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TLDR: How is the Galois action on étale path torsors defined?

Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\overline{x})$ denote the étale fundamental group of $\overline{X} := X\otimes\overline{k}$ based at the geometric point associated to $x$. Similarly we may consider $\pi_1^{\text{ét}}(\overline{X},\overline{y})$, and $\pi_1^{\text{ét}}(\overline{X},\overline{x},\overline{y})$, which denotes the set of étale paths from $\overline{x}$ to $\overline{y}$.

The Grothendieck $\pi_1$-exact sequence shows that there is a natural $G_k$ (absolute Galois) action on $\pi_1^{\text{ét}}(\overline{X},\overline{x})$, and $\pi_1^{\text{ét}}(\overline{X},\overline{y})$. I read in several places that there is a natural Galois action on $\pi_1^{\text{ét}}(\overline{X},\overline{x},\overline{y})$, does anybody know how it is defined?

My attempt: the way the Galois group acts on $\pi_1^{\text{ét}}(X,\overline{x})$, for example, is via conjugation by a loop in $\pi_1^{\text{ét}}(X,\overline{x})$, i.e. $g\in G_k, \gamma_x\in \pi_1^{\text{ét}}(X,\overline{x})$, then $g(\gamma_x) = \gamma_{g,x}^{-1}\gamma_x\gamma_{g,x}$. Similarly for $\pi_1^{\text{ét}}(X,\overline{y})$. So it appears that there is a natural action of $G_k$ on the arithmetic path space $\pi_1^{\text{ét}}(X,\overline{x},\overline{y})$, by sending a path $p\in \pi_1^{\text{ét}}(X,\overline{x},\overline{y})$ to $\gamma_{g,x}^{-1}p\gamma_{g,y}$. It would be very convenient if this action may be shown to restrict from an automorphism of the arithmetic path space $\pi_1^{\text{ét}}(X,\overline{x},\overline{y})$, to the geometric path space $\pi_1^{\text{ét}}(\overline{X},\overline{x},\overline{y})$. How do I proceed?

Another question, in case anybody is feeling generous, is if the étale path space has a geometric interpretation, similar to how the étale fundamental group has. I am familiar with a Tannakian description but do not find it extremely illuminating.

EDIT: I would be very appreciative of a Tannakian argument, however.

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    $\begingroup$ It might be easier to see what it looks like from the fiber functor perspective. The torsor of etale paths = Isom(F_x, F_y) for the fiber functors F_x, F_y. An iso. is essentially given by maps between the fibers Y_x and Y_y for all covers Y. Given such a map you can act by Gal on both sides and get a new isom. If I didn't make a dumb mistake that's basically the Galois action. Then you can always translate back from fiber functor picture to check what this action corresponds to. $\endgroup$
    – KristianJS
    Commented Mar 23 at 14:31
  • $\begingroup$ In your view, $g(p)$ is the path which satisfies $g(p) = g(F_y)pg^{-1}(F_x)$. I guess the point which I find unclear is: if I replace my Galois element acting on $F_y$ by another element, would I still get a path? For example, is $h(F_y)pg^{-1}(F_x)$ a valid etale path for any $g,h\in G_k$? $\endgroup$
    – kindasorta
    Commented Mar 23 at 15:19

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Following @KristianJS's comment, I devised a pedestrian way of understanding this fact: a geometric etale path is a compatible system of isomorphisms of fibres of geometric covers of my scheme. The Galois group doesn't just act compatibly on the fibres, but rather on the entire cover, and hence on the fundamental groupoid itself.

If I replace $\pi_1^{\text{et}}(\overline{X},\overline{x})$ by the Tannakian fundamental group of etale local systems on $\overline{X}$, the Galois action on covers translates to having a Galois action on all objects in this entire Tannakian category. On the other hand, since every path torsor is a pro-object of the category, it inherits an action as well.

In light of that I think it makes sense to think of this as: paths are also pro-covers.

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