition of with an equivalence of categories! What is invariant is existence of such that there is an isomorphism of and . This is the weaker universal property which is not always enough in practice (all the discussion above of affinity and Morita is about it). nLab, unlike Gabriel-Zisman and practice in algebra does the second version. This is OK with me, as long as one is aware that sometimes one needs stronger version, and such stronger choices exist always after certain choices are made. Few days ago I forgot distinguishing the two in a research on descent and Gabi produced a counterexample to a proposition which I have sent her.
Sorry, Zoran, you are right I just glanced over the first paragraphs of your message before commenting.
It is good to clarify subtleties in use of terminology, but I’d rather we do not give the non-invariant (“evil”) definition of localizatoin too much emphasis. It’s good to explain it on the page on localization (I hope you turn your nForum posts on this into a good entry eventually) but “localization” is too important a concept to make a non-invariant definition be the default one.
@Urs: I think you’re missing Zoran’s point. Localization is very important to homotopy theory as well, but it seems like this treatment of it excludes the homotopy-theoretic stuff to instead cover descent. It would be a lot easier if every localization had this property, but they do not.
I don’t know what you mean, Harry. There is a general (invariantly defined) notion of localization, and reflective subcategories are a special class.
Sure, but it seems like reflective subcategories are being covered at the detriment of the more general theory.
are being covered at the detriment of the more general theory.
Are being covered?? Show me the villain who covers them such!
In case this is secretly asking me to write something on the nLab, please contact me by private email and I send you my prize list. I charge by line.
I’m just going to stop talking, since I guess I’m being rude without knowing it. I apologize, Urs, and I don’t want to appear ungrateful.
It is good to clarify subtleties in use of terminology, but I’d rather we do not give the non-invariant (“evil”) definition of localizatoin too much emphasis.
Urs, I am working on this subject for long time and I really recall lots of applications in algebra for the strict definition. I do not think that the localization in strict sense is an evil construction. It is covariant, I mean if you take an equivalent category and take the corresponding localization in strict sense of the equivalent category it will be also equivalent to the original. So in the literal sense it is not evil. As far as its strong universal property, the localization of the equivalent category also has this strong universal property. Nothing evil in the forming categories of fractions in the strict sense and observing their universal property.
What you probably worry is that the image of the construction belongs to a non-replete sub-2-category of 2-category of functors, and that you do not like to consider that image, but rather the essential image of the construction, each characterized by a weaker universality property. But this kind of evilness of the category of all localizations is a different issue than talking about construction of a localization which is not evil and has its distinguished place among all functors from given category with given class of morphisms.
Edit: the imperative that one should consider the categorical constructions covariant under equivalence does not imply in my understanding that the strict commutation of diagrams apearing as output should not be considered. For example, that some functors strictly lifts another functor, or that some categories coming out of construction appear isomorphic.
To hint another case where it is sometimes useful to consider localizations in the strict sense, let me remind you that the localization is with respect to given class of arrows. Now what is often important (especially for understanding the compatibility of various categorical actions and localizations, which is essential for studying categories of sheaves on quotients for example, and for studying internally things like differential calculus), is to test and study various functors in terms of behaviour with respect to the class of arrows which the localization inverts, for example if it leaves it invariant, or leaves a complement invariant, or if a functor sends that set to something of specific nature in another category and so on. For such things tests are easier for localizations in the strict sense.
Of course, for each localization in the weak sense, at least if it has right adjoint, one can in principle find a localization in the strict sense equivalent to it.
Zoran,
thanks for all the input, but you should really write that out on the nLab. I can’t quite concentrate on this at the moment and won’t be a good discussion partner.
By the way, as I notified in the thread on “localization”, I did go through the entry localization now, promted by the discussion here, and tried to polish it a bit further.
To my surprise that entry did not emphasize the special case of reflective localizations at all, contrary to what I expected from this discussion here. I added in a remark highlighting this special case!
I ahve often seen the two distinctions of localisations (strict and ’up-to-equivalence’) and wondered why some say a localisation is such that
is an isomorphism
and some say it is such that
is an equivalence.
Zoran refers to the first as being Gabriel-Zisman, and useful for his purposes. However, coming to localisation from the peculiar direction of having to do it for (strict) 2-categories, where localisation is defined (by Pronk) to be up to equivalence of (weak) 2-categories, I wonder about the difference it would make to my own work. Pronk’s construction of a bicategory of fractions is rather unwieldy, and only defined up to some sort of equivalence, as it depends on a whole lot of choices of weak pullbacks. My construction of the localisation of (which is what Pronk’s aim was anyway) is a lot more determined, and ( a given subcanonical singleton pretopology) is pretty much determined up to isomorphism. Then there is the issue of saturated anafunctors, which are an even smaller model of the localisation, which in the guise of bibundles are pretty much de rigeur in the Lie groupoid community who use such things (the people who use the awkward phrase ’stacky Lie….’). Of course, this is all localisation of 1-arrows. Then there is the recent paper by Bacard where he localises a bicategory with respect to a class of 2-arrows. All these different levels of localisation need to be considered, and at least mentioned in the article, unless people feel that it should be only about 1-categorical localisation. I am happy to put all this in, I just want to gauge others’ reactions (and nForum comments are quicker to write than composing a nice piece for public consumption)
I just noticed that on the page there is a bit about higher category theory, but it is all (oo,1)-categories. When I get some time I will incorporate some of the about comments (#23) into the page.
It seems like it would be in line with all the rest of the nlab terminology to say “localization” for the up-to-equivalence notion and “strict localization” for the up-to-iso notion. Cf. n-category and strict n-category, 2-limit and strict 2-limit, 2-functor and strict 2-functor, etc.
In fact, a localization in this sense is actually a particular case of a 2-colimit; in that context it is often called a coinverter. So using the same strictness terminology as we do for 2-limits is more than just an analogy.
Right, Mike, I noticed that relation to limit terminology and content last night. Thanks, David for alerting about the bicategory of fractions. I had some thoughts recently about it. Do you know any work expanding on reflective bicategories ? I see to large extent how one can arrive from an ad hoc version of a reflective subcategory to Moerdijk-Projk conditions, but stating and proving all in full detail is more cumbersome than I can reach with my free time.
Urs 22. Do not worry. But I will continue alerting here people of things which I see differently about the localization before I revise the related pages. As you might have notices I returned seriously to some research questions related to descent and localization, so this will stay in my focus for few weeks to come; hence eventually it will appear in nlab. I will also mention in idea section when I get there that localization is not only about categories but also about many other algebraic structures.
Do you know any work expanding on reflective bicategories ?
No, I don’t. I feel like I need to explore localising 2-categories more, and prompted by some questioning here, look at conditions when one can construct a fairly easy localisation (beyond the Pronk axioms), but this is work I’m not being paid to do, and requires more time than I have.
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