Timeline for Basic questions about stacks
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 14, 2011 at 16:56 | comment | added | Harry Gindi | @Qiaochu: Those are roughly "stacks of $\infty-groupoids$". The thing you're attempting to model would be a stack of $\infty$-categories. Assuming we have a cartesian fibration $X\to N(C)$ over the nerve of an ordinary category $C$ (the $(\infty,1)$ analog of a fibered category), it suffices to show that it satisfies descent on the core groupoid. Anyway, regarding your question about slices, see the first or second chapter of Monique Hakim's thesis in the section about 2-stacks over the 2-category of toposes. | |
| Mar 14, 2011 at 16:51 | comment | added | Harry Gindi | I believe that Giraud introduced a legitimate notion of a 2-stack, and it should be in his book, cohomologie non-abelienne. It is also reproduced in the introductory chapter to Monique Hakim's thesis, topos annelés et schémas relatifs. | |
| Mar 14, 2011 at 16:49 | comment | added | Qiaochu Yuan | I don't actually know anything about stacks, but here's the relevant nLab page: ncatlab.org/nlab/show/infinity-stack | |
| Mar 14, 2011 at 16:41 | comment | added | Greg Muller | Hmm, this would be algebraic stacks over a fixed site $S$, and then a 'family of algebraic stacks over $X\in S$' would be a stack over the slice category $X\backslash S$ (ie, the category of maps into $X$)? So then the site of this moduli problem is really the 2-category of slice categories in $S$, and so the moduli space of algebraic stacks in $S$ should be a stack over a 2-site, I would guess. Is that what is meant by a 2-stack? | |
| Mar 14, 2011 at 16:36 | history | edited | Greg Muller | CC BY-SA 2.5 |
added 2 characters in body
|
| Mar 14, 2011 at 15:45 | comment | added | Qiaochu Yuan | If I wanted to study the "moduli space of algebraic stacks" satisfying certain properties, would it be enough for me to stay in the language of stacks or would I have to define whatever a $2$-stack is? The nLab tells me that such things exist, anyway. | |
| Mar 14, 2011 at 15:39 | comment | added | Greg Muller | Ok, it works for the site of manifolds too. Anything on which it makes sense to talk about a principal $G$-bundle. | |
| Mar 14, 2011 at 15:38 | history | edited | Greg Muller | CC BY-SA 2.5 |
edited body
|
| Mar 14, 2011 at 15:37 | comment | added | Martin Brandenburg | I'm not talking about schemes in question 2. | |
| Mar 14, 2011 at 15:29 | history | answered | Greg Muller | CC BY-SA 2.5 |