5 Galois Theory
The usual proof of the usual Fundamental Theorem of Galois Theory can pretty much be carried over to the general case if we can say the word “Galois”. Fortunately, the word is not too difficult to utter.
A Galois cover of is an element such that acts transitively on .
Let be Galois. Then splits as finitely many disjoint copies of , and acts freely and transitively over the copies, where acts as an automorphism of by pullback.
In Galois theory, we can construct Galois closures. We can do the same here.
Let be connected. Then there is a Galois cover with a map such that if is any Galois cover, then any map factors through .
Let , which gives rise to a geometric point . Let be the component of , mapping to by first projection. This is an étale cover since it is a connected component of one. We claim that this works.
If , then for all and .
If is Galois and , then it factors through .
Fix a lift . Then by composing with automorphisms of , we can pick maps such that . Then the image of includes a point in , namely , and hence maps into by connectedness of .□
Let be the poset of Galois covers of , ordered under if there is a morphism . For each , pick a point . Then if , we pick the morphism that sends to for use in the pro-system. Then this pro-system pro-represents .
is naturally equivalent to the category of continuous finite -sets.