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

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.
• No need for $\infty$-categories here :-) – David Roberts Mar 5 '19 at 19:40
• @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). – Denis Nardin Mar 5 '19 at 21:38