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New entries torsion theory and quotient category.
I added much material to localizing subcategory. I am sure there are some errors there, I am still very unhappy with the state of the entry. I would like to create in a way that it would be clear how the approaches in different treatments are equivalent and that it would enable the reader to use them. Of course, I trust mostly Pierre Gabriel and his thesis. Unfortunately some of the terminology and conventions are not followed in many of the later references, so it is difficult to spot when people are imprecise or the usage of the same term is slightly different. Besides some of the equivalences have their natural generality which is difficult to decide on the basis of the literature.
Added the following query box to quotient category: “The above definition of a quotient category appears to be different from that of CWM (p. 51 of the second edition), where a quotient category is obtained by identifying arrows. Perhaps there is a need for disambiguation?”
That is a different notion. Cat is taken as a 1-category and one takes an internal equivalence relation there and gets a quotient of the equivalence relation. This can be restated as certain coequalizer in Cat and also if one wishes called that way. Let me think how to disambiguate in a wise and modern way.
Thanks!
I would not expect that the notion of quotient category that comes from thinking of $Cat$ as a mere $1$-category would be the most useful one, although obviously it could be useful for some things. I also wouldn’t be at all surprised if we got two separate notions ($1$-quotient and $2$-quotient?). Compare the issues at subcategory.
In English-language ring theory, one says both “quotient field” and “field of quotients” for the same concept (the … of an integral domain); while the first term is older, the second is used to avoid confusion with quotient rings. (To be even clearer, one can even say “field of fractions”, which I usually do.)
We could do the same with Zoran’s quotient category (“category of quotients” or even “category of fractions”), which appears to be analogous.
Toby, the difference is not only weather it is 1- or 2-categorical but also the type of input data, so here we have rather different kinds of limits, not only different categorical dimension. In MacLane type quotient one morally identifies some morphisms witin morphism sets and in localizations one inverts some things in order to make certain morphisms isos. Firthermore when one says quotient category in the latter, although 2-categorical case, one really means a representative of ’quotient’ with strict universal property (strict localization vs. localization). Surely words like categories of fractions are also used. We should somehow clearly record the variants, while we can not rectify that field which is too old and with too many variants to be rectified.
Post #6 is not about the difference between localisation and quotienting; it is about quotienting only. (They seem to be almost completely different things to me, another reason to use different terminology.)
Well, 2-categorically, localization is a coinverter, which is a 2-colimit that at least bears some relationship to a coequalizer. I do agree, though, that the origin of the word “quotient” is different in the two cases: in one case it is the arrows in the output that are “quotients” (fractions) of the arrows in the input whereas in the other case it is the whole output category that is a “quotient” of the input category.
Perhaps the page quotient category should do mostly disambiguation, with the interesting material put on unambiguously named pages.
unambiguously named pages
Which would be what? category of quotients, 1-quotient category, and 2-quotient category? But we should also have a page, analogous to subcategory, that treats the naïve notion in CWM, even if it is evil, and only quotient category seems to make sense for that.
Toby #11: While I know nothing of $n$-categories for n>1 (and so I do not understand most of the above discussion), I hope it is OK to ask the following question: from “the naïve notion in CWM, even if it is evil” it sounds (to me) as if the notion of CWM is something “bad” or “old” that should be avoided, and deserves an explanation only for historical reasons.
But isn’t it common to identify arrows in categories in order to construct new categories? Isn’t this even part of the construction of categories of fractions (starting with the free category on the graph of $C$ augmented by formal inverses of arrows in $\Sigma$ and then identifying some arrows)? Thanks in advance for your answer.
The meaning of ‘evil’ here is a technical one, and you shouldn’t take it too seriously. (See evil for discussion.) Ignoring higher categories, what this amounts to is that, while we consider two groups to be essentially the same if they’re isomorphic, we consider two categories to be essentially the same if they’re equivalent, even if they’re not isomorphic.
The problem with the naïve notion of quotient category is not its answer to ‘What category do we get if we start with $C$ and identify these arrows?’; that’s fine. The problem is its answer to ‘Is $D$ a quotient of $C$?’. The answer to this question can change if $C$ or $D$ is replaced with an equivalent category; basically, the answer is No in some cases (where $D$ has too many isomorphic objects for $C$ too handle) where it really ought to be Yes.
There is some discussion of the dual question at subcategory.
Thanks very much, Toby! I’ve seen the notion of evil before here at the $n$Lab, but I needed the distinction you made between identifying arrows (which is fine) and asking which subcategories of a given categoty are entitled to be referred to as quotients. I have some more questions, but I’ll hold them until I read the discussion in subcategory.
By the way, “quotient category” is the second time that I see that Borceux avoids the definition of “classical” evil concepts (the first time was when I noticed that he avoids defining creation of limits).
Thanks again, Yaron
Edit after Zoran’s following remark: the quotient of course need not be a subcategory, and the above “..and asking which…” should be replaced by “which categories are entitled to be referred to as quotients of a given category.” (Sorry for the confusion)
I created entry strict localization. Terminology quotient category some use only for that case, but it is not uniform.
and asking which subcategories of a given categoty are entitled to be referred to as quotients
The quotient/localized category is not necessarily identifiable with a subcategory of the original category. For Borceux, and some others, they consider the exact reflective localizations on regular categories and call this localization by definition, but for a general localizations (i.e. just universal $W$-inverting functors, like in Gabriel-Zisman or in Kashiwara-Schapira) the existence of an adjoint to the localization (which is then automatically fully faithful) is not required.
Zoran: Sorry, I meant “which categories are entitled to be referred to as quotients of a given category.” (Don’t know why I wrote it as I did - sorry for the confusion).
Well, there is a factorization system on $Cat$ in which every functor $F:C\to D$ among small categories is decomposed (uniquely up to iso) into the composition of an iterated (possibly strict) localization followed by a conservative functor. Namely, one takes the class $W$ of all morphisms in $C$ which are inverted by $F$ and localizes at that class, this gives the decomposition into the composition of a localization followed by some functor; if the rest is conservative one is done and the whole functor is a localization, but it is also possible that in the localized category there are some new morphisms which get also inverted by the second part, so one needs to continue and the iterated localization is obtained in the limit. Cf. Factorisation+systems (joyalscatlab).
Thanks for your answer, Zoran. I am a little confused now (most surely because of my lack of knowledge): my comment (“which categories are entitled…”) was regarding a previous comment of Toby, and concerned CWM’s definition of quotient category (which is, as far as I understand, completely different from “category of fractions”). At a first glance I am not sure whether your answer refers to the same issue – but I will look into the links in your answer from which I will benefit anyway. Thanks again and sorry for all the fuss.
Sorry, I was not answering about the MacLane context, yet.
Toby #11: Hmm, I suppose you may be right.
Updates at thick subcategory (which now also redirects Serre quotient category, which is a specific localization at a (strongly) thick subcategory of an abelian category) and at torsion theory.
Added to quotient category the basic example (abelian groups)/(torsion groups). As soon as the Stacks Project website is updated with the most recent changes from the GitHub repository, I will add a reference.
I have tried to tidy up the wording a little in quotient category as it read awkwardly in some places.
I would presume that “there is a factorization” should be “there is a unique factorization”.
The page quotient category has caused confusion again, on MO. I don’t see the need for our current page quotient category at all. The last three paragraphs seem to be about Serre quotient categories, which already redirect to thick subcategory; so if there is any material there that isn’t also at thick subcategory it could just be moved to there. The first paragraph is just a redefinition of localization or strict localization, and the second paragraph explains the variance of terminology. So the only real content on this page is a remark that sometimes “quotient category” is used to mean a (strict) localization, particularly in the case of a Serre quotient; thus it seems it should be just a disambiguation page, with links to localization and Serre quotient category, as well as perhaps to other kinds of quotient categories.
What does the nLab call MacLane’s quotient categories (CWM, II.8), i.e., a category C is equipped with an equivalence relation on each hom-set, compatible with composition, for which we then form quotients of hom-sets, which results in a new category?
I don’t know if we have a name or a page for them currently.
Yes, it looks like this page should rather follow Wikipedia’s Quotient category: Give a definition of MacLane’s notion of quotient category and add a disambiguation warning that points the reader to the entries on “localization” and “thick subcategory”.
Ok, done in a stubby way.
In Abelian category community (nor triangulated category community) at least I do not know a single person or textbook (take Popescu for example) who calls by quotient category Mac Lane’s notion, everybody means Serre quotient category or its triangulated analogue (or enhanced/stable versions). It is a bit unfortunate as the MacLane’s 1-categorical notion, while more rarely needed in practice (I know of no deep theorems or deep applications, unlike for Serre’s procedure) is also a natural notion, but this is reality. I noted some time ago when we had a discussion about ideals and sieves that there is another notion of the ideal – precisely that needed for the obvious notion attributed to MacLane.
It is misleading to say that the Serre quotient and localized category are the same notion in mathematical culture. Localization is rather general universal functor, only in some special cases localizations can be obtained by quotienting procedure, say when you have Ore localization of ring you have equivalence classes of fractions (in ring theory people talk generally of the quotient rings of domains when it is possible to embed them in a skewfield, so Goldie quotient, Ore quotient, Martindale quotient etc.). Serre’s original procedure has some quotienting flavour (considering certain kernel in the procedure and nullifying morphisms which have some relation to this kernel) thus the name historically came. When I wrote a 2004 survey on nc localization I avoided term quotient in this context except at a single place where I mentioned Serre’s construction, which was out of my focus at the time, but partly because I was afraid of the confusion from quotienting by an ideal. There is lots of other terminological and methodological mess (multiple formalisms with elaborate baggage) in the field of localization in Abelian context (a little more order in triangulated community where the theory is also simpler).
To point out another complexity of the terminology when there are multiple procedures for constructing localization functors, both quotient-like and others (this time due homotopy as an equivalence relation), I quote from the recent paper
As is well-known, the localization of a category always exists and can be constructed by adding formal inverses, that is by identifying classes of zigzags; however this construction is unmanageable in practice. This is a motivation for the constructions in [5], where zigzags of length 2 suffice, and in [10], where the candidates for the inverses are already present in the model category and the localization can be constructed as a quotient (see also [4, 10.6], or [11, § 3.1] for a detailed explanation of this situation in an abstract context).
This paper deals with the situation analogous to that of [4], [11], that is the construction of the localization as a quotient, but in dimension 2. This amounts to constructing a localizing bicategory which has the same objects and arrows as the original bicategory. All the difficulty is thus in the 2-cells of this bicategory, which should in a sense include at the same time the original 2-cells of C and new 2-cells corresponding to a notion of homotopy.
quoted:
[4] Dwyer W.G., Hirschhorn P.S., Kan D.M., Smith J.H., Homotopy Limit Functors on Model Categories and Homotopical Categories, AMS Mathematical Surveys and Monographs 113 (2004).
[10] Quillen D., Homotopical Algebra, Springer Lecture Notes in Mathematics 43 (1967)
[11] Szyld M., The homotopy relation in a category with weak equivalences, arXiv:1804.04244 (2018).
Thus I think we could be a bit more sensitive to the complexity than just doing a redirect to a differently used term. I’ll try to add something in the future.
The wikipedia page is quite an inconsistent potpourri and it is not clear which of few mentioned notions are related there and which are subsets of which.
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