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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.
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.
Right, this is more directly intuitive in my opinion.
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.
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.
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.
Have added to the References-section at localization a pointer to Carlos Simpson’s implementation of Gabriel-Zisman localization in ZFC in Coq: Simpson05
As kind of a specialist in localization theory, I personally do not understand the term “Gabriel-Zisman localization”. GZ book is a pioneer work both for the general localization and for the more regular special case of calculus of left or right fractions in categories. Thus it always confuses me which one of the two is meant when choosing one of those to be named after Gabriel and Zisman (folklore tradition is not uniform in this). Of course, Gabriel treated at length and depth the localization in abelian context in his thesis (and Grothendieck earlier in Tohoku, about 6 years earlier. The calculus of fractions is on the other hand the trivial horizontal categorification of Ore methods for rings and monoids, coming from 1931 or so.
The localization of the product of two categories with weak equivalences is the product of the separate localizations.
One way to see this is by considering the universal property for the localization functor with one of the arguments restricted to any object, and then considering the behaviour of the naturality squares as the objects vary.
Is there a more elegant way to see this?
Localisation is a coinverter, so if one has that finite products commute with coinverters, you are done. This is probably an enriched/weighted limit analogue of finite limits commuting with filtered colimits (or better: finite products and sifted colimits, see commutativity of limits and colimits). A coinverter is in the class that is the colimit analogue of PIE limits, so perhaps that would help.
The easy way of seeing this is to note that localisation is a reflector into an exponential ideal. (Previously discussed here.)
Specifically, the reflector from relative categories into the subcategory of minimal ones, which is equivalent to $Cat$.
Could one of you be so kind to type this out in a coherent sentence (or two) at localization – Properties? Thanks!
@Zhen Lin/Mike
Ah, that’s much better :-)
I added a sentence.
Okay, thanks. Maybe I’ll find the time to add in something less telegraphic.
I don’t think it depends on foundations. I’m pretty sure that even in ZF you can perform that construction (suitably frobnicated) on a large category and get a large (non-locally-small) category as output.
I learned a new word from #21. :-)
It might be worth spelling out the details behind that claim, because I think a lot of people would believe that the construction is foundationally tricky. When you refer to a large category in ZF, I guess you mean not a formal object but something defined by a class formula, and then the construction is also given by a class formula?
Fair enough; it does require a bit of trickery. Yes, a large category is something defined by class formulas. If I have a class formula specifying the morphisms and weak equivalences in a category, I can define a new class formula specifying zigzags of such morphisms with backwards-pointing weak equivalences, and another one specifying sequences of generating equalities between such zigzags. Now we need to take the “quotient” of the first proper class by the second proper-class equivalence relation, which we can’t do directly because the “equivalence classes” may be proper classes and hence cannot be elements of another class, but we can use an axiom-of-foundation trick and consider the sub-sets of each equivalence proper-class consisting of all elements of least rank therein.
Is that detailed enough?
Thanks; yes, I think that’s probably detailed enough. I can get started on that later.
We should also have a little article on the trick at the end, aka “Scott’s trick”.
Yes, we should also have an article on Scott’s trick. (I can remember the trick itself, but never who it’s attributed to. According to Wikipedia the Scott is Dana?)
Yep!
Added some “foundational” remarks on the construction of localizations of large categories, involving especially Scott’s trick (a page we don’t have yet).
Thanks! I added a redirect to make replacement axiom go to the right place.
Removed the following comment, made by me some time ago, but never followed up
David Roberts: This could probably be described as the fundamental category of 2-dimensional simplicial complex with the directed space structure coming from the 1-skeleton, which will be the path category above. In that case, we could/should probably leave out the paths of zero length.
added pointer to:
added pointer to Ex. 4.15 in
as an example of a localization of a locally small category which is not itself locally small
Adjusted the lead-in to make it clear that “category of fractions” is a more standard terminology.
I propose renaming this entry to “category of fractions”, since this is a more standard terminology, and “localization” is ambiguous in this context and has other meanings, especially in quasicategories. “Localization” could be made into a disambiguation page then.
I would have assumed “category of fractions” is meant for a specific construction when a calculus of fractions is available. “Localization” is the term I always hear; flipping through my references folder, some examples
I don’t think the phrase “category of fractions” actually appears in any of the references I have. (but admittedly, I don’t collect references for basic 1-category theory)
I think it would be better to rename this page “localization of a category”, and have “localization” be a disambiguation page. A quick search of the nLab shows 58 different articles which are “localizations” of some sort or other:
I would have assumed “category of fractions” is meant for a specific construction when a calculus of fractions is available.
No, this is incorrect. The article cites (in the lead) definitions of a category of fractions in Gabriel–Zisman and Borceux, both of which do not assume any calculus of fractions. In a category of fractions $C[W^{-1}]$, morphisms are formal compositions of morphisms in $C$ and formal inverses of morphisms in $W$, i.e., “fractions”.
When a calculus of fractions is available, the category of fractions can be computed using fractions of a very specific type, e.g., $W^{-1}C$, $CW^{-1}$, or $W^{-1}CW^{-1}$.
Cisinski, Kan, and Riehl use “localization” to mean “category of fractions”.
Lurie, and lots of people following Lurie, use “localization” to mean “reflective localization”.
Lurie’s usage actually seems to be prevalent in the literature using quasicategories, notwithstanding the examples of Cisinski, Kan, and Riehl.
Re #34:
I created the articles localization of a category and quasicategory of fractions.
I would have assumed “category of fractions” is meant for a specific construction when a calculus of fractions is available.
No, this is incorrect.
If only there were “correct” and “incorrect” on matters of terminology.
In my experience, Hurkyl’s #33 is an accurate reflection of common use of terminology.
Here is another example:
who writes:
Under certain conditions, the localization of a category with respect to a class of morphisms can be described in terms of “formal fractions”. If this construction is possible, the resulting localization is a category of fractions.
Re #38: I have no objections to the new phrasing, which is also more detailed.
Not all sources are equally relevant, though: the books of Gabriel–Zisman and Borceux are widely cited (over 2000 citations each on Google Scholar), the paper of Fritz not so much. If there is a more widely cited source, it would be beneficial to add it.
Perhaps the confusion that you pointed out stems from the fact that in the commutative case, a “fraction” is an expression of the form $s^{-1}x=x s^{-1}$, whereas in the noncommutative case, there are several options: $s^{-1}x$, $x t^{-1}$, $s^{-1}x t^{-1}$, and the most general $s_1^{-1}x_1s_2^{-1}x_2s_3^{-1}\cdots$. The latter expression can also be called a (noncommutative) fraction, I would say.
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