Timeline for Derived functors out of an unbounded derived $\infty$-category
Current License: CC BY-SA 4.0
10 events
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| Jul 20, 2022 at 16:38 | comment | added | Tomo | To the first sentence in the last comment: I believe this is what I was alluding to in my original question; using e.g. stacks.math.columbia.edu/tag/06XN, we see that these systems are in fact essentially constant, so their (co)limits are representable. I was interested in Deligne’s formula because it allow one to use well-known, simple facts about triangulated categories to compute derived functors. I’m still confused about the importance of the cofibrations with cofiber isomorphic to a complex of flat modules mentioned above, but this discussion has clarified much for me; thank you! | |
| Jul 20, 2022 at 7:16 | comment | added | D.-C. Cisinski | Note that the description of derived functors via resolutions is also a proof that these possibly large colimits are in fact representable in our universe. Model structures for cochain complexes in Grothendieck categories are explained in Theorem 2.5 in this paper: intlpress.com/site/pub/pages/journals/items/hha/content/vols/… (see Remark 2.12 to see that what we call descent structures always exist). Cor. 3.5 and Prop. 3.7 deal with tensor structures. | |
| Jul 20, 2022 at 7:00 | comment | added | D.-C. Cisinski | There is a similar story with flat resolutions. The proofs that we can express all this via suitable model structures usually also give you the tools to deal with size issues (the fact that we can apply the small object argument is directly related to that). | |
| Jul 20, 2022 at 6:56 | comment | added | D.-C. Cisinski | The smallness up to cofinality comes from the fact that, in the 1-category of cochain complexes, the subcategory of quasi-isomorphisms is accessible (being the inversible Image of the accessible subcategory of isomorphisms through the small family of accessible functors $H^n$). It is not difficult but writing it down property in a convincing manner requires some effort. | |
| Jul 20, 2022 at 1:15 | comment | added | Tomo | I see why this class of cofibrations satisfies the required factorization condition (it follows from the argument of stacks.math.columbia.edu/tag/0G6M), but how does it help us? D(A) has all small limits so is it a matter of smallness of the indexing category if A isn’t Grothendieck? i.e. is the point that this calculus of fractions is small (while the one in the post may not be) so that the limits defining $\otimes^L$ are guaranteed to exist? When A is Grothendieck, is it easy to see the claim that the indexing categories in the post admit cofinal maps from small categories? | |
| Jul 19, 2022 at 9:35 | comment | added | D.-C. Cisinski | The left calculus of fractions induced by these cofibrations is what allows you to prove that the derived tensor product exists. If $A$ is Grothendieck, you can find cofinal functors with small domain with values in the indexing categories, which is why derived functors exist whenever the target has small (co)limits. Otherwise, everything we say above works for any small abelian category, if we consider derived functors with values in a (co)complete $\infty$-category. | |
| Jul 19, 2022 at 9:33 | comment | added | D.-C. Cisinski | If you have enough flat objects in $A$ (i.e. for any object $a$ in $A$, there is an epimorphism $x\to a$ with $x$ flat (in the sense the $x\otimes(-)$ is exact), then we can consider a class of cofibrations in $K(A)$: the maps with cofiber isomorphic in $K(A)$ (i.e. cochain homotopy equivalent) to a complex of flat objects in $A$ (no need of complicated conditons such as $K$-flatness: these are only there to get Quillen model structures and we do not care about that here). | |
| Jul 19, 2022 at 9:29 | comment | added | D.-C. Cisinski | We may consider several structures of categories with weak equivalences and fibrations on $K(A)$ (itself the localization of the $1$-category of cochain complexes by cochain homotopy equivalences, which is a model category if one takes degreewise split mono's as cofibrations): the weak equivalences are quasi-isomorphisms, and the fibrations are all maps (we can also take cofibrations to be all maps). The corresponding calculi of fractions is then the one you describe in the second part of your post. | |
| Jul 19, 2022 at 1:04 | history | edited | Tomo | CC BY-SA 4.0 |
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| Jul 18, 2022 at 21:05 | history | answered | Tomo | CC BY-SA 4.0 |