Do higher etale homotopy groups of spectrum of a field always vanish?












4












$begingroup$


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







share|cite|improve this question









$endgroup$

















    4












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      4












      4








      4


      3



      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Mar 5 at 18:47









      rorirori

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          11












          $begingroup$

          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.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            Mar 5 at 19:40






          • 3




            $begingroup$
            @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
            Mar 5 at 19:41








          • 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
            Mar 5 at 21:36






          • 2




            $begingroup$
            @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
            Mar 5 at 21:38








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            Mar 5 at 21:44












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          11












          $begingroup$

          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.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            Mar 5 at 19:40






          • 3




            $begingroup$
            @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
            Mar 5 at 19:41








          • 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
            Mar 5 at 21:36






          • 2




            $begingroup$
            @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
            Mar 5 at 21:38








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            Mar 5 at 21:44
















          11












          $begingroup$

          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.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            Mar 5 at 19:40






          • 3




            $begingroup$
            @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
            Mar 5 at 19:41








          • 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
            Mar 5 at 21:36






          • 2




            $begingroup$
            @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
            Mar 5 at 21:38








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            Mar 5 at 21:44














          11












          11








          11





          $begingroup$

          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.






          share|cite|improve this answer









          $endgroup$



          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.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Mar 5 at 19:16









          Denis NardinDenis Nardin

          9,01223564




          9,01223564








          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            Mar 5 at 19:40






          • 3




            $begingroup$
            @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
            Mar 5 at 19:41








          • 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
            Mar 5 at 21:36






          • 2




            $begingroup$
            @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
            Mar 5 at 21:38








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            Mar 5 at 21:44














          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            Mar 5 at 19:40






          • 3




            $begingroup$
            @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
            Mar 5 at 19:41








          • 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
            Mar 5 at 21:36






          • 2




            $begingroup$
            @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
            Mar 5 at 21:38








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            Mar 5 at 21:44








          1




          1




          $begingroup$
          No need for $infty$-categories here :-)
          $endgroup$
          – David Roberts
          Mar 5 at 19:40




          $begingroup$
          No need for $infty$-categories here :-)
          $endgroup$
          – David Roberts
          Mar 5 at 19:40




          3




          3




          $begingroup$
          @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
          Mar 5 at 19:41






          $begingroup$
          @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
          Mar 5 at 19:41






          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
          Mar 5 at 21:36




          $begingroup$
          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
          Mar 5 at 21:36




          2




          2




          $begingroup$
          @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
          Mar 5 at 21:38






          $begingroup$
          @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
          Mar 5 at 21:38






          2




          2




          $begingroup$
          I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
          $endgroup$
          – David Roberts
          Mar 5 at 21:44




          $begingroup$
          I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
          $endgroup$
          – David Roberts
          Mar 5 at 21:44


















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