Timeline for Does the language of fibred categories gives the commutativity of the diagrams in Residues and Duality?
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13 events
| when toggle format | what | by | license | comment | |
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| Sep 27, 2021 at 8:51 | vote | accept | Gabriel | ||
| Sep 27, 2021 at 8:43 | comment | added | D.-C. Cisinski | Well, if you can prove the formula without refering to how the functor was constructed, this solves the problem of proving compatibilities appearing in the constructions. Many difficulties appear because we sometines mix the definition/construction with the actual computation (all of these aspects being important). For classical constructions, a rather good balance is reached in Neeman's treatment of Grothendieck duality made simple: arXiv:1806.03293 | |
| Sep 26, 2021 at 15:04 | comment | added | Gabriel | Thank you for the references. I'm still not convinced that ignoring the fact that some functors may be derived from non-derived functors solves all the problems, though. For example, the compatibility of the trace map with base change (diagram 1.1.3 in B. Conrad's Grothendieck Duality and Base Change) still seems hard. | |
| Sep 26, 2021 at 12:01 | comment | added | D.-C. Cisinski | References are bookstore.ams.org/surv-221 (here for free: people.math.harvard.edu/~gaitsgde/GL/Vol1.pdf) The on going work of Harpaz et al (arXiv:2006.14495 and arXiv:2012.04537) solves many of the issues raised by the work of Gaitsgory and Rozenblyum (check though a pdf version of their book how many times they write "we could not find a proof in the literature"). | |
| Sep 26, 2021 at 11:56 | comment | added | D.-C. Cisinski | What I meant is: I only consider functors defined at the level of derived categories and do not consider the problem of knowing whether they are derived from non-derived functor or not as part of the six functor formalism. | |
| Sep 26, 2021 at 11:07 | history | bounty ended | CommunityBot | ||
| Sep 26, 2021 at 10:21 | comment | added | Gabriel | Do you have some references for Gaitsgory and Rozenlbyum work and for "the axiomatization of the six functor formalism is indeed a way to make this precise and to solve all coherence issues of interest"? | |
| Sep 26, 2021 at 10:19 | comment | added | Gabriel | I don't understand what you mean by "I would only consider derived functors". Often, some functors on six functor formalisms are not derived functors (the $f^!$ in Verdier or Grothendieck duality, for example). Moreover, I indeed agree that defining properly what are those compatibilities is challenging. | |
| Sep 26, 2021 at 9:59 | comment | added | D.-C. Cisinski | Finally, there are more coherence issues raised by higher category theory, if we consider the $(\infty,1)$-categorical version of derived categories. In the wok of Gaitsgory and Rozenlbyum, there is a description of the six functor formalism using the language of symmetric monoidal $(\infty.2)$-categories which certainly subsumes every other coherence issue from Residues and Duality. Gaitsgory and Rozenlbyum gave precise statements, and proving them in full is part of current research in higher category theory (already at a rather advanced stage). | |
| Sep 26, 2021 at 9:55 | comment | added | D.-C. Cisinski | The ingredients do not go beyond cartesian fibrations as in Giraud's thesis, though. | |
| Sep 26, 2021 at 9:55 | comment | added | D.-C. Cisinski | The kind of diagrams discussed above (non-derived vs derived functors) are not part of the six functor formalism: I consider this as obvious, and I would only consider derived functor so that such question would not even occur. Your question is not well defined, then: what would be "all compatibilities": being able to define precisely what that means is already chalenging. That said, the axiomatization of the six functor formalism is indeed a way to make this precise and to solve all coherence issues of interest. | |
| Sep 26, 2021 at 9:32 | comment | added | Gabriel | Dear @Denis-Charles Cisinski, I'm not sure if what you said answers my question; I already knew that my diagram commutes. What I want to know is if there's some way to obtain all compatibility diagrams in six functor formalisms at once, without needing to do all of them one by one. | |
| Sep 26, 2021 at 7:17 | history | answered | D.-C. Cisinski | CC BY-SA 4.0 |