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• CommentRowNumber1.
• CommentAuthorJohn Baez
• CommentTimeDec 7th 2009
I added some basic explanatory stuff to the "Idea" section of localization. The simple idea of localizing a commutative ring really does, I think, correspond to some fancier operation you can do in topos theory (namely the topos of sheaves on the spectrum of that ring?). So, someone should try to explain that... but I can't.

Someone should also add links from "localization" to various fancier pages on fancier kinds of localization. I did so for Bousfield localization. I also added a bit more explanation to that page.

(I think my job at the nLab is to make pages easier for beginners to follow. This is a gargantuan, almost hopeless task... but I don't see anything else I'm dying to do here. So I'll just do a little. It makes sense for me to add more explanation to an nLab page whenever I plan to cite it on This Week's Finds.)
• CommentRowNumber2.
• CommentAuthorJohn Baez
• CommentTimeDec 7th 2009
Actually there were links to fancier kinds of localization on the localization page. So never mind that.
• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeDec 7th 2009
• (edited Dec 7th 2009)

I dislike the emphasis in the introduction on IDEALS corresponding to the points as the stuff at which we localize: intutition is that we localize AWAY from points, so intuitively we have to start with the set to which we localize not the bloody complement. It is more direct to understand the intuition behind the inversion: we can not invert by ZERO. Therefore if we talk about some algebra of GLOBAL functions then adding inverses of some global function can be done only AWAY from SINGULARITIES of that inverse, i.e. away from nullpoints. This commutative intuition does not care which functions we talk about: continuous, polynomial or holomorphic...Localizing means passing to smaller set, which is the complement of singularities. Now it is long way till asserting that in some algebraic situations like when wroking with regular functions in algebraic geometry, the complement of the set of functions which are inverted is an ideal. So putting ideal at the first place is difficult.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeDec 7th 2009
• (edited Dec 7th 2009)

So you'd rather talk about localising at a multiplicative set (that is, a submonoid under multiplication) than away from an ideal? Come to think of it, so would I; that's how I learnt it in first-year algebra.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeDec 7th 2009

Right, this is more directly intuitive in my opinion.

• CommentRowNumber6.
• CommentAuthorJohn Baez
• CommentTimeDec 8th 2009
I've done some more work trying to make this article user-friendly.

If someone wants to rewrite the intro using multiplicative sets, go ahead. Just keep it simple and intuitive!
• CommentRowNumber7.
• CommentAuthorTobyBartels
• CommentTimeDec 9th 2009

Spammer wrote:

As zskoda said, putting ideas in first place is really hard.

(And then linked to an online poker site.)

I'll delete it, since the link makes it clear that it's spam (and I don't know any other way to disable the link), but I wanted to preserve the comment, which is just hilariously wrong.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 4th 2010
• (edited Oct 4th 2010)

I edited localization a little, in an attempt to polish it.

• added to the useful but lengthy Idea-section a quick one-sentence summary of what localization is.

• moved the remarks about localization of higher categories to their dedicated subsection;

• added an explicit section on reflective localizations to the Definition-section.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeJun 8th 2011

The folowing discussion, archived from the entry localization was prompted by a remark that the terminology “localization” was confusing.

Zoran Skoda Mike, why do you say confusing? First of all localization of a ring induces localization of categories of 1-sided modules by tensoring with the localized ring over original ring and conversely applying localization functor to a ring itself produces the localized ring. The canonical morphism from the ring to its localization, sometimes also called localization, is the adjunction morphism indexed by the ring. The localization functor is just a natural extension of the localization from a ring to all modules over the ring not just the ring itself. The same for corresponding (components of) the adjunction morphisms in that case.

Second, if one does special case when localizations can be made via categories of fractions then Ore conditions are literally Ore conditions from the theory of localization of monoids or rings (besides a monoid is just a category with one object). Third, Bousfield localization in triangulated setup is a localization associated to an idempotent monad just like usual flat localization (or any localization having faithfully flat right adjoint), just the functors are triangulated, and the monad is Z-graded. Finally Cohn universal localization is just H_0 of Bousfield localization and in the matrix form one is essentially solving the Ore condition, as it is shown by Malcolmson and independently and earlier Gerasimov. In fact when one restricts the Cohn localization to finitely generated projectives one has a flat localization. So it is not just an analogy – these are all special cases of the same picture and mechanism.

Mike: The thing that is confusing to me is when one extends the use of “localization” beyond the context of localization of things like modules and sheaves. I do not see any “locality” involved in the process of inverting an arbitrary class of morphisms in a category. Of course there is just one concept here, but I do not like the choice of the word “localization” to describe it.

Tim: Might I suggest that a little historical note tracing the origin of the term (including local ring, as well) might be a good point. Sometimes such a look back to the origins of a term can show old light on new concepts and help one ’create’ good new concepts or to view the concepts in a new light.

Mike: Just to clarify, I do definitely see the reason why inverting a multiplicative system on a ring is called “localization.” I still might prefer it if people had chosen a word that describes what happens to the ring itself, rather than its spectrum, but I understand the motivation behind the term.

Tim: My point was just that others (i.e. more ’debutant’ in the area) might benefit from a few lines on the geometric origin of the term.

Actually I agree with what you, Mike, sort of imply namely that some ’local’ example would be good to see. Is there something in the geometric function theory area that would provide a nice example say of a naturally occurring ’prestack’ where the passage to the corresponding stack is clearly restricting to ’germs’ of the categories involved? I have never thought about that point, any ideas?? It may be easy, I just don’t know.

Urs: sounds good. I have now tried to rework the entry a bit reflecting this discussion. We could/should still add a more detailed historical note.

Also, it would be good to arrange the points that Zoran mentions into a coherent bulleted list in an examples section. Maybe somebody can do that.

Type the two words.

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