Do higher etale homotopy groups of spectrum of a field always vanish?Fibration sequences in étale homotopy...
Do higher etale homotopy groups of spectrum of a field always vanish?
Fibration sequences in étale homotopy theory arising from geometric fibresThe etale fundamental group of a fieldEtale topos as a classifyng topos ?“Étalification” of a schemealternate interpretations of Galois action on Tate modulequestion about the induced homomorphism of etale fundamental groupsClassifying Spaces and Eilenberg-Maclane objects in the category of simplicial ringsHow is a universal deformation ring in a 1-dimensional DM stack related to the complete etale local ring of coarse moduli scheme?Galois descent for absolute Galois groupLines in upper half-space
$begingroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
New contributor
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$begingroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
New contributor
$endgroup$
add a comment |
$begingroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
New contributor
$endgroup$
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?
If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.
I am not sure what happens for general fields.
ag.algebraic-geometry etale-covers
ag.algebraic-geometry etale-covers
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New contributor
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asked yesterday
rorirori
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1 Answer
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$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
$endgroup$
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
yesterday
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
yesterday
1
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
yesterday
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
yesterday
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
yesterday
add a comment |
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$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
$endgroup$
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
yesterday
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
yesterday
1
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
yesterday
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
yesterday
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
yesterday
add a comment |
$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
$endgroup$
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
yesterday
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
yesterday
1
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
yesterday
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
yesterday
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
yesterday
add a comment |
$begingroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
$endgroup$
The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.
Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.
answered yesterday
Denis NardinDenis Nardin
8,64723562
8,64723562
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
yesterday
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
yesterday
1
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
yesterday
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
yesterday
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
yesterday
add a comment |
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
yesterday
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
yesterday
1
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
yesterday
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
yesterday
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
yesterday
1
1
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
yesterday
$begingroup$
No need for $infty$-categories here :-)
$endgroup$
– David Roberts
yesterday
2
2
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
yesterday
$begingroup$
@DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
$endgroup$
– Denis Nardin
yesterday
1
1
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
yesterday
$begingroup$
I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
$endgroup$
– David Roberts
yesterday
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
yesterday
$begingroup$
@DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
$endgroup$
– Denis Nardin
yesterday
2
2
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
yesterday
$begingroup$
I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
$endgroup$
– David Roberts
yesterday
add a comment |
rori is a new contributor. Be nice, and check out our Code of Conduct.
rori is a new contributor. Be nice, and check out our Code of Conduct.
rori is a new contributor. Be nice, and check out our Code of Conduct.
rori is a new contributor. Be nice, and check out our Code of Conduct.
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