Timeline for What does the Zariski topos of $\mathbb{P}^1$ classify?
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10 events
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May 16, 2017 at 22:02 | comment | added | Ingo Blechschmidt | @h__: I'd like to credit you in my PhD thesis, since your question prompted a short section of it. Please contact me by mail at [email protected] with your realname if you feel comfortable with this. | |
May 13, 2017 at 21:06 | answer | added | Ingo Blechschmidt | timeline score: 11 | |
Jan 2, 2017 at 4:25 | comment | added | h__ | @ChristopherTownsend Check out stacks.math.columbia.edu/tag/020N. By Zariski topos over $S$ I mean $\text{Sh}((\text{Sch}/S)_{\text{Zar}}) $. When $S = \text{Spec}\mathbb{Z}$ this is just the category of sheaves on big Zariski site of all schemes. | |
Jan 1, 2017 at 22:22 | comment | added | Christopher Townsend | I am trying to understand your question. The Zariski topos is a topos and it classifies local rings. So I don't understand what 'over SpecZ' bit means. | |
Jan 1, 2017 at 9:56 | comment | added | Will Sawin | I don't think there's any particular reason to expect a description in terms of a logical theory to be the best tool to see if the two functors are isomorphic. | |
Jan 1, 2017 at 8:52 | comment | added | h__ | @WillSawin Yes of course. One can use this for functors represented by projective schemes. But that's very ad hoc. For example it's hard to see if two functors are isomorphic this way. | |
Jan 1, 2017 at 8:37 | comment | added | Will Sawin | It seems to me that one can do a general projective variety by the same trick with a locally free module of rank $1$ - one has an invertible module and $n$ elements such that at least one of the elements invertible and such that the $n$ elements satisfy certain homogenous relations in some tensor powers of the model. This handles algebraic curves, at least. | |
Jan 1, 2017 at 8:16 | comment | added | h__ | @WillSawin Here by "classify" I mean "how does it serve as a classifying topos of some particular logic theory" so I'd like to see a few axioms for it. (e.g. local rings can be described by the axioms of rings + that for all x, either $x$ or $1-x$ is invertible.) The axioms would look like some extra data on $R$. Here I guess $\mathbb{P}^1$ is easy, as it's homogeneous, but for general X it won't be so easy, for reasons similar to that people usually don't have a good way to describe a general scheme as a representable functor. | |
Jan 1, 2017 at 6:43 | comment | added | Will Sawin | Because it's a local construction, the difference between the affine case and the general case shouldn't matter much. I think in general, the big Zariski topos of $X$ classifies pairs of a local ring $R$ and an $R$-point of $X$. What is bad about your description of $\mathbb P^1$? | |
Dec 31, 2016 at 21:26 | history | asked | h__ | CC BY-SA 3.0 |