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This last, in particular, has no analogue at all in the Galois group approach to fundamental groups. When $X$ is a variety over $\mathbb{Q}$, it becomes possible, for example, to study $\pi^{et}_1(X,b)$ and $\pi^{et}_1(X;b,c)$ as sheaves on $Spec(\mathbb{Q})$, which encode rich information about rational points. This is a long story, which would be rather tiresome to expound upon here (cf. lecture at the INI ). However, even a brief contemplation of it might help you to appreciate the arithmetic perspective that general $\pi^{et}_1$'s are substantially more powerful than Galois groups. Having read thus far, it shouldn't surprise you that I don't quite agree with the idea explained, for example, in this post that a Galois group is only a 'group up to conjugacy'. To repeat yet again, the usual Galois groups are just fundamental groups with specific large base-points. The dependence on these base-points as well as a generalization to small base-points is of critical interest.

This last, in particular, has no analogue at all in the Galois group approach to fundamental groups. When $X$ is a variety over $\mathbb{Q}$, it becomes possible, for example, to study $\pi^{et}_1(X,b)$ and $\pi^{et}_1(X;b,c)$ as sheaves on $Spec(\mathbb{Q})$, which encode rich information about rational points. This is a long story, which would be rather tiresome to expound upon here (cf. lecture at the INI ). However, even a brief contemplation of it might help you to appreciate the arithmetic perspective that general $\pi^{et}_1$'s are substantially more powerful than Galois groups. Having read thus far, it shouldn't surprise you that I don't quite agree with the idea explained, for example, in this post that a Galois group is only a 'group up to conjugacy'. To repeat yet again, the usual Galois groups are just fundamental groups with specific large base-points. The dependence on these base-points as well as a generalization to small base-points is of critical interest.

This last, in particular, has no analogue at all in the Galois group approach to fundamental groups. When $X$ is a variety over $\mathbb{Q}$, it becomes possible, for example, to study $\pi^{et}_1(X,b)$ and $\pi^{et}_1(X;b,c)$ as sheaves on $Spec(\mathbb{Q})$, which encode rich information about rational points. This is a long story, which would be rather tiresome to expound upon here (cf. lecture at the INI ). However, even a brief contemplation of it might help you to appreciate the arithmetic perspective that general $\pi^{et}_1$'s are substantially more powerful than Galois groups. Having read thus far, it shouldn't surprise you that I don't quite agree with the idea explained, for example, in this post that a Galois group is only a 'group up to conjugacy'. To repeat yet again, the usual Galois groups are just fundamental groups with specific large base-points. The dependence on these base-points as well as a generalization to small base-points is of critical interest.

This question reminds me to add another very basic reason to avoid the Galois group as a definition of $\pi_1$. It's rather tricky to work out the functoriality that way, again because the base-point is de-emphasized. In the $Aut(F_b)$ approach, functoriality is essentially trivial.

This last, in particular, has no analogue at all in the Galois group approach to fundamental groups. When $X$ is a variety over $\mathbb{Q}$, it becomes possible, for example, to study $\pi^{et}_1(X,b)$ and $\pi^{et}_1(X;b,c)$ as sheaves on $Spec(\mathbb{Q})$, which encode rich information about rational points. This is a long story, which would be rather tiresome to expound upon here (cf. lecture at the INI ). However, even a brief contemplation of it might help you to appreciate the arithmetic perspective that general $\pi^{et}_1$'s are substantially more powerful than Galois groups. Having read thus far, it shouldn't surprise you that I don't quite agree with the idea explained, for example, in this post that a Galois group is only a 'group up to conjugacy'. To repeat yet again, the usual Galois groups are just fundamental groups with specific large base-points. The dependence on these base-points as well as a generalization to small base-points is of critical interest.

This question reminds me to add another very basic reason to avoid the Galois group as a definition of $\pi_1$. It's rather tricky to work out the functoriality that way, again because the base-point is de-emphasized. In the $Aut(F_b)$ approach, functoriality is essentially trivial.

Added, 5 October:

I was asked by a student to give away the answer to the puzzle. The crux of the matter is that any continuous map $$B:S\rightarrow M$$ from a simply connected set $S$ can be used as a base-point for the fundamental group. One way to do this to use $B$ to get a fiber functor $F_B$ that associates to a covering $$N\rightarrow M$$the set of splittings of the covering $$N_B:=S\times_M N\rightarrow S$$ of $S$. If we choose a point $b'\in S$, any splitting is determined by its value at $b'$, giving a bijection of functors $F_B=F_{b'}=F_b$ where $b=B(b')\in M$. Now, when $$B:M'\rightarrow M$$ is the universal covering space, I will really leave it as a (tautological) exercise to exhibit a canonical anti-isomorphism $$Aut(F_B)\simeq Aut(M'/M).$$ The 'point' is that $$F_B(M')$$ has a canonical base-point that can be used for this bijection.