# 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 $X$ is an element $Y \in \mathscr {FE}t_{X}$ such that $\operatorname{Aut}(Y)$ acts transitively on $F(Y)$.

Let $Y \in \mathscr {FE}t_{X}$ be Galois. Then $Y \times _X Y$ splits as finitely many disjoint copies of $Y$, and $\operatorname{Aut}(Y)$ acts freely and transitively over the copies, where $\operatorname{Aut}(Y)$ acts as an automorphism of $\pi _1: Y \times _X Y \to Y$ by pullback.

By definition of the pullback, the lifts of $z$ to $Y \times _X Y$ bijects with the lifts of $x$ to $Y$, and this preserves the action of $\operatorname{Aut}(Y)$. Since $\operatorname{Aut}(Y)$ acts freely on the lifts of $z$, it must act freely on $Y \subseteq Y \times _X Y$. So $Y \times _X Y$ splits as $n$ copies of $Y$, and there is nothing left since $Y \times _X Y \to Y$ has degree $n$.

In Galois theory, we can construct Galois closures. We can do the same here.

Let $Y \in \mathscr {FE}t_{X}$ be connected. Then there is a Galois cover $Y'$ with a map $Y' \to Y$ such that if $Z \in \mathscr {FE}t_{X}$ is any Galois cover, then any map $Z \to Y$ factors through $Y'$.

If $(x_1, \ldots , x_n) \in Y'$, then $x_i \not= x_j$ for all $i$ and $j$.

$Y' \to X$ is Galois.

If $Z$ is Galois and $q: Z \to Y$, then it factors through $Y'$.

Fix a lift $z \in F(Z)$. Then by composing with automorphisms of $\operatorname{Aut}(Z)$, we can pick maps $q_1, \ldots , q_n: Z \to Y$ such that $q_i(z) = y_i$. Then the image of $(q_1, \ldots , q_n): Z \to Y^n$ includes a point in $Y'$, namely $y$, and hence maps into $Y'$ by connectedness of $Z$.

Let $(X_\alpha )_\alpha \subseteq \mathscr {FE}t_{X}$ be the poset of Galois covers of $X$, ordered under $X_\alpha \leq X_\beta$ if there is a morphism $X_\beta \to X_\alpha$. For each $X_\alpha$, pick a point $x_\alpha \in F(X_\alpha )$. Then if $X_\alpha \leq X_\beta$, we pick the morphism that sends $x_\alpha$ to $x_\beta$ for use in the pro-system. Then this pro-system pro-represents $F$.

that sends a map $q: X_\alpha \to Y$ to $q(x_\alpha )$. Conversely, given any $y \in F(Y)$, we may restrict to the component of $y$ and assume $Y$ is connected. Then since $X_\alpha \to Y$ is surjective and $\operatorname{Aut}(X_\alpha )$ acts transitively on $F(X_\alpha )$, precomposing with an automorphism gives a morphism $X_\alpha \to Y$ that sends $x_\alpha$ to $y$, and such a map is unique since $\operatorname{Aut}(X_\alpha )$ acts freely. This gives the desired inverse map.

$\mathscr {FE}t_{X}$ is naturally equivalent to the category of continuous finite $\pi _1(X, x)$-sets.

^{1}. Functions can be constructed similarly.