Timeline for Why do we need the axiom MS3 for localizing categories?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 5, 2021 at 9:17 | vote | accept | Gabriel | ||
| Aug 4, 2021 at 23:10 | answer | added | Zhen Lin | timeline score: 5 | |
| Aug 4, 2021 at 18:20 | comment | added | Gabriel | @TimCampion as you actually know, I'm writing my own notes on the subject, so I assure you that I've read all the presentations of localization of categories that I could find haha. And none explained why this axiom is important. | |
| Aug 4, 2021 at 18:18 | comment | added | Gabriel | @Denis-CharlesCisinski I would be very glad if you could explain more your point. For example, how exactly is the localization "better behaved" when we suppose this axiom? I don't quite understand how this relates to (co)limits. | |
| Aug 4, 2021 at 18:05 | comment | added | Tim Campion | One commonly-used term for this sort of condition is "right Ore condition". See also the nlab page. The point is well-taken -- it would be valuable to explicitly spell out exactly what is achieved by such conditions. | |
| Aug 4, 2021 at 18:01 | comment | added | D.-C. Cisinski | @Gabriel Axioms of the form LMS3 are part of the notion of calculus of fractions. They ensure that the localization will have as many finite (co)limits as $C$. This kind of axiom is not necessary to understand localizations and only gives useful sufficient conditions ensuring the localization is well behaved. | |
| Aug 4, 2021 at 17:57 | comment | added | Gabriel | @TimCampion When we consider some axiom, we hope to gain something from it, don't we? That's my question. It is clear to me why someone would consider the other axioms or saturated classes. The axiom LMS3 is not as clear. | |
| Aug 4, 2021 at 17:56 | comment | added | D.-C. Cisinski | A related discussion about the construction of the additive localization by closing $S$ under finite sums: mathoverflow.net/q/44047/1017 | |
| Aug 4, 2021 at 17:51 | comment | added | Tim Campion | One way to see that $S^{-1}C$ and $S^{-1}_{add} C$ exist is to invoke the adjoint functor theorem. I guess what I'm driving at, though, is that your original question (the bolded text) is ambiguous: if all you want out of the localization is its defining universal property, then you don't "really need" anything at all. So it seems to me that one should first ask the preliminary question "What does one 'really need' out of a localization construction?". One property often considered desirable is that $S$ should be saturated, i.e. that no morphism not in $S$ become invertible in $S^{-1}C$. | |
| Aug 4, 2021 at 17:31 | comment | added | Gabriel | Cool! I'm still interested in a reference for that and in my original question, though. | |
| Aug 4, 2021 at 17:28 | comment | added | Tim Campion | Yes, that's what I mean. | |
| Aug 4, 2021 at 17:27 | comment | added | Gabriel | @TimCampion so you mean that $S^{-1}_{add}C$ is universal among only the additive functors, but not all functors. Is that it? | |
| Aug 4, 2021 at 17:20 | comment | added | Tim Campion | I don't claim that $S^{-1} C = S^{-1}_{add} C$ in general. | |
| Aug 4, 2021 at 17:17 | comment | added | Gabriel | @TimCampion firstly, I didn't know that a localization of an additive category is always additive. Do you have some reference for that? Anyway, I think that "having a simple structure for the sum of morphisms on the localization" is a good motivation for considering the axiom LMS2. But I don't really see what we gain with the axiom LMS3. | |
| Aug 4, 2021 at 17:10 | comment | added | Tim Campion | If C is a small category, and $S \subseteq Mor C$ is any set of morphisms, there always exists a small category $S^{-1}C$ and a functor $C \to S^{-1} C$ which universally forces the morphisms of $S$ to become invertible. Likewise, if $C$ is a small additive category and $S \subseteq Mor C$, there is always a small additive category $S^{-1}_{add} C$ universally inverting the morphisms of $S$. So in some sense, there is no "real need" for any requirements whatsoever on $S$, without further clarification of what one's "real needs" are. | |
| Aug 4, 2021 at 16:59 | history | asked | Gabriel | CC BY-SA 4.0 |