Timeline for Derived functors and functorial resolutions/(co)fibrant replacements
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| 1 hour ago | comment | added | Zhen Lin | I did not say that. Proving that it cannot be done is hard too. | |
| 1 hour ago | comment | added | Carl-Fredrik Lidgren | @ZhenLin In other words, the data written in 2? My question was basically under what conditions one could upgrade such a thing to a functorial choice. I'm also aware that e.g. assuming the weak equivalences form a right/left multiplicative system, one can follow e.g. Kashiwara–Schapira to construct one's desired absolute Kan extensions (this is all written in the original question). My take-away is that there is no way of doing what I want, then? | |
| 1 hour ago | comment | added | Zhen Lin | One weakens the definition of deformation so that (in the same notation of your post) $Q$ is not a functor but rather only an assignment of objects and morphisms and similarly we do not demand naturality of the weak equivalences; of course, one has to compensate by adding other conditions. The point is to weaken the definitions just enough that we can construct deformations using non-functorial factorisation and still construct absolute Kan extensions. | |
| 2 hours ago | comment | added | Carl-Fredrik Lidgren | @ZhenLin Well, the reason I'm "fixating" on them is because I like the proofs in that context and specifically want to know how practical it is to exclusively use them because of that. I don't think this is so strange. With regards to the last thing you said, out of curiosity, what are you referring to? Just that one can construct derived functors in the context of model categories, or something else? | |
| 2 hours ago | comment | added | Zhen Lin | It's not about working or not working. It's about being meaningful and having significance. You can play the K-injectives game in $\mathbf{D} (\mathcal{A})$ rather than $\mathbf{K} (\mathcal{A})$ but it becomes trivial and utterly devoid of meaning. On the other hand, there are not enough K-injectives in $\textbf{Ch} (\mathcal{A})$ to enjoy the functoriality-for-free property. I also don't see why you are fixating on functorial deformations. They are convenient but they are not optimal. There is a not-quite-functorial version that works in model categories without functorial factorisation. | |
| 2 hours ago | comment | added | Carl-Fredrik Lidgren | @ZhenLin Perhaps, but on the other hand, if it works then it works (and I don't see anything that would prevent it from working). It's also somewhat besides the point, because as I said, if there is a way to do what I want on the level of $\mathbf{Ch}(\mathcal{A})$ (even better: in some more arbitrary homotopical category) then that would leave me very satisfied. To be as clear as I can about what I want: fix some $\mathcal{C}'\subseteq\mathcal{C}$ as in 1, and assume that one can do 2 as stated (i.e. non-functorially). I want a way to guarantee that 2 can be upgraded to a functorial choice. | |
| 2 hours ago | comment | added | Zhen Lin | Functoriality at the level of $\mathbf{K} (\mathcal{A})$ is not really functoriality – I would wager it is not what DHKS were thinking of when they introduced deformations. Authors like May and Riehl are quite clear that derived functors should be defined at the "point set" level (i.e. $\textbf{Ch} (\mathcal{A})$, as opposed to the homotopy level, which for them means $\mathbf{D} (\mathcal{A})$ because there is no general analogue of $\mathbf{K} (\mathcal{A})$) and that is why we have to introduce the concept of deformation – these things are not canonical at the "point set" level. | |
| 2 hours ago | comment | added | Carl-Fredrik Lidgren | @ZhenLin I'm satisfied with "anything" which can give functoriality, in the sense that I'll take what I can get. If something better exists, then I'd rather have that (for example, while the applications I care about are in contexts like $\mathbf{Ch}(\mathcal{A})$ and $\mathbf{K}(\mathcal{A})$, I'd prefer methods that work in broader contexts as well). As an additional note: if one can get full model structures with functorial factorizations, that's nice, but I also do not demand it at all (I only want the deformation part that could, for example, come from such a thing). | |
| 3 hours ago | comment | added | Zhen Lin | @Carl-FredrikLidgren I see you are satisfied with functoriality at the level of $\mathbf{K} (\mathcal{A})$. This is much easier than functoriality at the level of $\textbf{Ch} (\mathcal{A})$, which is what model-categorical functorial factorisations give. | |
| 6 hours ago | comment | added | Leo Alonso | @Carl-FredrikLidgren Sorry, I meant section 2.7. Perhaps you are looking for an enhancement. I recall this was studied by Schnürer, see his arXiv:1507.08697. | |
| 7 hours ago | comment | added | Carl-Fredrik Lidgren | @LeoAlonso Are you sure you mean section 2.9? In both the copies I have, it only goes up to 2.7. Though, on a separate note, I don't think I remember Lipman discussing anything about functoriality, and since I'm interested in the approach using deformations, functoriality is important. If one excludes functoriality, then I think all my questions are answered by the contents of Kashiwara–Schapira. (Maybe that also clarifies what I want? I don't know) | |
| 7 hours ago | comment | added | Leo Alonso | @Carl-FredrikLidgren Let $\mathcal{A}$ be an abelian category. Let $\mathcal{Z} \subset K(\mathcal{A})$ the category of acyclic complexes as complexes in the homotopy category of complexes of $\mathcal{A}$. Then K-injectives is just $\mathcal{Z}^\perp$ and K-projectives is $^\perp\mathcal{Z}$, completely canonical. If you look for flasque, flat or in general acyclic resolution look at Lipman's "Notes on Derived Functors and Grothendieck Duality" in Springer Lecture Notes, no. 1960, specially section 2.9. | |
| 7 hours ago | comment | added | Carl-Fredrik Lidgren | @DenisT To be honest, I don't really have any particularly exotic things in mind, and the already mentioned methods are perfectly fine (e.g. that of Spaltenstein). The linked answer on K-flat resolutions is a good example of the kind of situation I'm talking about (where one is interested in getting the derived tensor product). Mainly, I just find the explanation there a bit unsatisfying, and was hoping there was something which didn't rely on such an explicit situation, if that makes sense. | |
| 7 hours ago | comment | added | Carl-Fredrik Lidgren | @LeoAlonso What I mean is that I can't start with some random/arbitrary choice for $\mathcal{C}'$. There are methods that work when you have projectives, or K-projectives, etc., but what I'm asking for is something that will work more generically. I'm prepared for such a thing not to exist, but I have a wish that it does. | |
| 7 hours ago | comment | added | Denis T | @Carl-FredrikLidgren It would be really helpful if you produce at least one specific situation of two categories-with-equivalences, and specific properties which you want from would-be derived functor you want from them. | |
| 8 hours ago | comment | added | Leo Alonso | I'm really puzzled at what you mean but not quite uniform. Spaltentenstein's method (or its simpler variant by Bökstedt-Neeman when there are enough projectiles) seem very reasonable and doesn't need the machinery of model categories. Also, the category of K-injectives is equivalent to the derived category. We can interpret functor resolutions as a Bousfield localization of the homotopy category. Isn't this uniform enough? | |
| 8 hours ago | comment | added | Carl-Fredrik Lidgren | I made an edit to add some more context/explanation, which also hopefully shows what kind of unreasonable thing I'm really asking for (but still hoping is possible in restricted but practically applicable scenarios). | |
| 8 hours ago | history | edited | Carl-Fredrik Lidgren | CC BY-SA 4.0 |
Added a bit more explanation about what's being asked/looked for.
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| 9 hours ago | comment | added | Carl-Fredrik Lidgren | The problem I have with the first part is that it assumes one has a model structure, and that one is working with functors suitably compatible with that model structure. In practice, this can be really troublesome to arrange for, regardless of whether one wants the model structure to have functorial factorizations (but on the other hand, it seems hard to arrange for anything). | |
| 14 hours ago | comment | added | Zhen Lin | In practice, model categories often come with functorial factorisations, so much so that some authors make it part of the definition. On the other hand, the functorial variant of the multiplicative system is much less commonly discussed. I think it might appear implicitly in the monograph of Dwyer, Hirschhorn, Kan, and Smith (which is where deformations were introduced). | |
| yesterday | comment | added | Carl-Fredrik Lidgren | Preferably, I'd want situations in which one can ensure that 1 and 2 can be done functorially, to the extent that one can realistically expect that to be possible. This is mostly because I'm interested in seeing what can be done if one is forced to use only the deformations approach; in reality, one can of course use something like Kashiwara–Schapira's results to sidestep this need entirely. | |
| yesterday | comment | added | Zhen Lin | As you say yourself, functoriality is hard to obtain. So what is your actual question? Do you want stronger versions of 1 or 2 that give functoriality? Or would you be satisfied with a variant of 3 that does not use functoriality? | |
| S yesterday | review | First questions | |||
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| S yesterday | history | asked | Carl-Fredrik Lidgren | CC BY-SA 4.0 |