Timeline for Derived functors out of an unbounded derived $\infty$-category
Current License: CC BY-SA 4.0
9 events
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| Jul 13, 2022 at 10:04 | comment | added | D.-C. Cisinski | In the comment above, the reference to 7.3.16 is typo: should be 7.2.16 | |
| Jul 12, 2022 at 21:30 | vote | accept | Tomo | ||
| Jul 12, 2022 at 19:59 | comment | added | D.-C. Cisinski | @DenisNardin In fact Corollary 7.5.17 says that, in general, in the case of ∞-category with weak equivalences and fibrations, the right derived functor à la Deligne always coincides with the right derived functors of the restriction to fibrant objects, and Lemma 7.5.24 tells us that a functor which sends weak equivalences to isomorphisms has a trivial derived functor. This explains why we have derived functors using resolutions (7.5.25). Derived functors defined using resolutions are in fact absolute (7.5.25.3) and this explains why we have derived adjunctions (Theorem 7.5.30). | |
| Jul 12, 2022 at 19:31 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 436 characters in body
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| Jul 12, 2022 at 14:08 | comment | added | Denis Nardin | @D.-C.Cisinski Ah sorry, I was looking for it in the wrong section (7.5), I guess it could be helpful to add the reference to 7.2.9 in this answer so that it is more apparent where to look for it... | |
| Jul 12, 2022 at 12:32 | comment | added | D.-C. Cisinski | To be more precise: Deligne's formula is essentially a combination of Corollary 7.2.9 and Theorem 7.3.16 in my book (in the case of Verdier quotients, we have a filtered colimit, so that this formula is even nicer). Paragraph 7.5.25 constructs derived functors using resolutions and compares with Deligne's formula (in order to prove the expected universal property). | |
| Jul 12, 2022 at 12:26 | comment | added | D.-C. Cisinski | @DenisNardin I do define in my book derived functors as Kan extensions along the localization of any marked ∞-category. The general theory of derived functor simply is the general theory of Kan extensions. The case of ∞-categories with weak equivalences and fibrations covers at the very least cochain complexes in an abelian category, and the point is to construct these Kan extensions using resolutions. Deligne's formula is a variation on the fact that such derived functors are pointwise, so that we can understand them after embedding the codomain in a suitable presheaf category. | |
| Jul 12, 2022 at 9:39 | comment | added | Denis Nardin | While this is a great reference, I don't think it covers the general notion of derived functors (it always assumes that the source is an $\infty$-category with weak equivalences and fibrations), and in particular it does not prove the formula asked by the OP (although it does prove it in this special case, where it is substantially simpler) | |
| Jul 12, 2022 at 0:09 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |