Timeline for what is $\mathrm{Bun}(G)$?
Current License: CC BY-SA 3.0
16 events
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| May 1, 2017 at 2:01 | comment | added | nfdc23 | @JoshuaGrochow: The purpose of Artin's criteria is to precisely to provide a list of checkable conditions sufficient to ensure existence of the required "scheme charts" (e.g., etale or smooth cover by a representable functor, in a suitable sense of such kind of "cover"). For this, the geometric substance of the definition of the given functor/groupoid needs to be understood. One can also sometimes get by without Artin's criteria, using that other things are known to be algebraic spaces/stacks. But it is technically important to recognize that one is not trying to make a ringed space. | |
| Apr 30, 2017 at 21:05 | comment | added | Joshua Grochow | @Qfwfq: No need to apologize - thanks for the helpful comments! | |
| Apr 30, 2017 at 20:25 | comment | added | Qfwfq | @JoshuaGrachow: see also the very good answer by Angelo here mathoverflow.net/questions/77110/… (if you only have a FOP -i.e. don't have algebraicity- then it's more difficult to make "geometry" with it) and by TorstenEkedahl here: mathoverflow.net/questions/56962/… . Sorry for the long chain of comments! | |
| Apr 30, 2017 at 19:46 | comment | added | Qfwfq | (...) On the other hand: A) you can also say an analogous thing about, say, $\mathrm{Spec}(R)\mapsto \mathrm{GL}_n(R)$, or $S\mapsto\mathrm{Hilb}_X^n(S)$, B) as soon as you want to prove something is algebraic, you have to construct an atlas i.e. a presentation as a groupoid in schemes, C) also, many moduli problems admit a (not very intrinsic) description as global quotient stacks: here you don't need the F.o.P.. | |
| Apr 30, 2017 at 19:46 | comment | added | Qfwfq | @JoshuaGrochow: 2) yes, that was my point. - 1) I agree, one important feature of the FOP is that when you have a moduli problem what Nature gives you is indeed a functor/pseudofunctor/fiberedcategory, and you can already think of it "tautologically" as some sort of space (no need of representability proofs!). (...) | |
| Apr 30, 2017 at 18:51 | comment | added | Joshua Grochow | @Qfwfq: Thanks! I agree with the geometric viewpoint, and I appreciate your comments about the functorial one. But how do you reconcile those with this answer, which seems to be saying that part of the value is that you can often easily define a functor and do geometry with it, but then if you can show that it has particularly nice geometry (e.g. is representable, is a variety, etc.) then that's icing on the cake? Is your point that, functorial viewpoint or not, stacks/alg. spaces allow more permissive notions of gluing than schemes, so they are often easier to construct, etc? | |
| Apr 30, 2017 at 14:54 | comment | added | Qfwfq | (...) if one was willing to, one could entirely talk of group varieties with the language of functor of points, and of group schemes with the language of ringed spaces (though, in the latter case, I doubt one would gain anything interesting). | |
| Apr 30, 2017 at 14:51 | comment | added | Qfwfq | (...) so, if what one really wants to do is dealing with the quotients intrinsically, she has to allow for the possibility of "changing presentation" and this is taken account in the definition of Morita morphism. Now, the functorial viewpoint is very powerful and can (and is) used to introduce stacks etc from scratch, but it's totally orthogonal to the motivation behind algebraic spaces and stacks. Even for group schemes, some books/notes seem to suggest that the main difference with usual algebraic groups (group varieties) is the functorial approach; this is wrong: (...) | |
| Apr 30, 2017 at 14:46 | comment | added | Qfwfq | (...) The point is, rather, that you want to patch together local (affine in the case of AG) data to get a well behaved "space" of some sort; for example, the patching could be done by an étale equivalence relation, or by quotienting an existing scheme by a higly non-free action. It turns out such a patching cannot be done well in the category of schemes, but it can be performed in groupoids (in schemes). But groupoids are not very "intrinsic", in the same sense as an action is not an intrinsic datum attached to its quotient: (...) | |
| Apr 30, 2017 at 14:46 | comment | added | Qfwfq | @JoshuaGrochow: I've upvoted this answer too - but the "functorial viewpoint" is not the point of stacks, pretty much as it's not the point of algebraic spaces (at least as long as you consider geometric stacks and so on, i.e. things with an atlas) or even schemes. (...) | |
| Apr 28, 2017 at 15:01 | vote | accept | john mangual | ||
| Apr 28, 2017 at 4:20 | comment | added | Joshua Grochow | This would also serve as an answer to some much more general question about the relationship between schemes (which I think of as "level n" of AG abstraction for some small n, affine algebraic varieties over C being level 0) and stacks, algebraic spaces, and the functorial viewpoint on AG ("level n+1")...and it's great. I'd up vote multiple times if I could! | |
| Apr 28, 2017 at 3:24 | history | edited | nfdc23 | CC BY-SA 3.0 |
fixed some typos
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| Apr 27, 2017 at 15:59 | history | edited | nfdc23 | CC BY-SA 3.0 |
added 1758 characters in body
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| S Apr 27, 2017 at 15:18 | history | answered | nfdc23 | CC BY-SA 3.0 | |
| S Apr 27, 2017 at 15:18 | history | made wiki | Post Made Community Wiki by nfdc23 |