Timeline for Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology
Current License: CC BY-SA 4.0
9 events
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| Apr 3, 2020 at 22:25 | comment | added | dorebell | If you care mostly about sheaf cohomology (as opposed to the general theory of derived functors, which I'd argue is essentially the same thing as the general theory of injective resolutions etc or of derived categories), then Cech cohomology is pretty concrete and conceptually simple. (The fact that these have long exact sequences is then a theorem rather than a definition). | |
| Apr 3, 2020 at 17:41 | comment | added | Dmitri Pavlov | Theoretically, some time in the future, homotopy type theory may be able to provide enough tools to state and prove comparison theorems without referring to injective resolutions, triangulated categories, or similiar tools. But right now, in the current state of the field, there is hardly anything simpler than injective resolutions, model categories, triangulated categories, stable ∞-categories, etc., all of which inevitably pass through some form of injective or projective resolutions. | |
| Apr 3, 2020 at 17:36 | comment | added | Denis Nardin | @DmitriPavlov Yeah, I decided that it was not a suitable approach, or I'd have written an answer (although there's no need of injective resolutions to define the derived ∞-category of an abelian category, and it's arguably not even the most natural approach). That said, I don't think you can go much lower tech than $\delta$-functors... | |
| Apr 3, 2020 at 17:30 | comment | added | Dmitri Pavlov | @DenisNardin: If the OP already consider injective resolutions to have "too many constructions", then suggesting stable ∞-categories is a bit weird. Recall that Lurie's definition of the derived ∞-category (Definition 1.3.2.7 in Higher Algebra) uses projective resolutions, for example. So you suggested approach definitely violates OP's requirement of not using resolutions. | |
| Apr 3, 2020 at 9:11 | comment | added | Arrow | Thanks for the link. I'm happy to take basic topology and category theory for granted, but I don't want to assume something like the Quillen equivalence between simplicial sets and topological spaces. Sorry I can't be more specific - I just have no idea how the $\infty$-category approach looks like. | |
| Apr 3, 2020 at 9:04 | comment | added | Denis Nardin | See my answer here for a quick rundown of stable ∞-categories: mathoverflow.net/questions/344219/… . Admittedly you need to develop some theory before working with them. And what do you mean by "write down all details"? Starting from ZFC set theory (I assume not)? What kind of things are you willing to take for granted? | |
| Apr 3, 2020 at 9:01 | comment | added | Arrow | I don't know what stable $\infty$-categories are but I have nothing against them. My fear is that they don't admit a "reasonably quick path" to concrete comparison theorems, but I'd be happy to discover otherwise. I should emphasize that I want to be able to write down all the details. | |
| Apr 3, 2020 at 8:50 | comment | added | Denis Nardin | Does your desire to avoid triangulated categories extend to stable ∞-categories? If so I cannot see how you could avoid $\delta$-functors. | |
| Apr 3, 2020 at 8:34 | history | asked | Arrow | CC BY-SA 4.0 |