This question is a follow-up to https://mathoverflow.net/questions/372349/are-there-infinitely-many-l-rigs?r=SearchResults and to https://mathoverflow.net/questions/355240/is-an-automorphic-form-of-operatornamegl-n-mathbba-mathbbq-deter.

I copy paste a deepl translation of an old answer I got to a question of mine on a French math forum:

"The idea is that to any "space" X one can associate the category of its coverings R(X). If we choose a "geometric point", we have a "fiber in x" functor: ωx:R(X)→Ens which to a covering f:Y→X associates f-1(x). If we are given a path γ:x0→x1, then, for any y0∈f-1(x0), it rises to a single path γ~ of origin y0. Considering the end of this path, we get a point y1∈f-1(x1). And this depends only on the homotopy class of γ. So we have an application that to a homotopy class of paths associates an isomorphism of fiber functors: π1(X;x0,x1)→Isom(ωx0,ωx1). Grothendieck's brilliant remark is that this application is an isomorphism! In particular, π1(X,x)=Aut(ωx) and the right-hand side term thus gives a purely algebraic definition (we do not use the topology of R) of the fundamental group.

Where we join Galois theory is that, if we take - space X" = a field K (we note X=Spec(K) - "covering Y of X" = finite separable extension L/K. - "geometric point" = embedding x:K→Ξ into an algebraically closed field. Then the associated "fiber in x" functor is the functor ωx:L/K↦HomK(L,Ξ) which to L/K associates its K-embeddings in Ξ. The theory applies and we find that the "fundamental group" of K (i.e., the group of automorphisms of the fiber functor ωΞ is the absolute Galois group Gal(Ksep/K) of K (Galois group of a separable closure)."

So I would like to know if, provided that an L-rig can be extended into a field, one can establish a parallel between the map that sends an automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ to the associated L-function and such a "fiber functor" that would allow to say that the automorphism group/"fundamental group" of $\mathcal{M}$ is an absolute Galois group. If yes, is it provably the absolute Galois group of the rationals?