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 $\mathrm{Spec}\,k^{sep}\rightarrow\mathrm{Spec}\,k$ 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.


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 $B\mathrm{Gal}(k)/H$, and so it is the profinite space usually written $B\mathrm{Gal}(k)$ or $K(\mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.

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    $\begingroup$ No need for $\infty$-categories here :-) $\endgroup$ – David Roberts Mar 5 '19 at 19:40
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    $\begingroup$ @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 '19 at 19:41
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    $\begingroup$ @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 '19 at 21:38
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    $\begingroup$ I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-) $\endgroup$ – David Roberts Mar 5 '19 at 21:44
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    $\begingroup$ It is not the case that the homotopy type respects colimits in the 2-category of 1-topoi in general. It happens to be true for this example, but one has to give an ad hoc argument for that. So, on the contrary, the kind of argument Denis is giving is precisely where the ∞-categorical point of view has a nontrivial conceptual advantage. $\endgroup$ – Clark Barwick Jul 15 '19 at 15:41

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