Timeline for The étale topos of a scheme is the classifying topos of which groupoid?
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14 events
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| Apr 8, 2021 at 7:21 | comment | added | Ingo Blechschmidt | @user20948: Yes! If $\mathcal{E}$ classifies a theory $\mathbb{T}$, the slice $\mathcal{E}/X$ classifies "$\mathbb{T}$ adjoined by a constant $x:X$". This makes sense because the object $X$ of $\mathcal{E}$ can be obtained from the generic $\mathbb{T}$-model by geometric constructions (small colimits, finite limits). Hence a sort "$X$" together with appropriate auxiliary sorts, function symbols and axioms can be added to $\mathbb{T}$ in such a way that the interpretation of this sort "$X$" in any $\mathcal{E}$-topos $\mathcal{F}$ is precisely (the pullback to $\mathcal{F}$ of) the object $X$. | |
| Mar 20, 2021 at 9:08 | comment | added | user20948 | @IngoBlechschmidt Is it in general true that if a topos classifies a theory, then for each object, the slice topos classifies a relative theory with respect to that object? | |
| Jul 3, 2018 at 8:33 | comment | added | W.Rether | @TimCampion How do you see that a set of models can be that of models with bounded cardinality (point 1 of your answer)? | |
| Jul 2, 2018 at 21:46 | comment | added | W.Rether | Of course the internal point of view is much more natural and clear here! Related question: does anyone know how the universal henselian $R$-algebra looks like? This is in order to rebuild Butz and Moerdijk's proof using Tim's suggestion. Cfr. also this other article by Butz and Moerdijk which constructs the groupoid like we were discussing about. | |
| Jul 2, 2018 at 9:05 | comment | added | Ingo Blechschmidt | I can confirm your guesses regarding what is classified by the étale topos. A reference in the case of $X = \mathrm{Spec}(R)$ is a very nice paper by Mathieu Anél. What Tim is saying doesn't have an external meaning, since $\mathcal{O}_S$ is not a ring, but a ring object in $\mathrm{Sh}(S)$. However, it has from the internal point of view of $\mathrm{Sh}(S)$; it correctly characterizes the étale topos in the world of toposes over $\mathrm{Sh}(S)$. This doesn't seem to be explicitly written down anywhere. It might be the case that my thesis comes closest. | |
| Jul 1, 2018 at 21:08 | comment | added | Tim Campion | Probably, but I don't actually know enough algebraic geometry to say for sure. | |
| Jul 1, 2018 at 16:25 | comment | added | W.Rether | Anyway, if $X={\rm Spec}\ R$, this could be the theory of strictly henselian $R$-algebras? Or is this true only in the case $R=k$ field? | |
| Jul 1, 2018 at 16:08 | comment | added | Tim Campion | I'm actually not clear on what theory exactly is classified by the etale topos $\mathcal E_S$ over a base scheme $S$. I think it might be "the theory of strictly henselian algebras over $\mathcal O_S$", whatever that means. But whatever it is, the topological groupoid one gets will still be the groupoid of points of $\mathcal E_S$, with some appropriate topology. | |
| Jul 1, 2018 at 13:34 | comment | added | W.Rether | This would really seem to work! I just don't understand the last sentence: what do you mean by "this"? The objects of the groupoid? And if yes, in what sense? Thanks. | |
| Jun 21, 2018 at 0:12 | comment | added | Tim Campion | Tentatively, that seems right -- this should be the topological groupoid of ($I$-enumerated) strictly henselian local rings (with a cardinality bound) and isomorphisms, with topology such that for every definable set $D$ in the coherent theory of strictly henselian rings (in particular, this is an intuitionistic theory), the set of enumerated models $f: I\to R$ such that $f(i) \in D$ is an open set. Working over a base scheme $S$, this should all be internal to sheaves on $S$, I suppose. | |
| Jun 20, 2018 at 15:07 | comment | added | W.Rether | Thanks. Yes, the construction is pretty clear in abstract. The point of view that you enlighten could perhaps suggest that, knowing the theory that the étale topos classifies (those "strict local rings"), one could move from that and construct the groupoid...? | |
| Jun 19, 2018 at 4:56 | comment | added | Tim Campion | I should point out that I think Butz and Moerdijk explain this pretty clearly and in a bit more detail in section 2 of their paper. | |
| Jun 19, 2018 at 4:52 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 155 characters in body
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| Jun 19, 2018 at 4:47 | history | answered | Tim Campion | CC BY-SA 4.0 |