# Localizations or quotients of categories?

Motivation: In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the quotient category $\mathcal{K}$ of $\mathcal{A}$-chain complexes modulo homotopy and observes this is triangulated; finally, the derived category $\mathcal{D}$ is the localization of the above triangulated category with respect to some morphisms defined in terms of the triangulated structure.

I have always thought as the first process of modding out by homotopy equivalences as being a way to make homotopy-equivalent maps look the same or inverting things which are homotopy equivalent to invertible arrows. Then realized that this is also the aim of localising: force some arrows to become isomorphisms.

Here the question: is there a conceptual frame in which we can see both localisation and modding out by a relation as special cases? Or, are the two concepts better related that my vague interpretation that they both make some arrows invertible? I have seen someone in nLab saying that they are entirely different constructions but that left me a bit disappointed since I still believe a bit in the feeling that at the end of both days I find myself with a new category having the same objects as the old one, and some new isomorphisms. More generally: I realise that if I had a category in my hands and were willing to see some arrows become isomorphisms, I would not know which of the two approaches would be natural in my case, whence the feeling I still have some fog in my mind.

I should add that a colleague of mine observed that he would expect the first construction to be reminiscent of taking closed immersions, and the second to open immersions; which I fully agree to, but would again like to know how to think about these parallels when dealing with a category.

• Sure. They're both colimits. (I mean, it's not as if you're trying to compare a limit and a colimit. Localizations and quotients both have universal properties involving maps out rather than maps in, so saying entirely unrelated seems a bit unfair to me.) Nov 27, 2014 at 9:58
• In your specific example, $\mathcal{K}$ is actually isomorphic to the localisation of $\mathbf{Ch}(\mathcal{A})$ with respect to chain homotopy equivalences. This is true more generally when you have suitable cylinder objects or path objects. Nov 27, 2014 at 11:42
• @QiaochuYuan: Uhm... could you explain a bit more? In which world are you considering this colimit(s)? They are objects having universal properties "out" but as Categories, so you are working in $Cat$ or something? Nov 27, 2014 at 12:49
• @Filippo: yes. More specifically, localizing a category by inverting some collection of morphisms can be described as a pushout in $\text{Cat}$, and quotienting a category by an equivalence relation on morphisms can be described as a coequalizer in $\text{Cat}$ (I think). Nov 27, 2014 at 21:23

They are completely different constructions which are in general absolutely unrelated but might coincide on some rare occasion. the image of closed immersion vs open immersion is quite good I think, especially if you think about it in terme of localization of a ring: it might happen that localizing a ring is the same as taking a quotient, but this is rather exceptional...

(Edit: this being said, I totally agree with Qiaochu Yuan comments and "absolutely unrelated" was slightly exaggerated )

The reason why they give the same things in a lot of example (including chain complexes), giving the idea that they should be related, is because these examples are Quillen model categories and that it is the main result of Quillen's "Homotopical Algebra" (where he defined model categories) that for Quillen model category the localization by weak equialence can be constructed as a quotient of the full subcategory of fibrant-cofibrant objects.

Edit for References: The orginal paper of Quillen (Homotopical algebra) is actually a really good reference on the subject, which give both the general theory and the examples of simplicial sets and of chain complex. There has been also several book on the subject, I would advise to take a look to Wikipedia which contains already most of the relevant information as well as references to the main books on the subject.

• For a finitely generated algebra, it might seem odd to have localization and quotients coincide, but consider the case $R[x_i\mid i\in\mathbb N]$, i.e. you have infinitely many variables. Localizing in some polynomial $f$ is the same as picking a variable $x_i$ that does not appear in $f$ and taking the quotient by $x_if-1$. This was my initial thought when I saw the question: Maybe categories are so "large" that quotients and localizations could coincide more easily. Nov 27, 2014 at 10:36
• I'm not completely agreeing with your remarks for the following reason: the two constructions you are giving indeed give the same ring, but in the case of a quotient or a localization of a ring $R$, what is important is not the ring, but the $R$-algebra you obtain (the universal property you obtain involve the $R$-algebra structure) and in your situation, the two $R[(x_i)]$ algebras are not isomorphic. Nov 27, 2014 at 10:44
• The way I see the difference in the categorical context can be formulated as follows. In any case, identifying certain objects is not natural; one may speak about forcing non-isomorphic objects to become isomorphic. Now the latter can be achieved in two different ways - either by turning certain morphisms between them into isomorphisms, or adding completely new isomorphisms. The former is more on the quotient side and the latter more on the localization side... Nov 27, 2014 at 11:31
• The terme "quotient" is used here for quotient of a category by an equivalence relation on morphisms. Not for trying to identify object. Nov 27, 2014 at 11:34
• @FilippoAlbertoEdoardo, consider an antisimmetric category (so that for distinct objects $x$ and $y$ at most one of the sets $\hom(x,y)$ and $\hom(y,x)$ is nonempty,and there are no nonidentity endomorphisms) and consider the equivalence relation which identifies all elements in each nonempty $\hom$ set. This is not a lozalization, because there are no isomrphisms in the resulting category. Dec 30, 2014 at 17:02

The term you're looking for is 'coinverter' (dual of the notion of inverter). This is a weighted colimit in the $Cat$-enriched category $Cat$. See this page on the nLab.

• I tried to follow the link, but it seems to me thatit only describes localizations and I find no reference to quotients. Nov 27, 2014 at 23:39
• One can take quotients of categories to calculate localisations in certain cases. For instance (the following may be roughly right) when one takes the fibrant and cofibrant objects in a model category with a cylinder object, then taking the quotient of the full subcategory by the relation of homotopy gives the localisation at the weak equivalences. This is a serious theorem of Quillen, and showed that localisations of model categories are locally small. Chain complexes have a model structure with weak equivalences the quasi-isomorphisms (and this was one of the motivating examples for Quillen) Nov 28, 2014 at 4:11
• You might want to look at Quillen's monograph Homotopical Algebra link.springer.com/book/10.1007%2FBFb0097438, if you have access. Nov 28, 2014 at 4:14