Timeline for What is the point of pointwise Kan extensions?
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21 events
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Jul 29 at 16:54 | comment | added | varkor | @TimCampion: do you have a reference for (5)? (It's easy to prove, but it would be useful to be able to cite something.) | |
Apr 12, 2018 at 19:02 | history | edited | Tim Campion | CC BY-SA 3.0 |
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Apr 12, 2018 at 18:55 | vote | accept | Tim Campion | ||
Oct 19, 2015 at 12:04 | history | edited | Tim Campion | CC BY-SA 3.0 |
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Oct 19, 2015 at 12:04 | answer | added | Tim Campion | timeline score: 6 | |
Oct 17, 2015 at 14:13 | answer | added | Daniel Barter | timeline score: 7 | |
Oct 17, 2015 at 13:50 | answer | added | Tim Campion | timeline score: 11 | |
Oct 17, 2015 at 13:38 | history | edited | Tim Campion | CC BY-SA 3.0 |
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Oct 9, 2015 at 2:12 | comment | added | pro | @TimCampion I'm not sure, I've never really used the formal theory of Kan extensions, so it may very well that in cases where I thought the formula was fundamental I could've got away with the more general Kanexts theory. If you come across anyone from the Gaitsgory derived Langlands school, make sure to ask them. | |
Oct 9, 2015 at 2:02 | comment | added | Tim Campion | @pro I guess this is kind of silly of me to ask, but what's an example of something you can do with this better understanding of what a quasi-coherent modules are that you couldn't do just from knowing it's a Kan extension? | |
Oct 7, 2015 at 13:37 | comment | added | pro | I guess I didn't really need to go full derived. One can run the same story for ordinary rings R and the abelian category R-Mod. The issue is that for an arbitrary presheaf F, that construction does not yield an abelian category in general (while the derived counterpart spits out a stable category, which is what you want). | |
Oct 7, 2015 at 13:35 | comment | added | pro | but I don't really understand what this means. On the other hand, given that it's ptwise Kan extension, we can use the colimit formula. Unpacking everything (unless I'm confusing myself), this boils down to saying that a quasi-coherent sheaf M on F (or, rather, the derived version of that) is a gadget which to every map Spec R ---> F (or, by Yoneda, every "element" of F(R)) assigns an R-module M(Spec R--->F) in a coherent way. | |
Oct 7, 2015 at 13:32 | comment | added | pro | @TimCampion silly example from DAG: let's work over the complex numbers, if R is a cdga (say in negative degrees) there is an $\infty$-cat of R-modules. Say we are interested only in the maximal groupoid (because we want to do stacks) and call it R-Mod (it's an $\infty$-groupoid, ie a space). The construction extends to a whole functor cdga --> Spaces. Now, say you have an arbitrary prestack F on cdga's (ie a functor cdga --> Spaces), how do you define the category of quasi-coherent modules over it? Well, it's a Kan extension of the previous functor along the inclusion cdga--> Fun(cdga,Spaces) | |
Oct 7, 2015 at 3:55 | comment | added | Tim Campion | @pro On second thought, if you could give a specific example, I think that would qualify as the sort of thing I'm looking for. | |
Oct 7, 2015 at 3:30 | comment | added | Tim Campion | @pro I suppose I'm worried that this is the answer. Since pointwise Kan extensions are defined at a high level of abstraction, I'm hoping that there's a similarly abstract story about their significance. It's not guaranteed that my hopes will be fulfillable. | |
Oct 7, 2015 at 3:28 | comment | added | Tim Campion | @ViditNanda :) I added a paragraph with the Kelly quote I was thinking of. One of the more surprising instances of this phenomenon (emphasized by Emily Riehl in Categorical Homotopy Theory) is the fact that derived functors, when computed via functorial resolutions, are not only pointwise, but absolute Kan extensions (preserved by any functor whatsoever) -- so they can be computed via the (co)limit formula even though derived categories don't have many (co)limits. | |
Oct 7, 2015 at 3:26 | comment | added | pro | circular comment: isn't the point of pointwise Kan extensions precisely the fact that they are pointwise? I mean, the colimit formula is for me really useful. Not sure if I can pinpoint a precise example, but when I was learning some basics of "functorial" algebraic geometry, having a formula to compute Kan extensions was important (both for understanding and computing). | |
Oct 7, 2015 at 3:25 | history | edited | Tim Campion | CC BY-SA 3.0 |
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Oct 7, 2015 at 3:21 | comment | added | Vidit Nanda | "I've also heard it claimed that all mathematically important Kan extensions are pointwise." [Citation Needed] | |
Oct 7, 2015 at 3:19 | history | edited | Tim Campion | CC BY-SA 3.0 |
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Oct 7, 2015 at 3:10 | history | asked | Tim Campion | CC BY-SA 3.0 |