Timeline for When (or why) is a six-functor formalism enough?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 14, 2023 at 16:50 | vote | accept | Will Sawin | ||
| Mar 13, 2023 at 8:53 | history | edited | Matthieu Romagny | CC BY-SA 4.0 |
added "?"
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| Mar 13, 2023 at 7:34 | answer | added | Peter Scholze | timeline score: 30 | |
| Mar 11, 2023 at 19:53 | comment | added | LSpice | @PeterScholze, there's surely no better answer to what was meant by a sentence in your notes than your own, so maybe you could promote your comments to an answer? | |
| Mar 11, 2023 at 17:06 | comment | added | Peter Scholze | That said, there is definitely room for some kind of algorithm or graphical calculus or such that would help one check expected commutative diagrams involving all six functors. | |
| Mar 11, 2023 at 16:52 | comment | added | Peter Scholze | I'm not sure what a "theorem" justifying the assertion would be, as the "classical" notion of a 6-functor-formalism is somewhat ill-defined, consisting of some slightly random collection of expected isomorphisms, coherences, and maps. | |
| Mar 11, 2023 at 16:51 | comment | added | Peter Scholze | What I meant is just that passing to adjoint functors has extremely good functoriality properties (they are unique up to contractible choice and "functorial"), and that all the "formulas" one knows that involve some of those right adjoints can be deduced (like $f^! Hom(A,B)=Hom(f^*A,f^!B)$). So far I have not seen expected coherences that did not follow automatically from the given datum. This includes, notably, the identifications $f^!=f^*$ for etale $f$, and $f_!=f_*$ for proper $f$ (as well as the map $f_!\to f_*$ for separated $f$). | |
| Mar 11, 2023 at 16:11 | history | asked | Will Sawin | CC BY-SA 4.0 |