Timeline for Using higher topos theory to study Cech cohomology
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14 events
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| Jan 20, 2023 at 22:55 | comment | added | Zhen Lin | No, I do not have a reference. But I'm sure someone does and will give you one if you ask a separate question specifically about it. At the most basic level this is a question about strictification: if we forget about coverages for a moment this reduces to showing that the homotopy $(\infty, 1)$-category of the category of presheaves valued in a model category $\mathcal{M}$ is equivalent to the $(\infty, 1)$-category of presheaves valued in the homotopy $(\infty, 1)$-category of $\mathcal{M}$. | |
| Jan 20, 2023 at 13:28 | comment | added | Markus Zetto | @Z.M Your comment also seems to be related to that (and you are of course right, Dold-Kan only works like this in the bounded case), isn't it? Making an object of the derived category of chain complexes of sheaves into a sheaf in the derived category seems to rely on this construction. | |
| Jan 20, 2023 at 13:23 | comment | added | Markus Zetto | Sorry for the late reply, I got a bit sick the last days. @ZhenLin Yes, it seems like this is what I am looking for - SAG 2.1.2.2 seems to go in that direction, is it a good idea to follow the proof there or should I look for a more specific reference? | |
| Jan 16, 2023 at 15:12 | comment | added | Zhen Lin | @MarkusZetto So you are asking for a proof that the derived category of chain complexes of sheaves is equivalent to the category of sheaves of objects in the derived category of chain complexes (inserting $\infty$ appropriately)? That is actually a non-trivial fact, but the direction you are interested in is the easy one. | |
| Jan 16, 2023 at 13:49 | comment | added | Z. M | I don't know any Dold–Kan correspondence for unbounded chain complexes. | |
| Jan 16, 2023 at 13:40 | comment | added | Z. M | The point is that you have to give an ($\infty$-categorical) definition of $R\Gamma$. There are some tricky point such as hypercompleteness. | |
| Jan 16, 2023 at 12:59 | comment | added | Markus Zetto | You are right, what I propose in my last comment is definitely wrong, as it is not enough to replace $A$ by an injective on each open, but has to replace the whole sheaf. But I still think $R \Gamma ( - , A)$ should be an $\infty$-sheaf in $D(\mathbb{Z})$, since this is essentially what the correspondence between Cech and sheaf cohomology in nlab 3.7 says. As long as we work over $\mathbb{Z}$, we can apply the Dold-Kan correspondence as in the first link and the statement should become, as you say, tautological; but for any ring I still think one should work in $D(R)$-valued $\infty$-sheaves? | |
| Jan 16, 2023 at 12:19 | comment | added | Zhen Lin | Ah. Actually, I think that definition probably does not work at all. You need to think of $A$ as an object in the derived category of chain complexes of sheaves. | |
| Jan 16, 2023 at 12:01 | comment | added | Markus Zetto | @ZhenLin Thanks for your comment! I am a bit confused, wouldn't taking sections of $A$ require that we first show that $A$ is an object of the topos? In other words, if we let $L: \operatorname{Ch}(\mathbb{Z}) \to D( \mathbb{Z} )$ be the localization functor into the derived $\infty$-category, we would have to show that $L \circ A$ is a sheaf. This does not seem clear, as $L$ doesn't preserve limits - but maybe I use the wrong definitions/ point of view, as you say. | |
| Jan 15, 2023 at 11:45 | comment | added | Zhen Lin | The first statement is basically a tautology, depending on your definitions. The point is that every covering sieve becomes a colimit diagram in the topos, so taking sections yields a sheaf. | |
| Jan 15, 2023 at 11:01 | history | edited | Markus Zetto | CC BY-SA 4.0 |
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| Jan 10, 2023 at 0:47 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Jan 9, 2023 at 22:16 | history | edited | YCor | CC BY-SA 4.0 |
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| Jan 9, 2023 at 21:40 | history | asked | Markus Zetto | CC BY-SA 4.0 |