# 1 Introduction

The fundamental theorem of Galois theory says

Let $k$ be a field and $G = \operatorname{Gal}(\bar{k}/k)$ be the absolute Galois group
^{1}
. Then there is a contravariant equivalence of categories between

the category of finite separable extensions of $k$; and

the category of finite sets with a continuous transitive action of $G$.

There is a very similar theorem in algebraic topology.

Let $X$ be a path connected, locally path connected and semi-locally simply connected topological space. Then there is an equivalence of categories between

the category of finite connected covering spaces of $X$; and

the category of finite sets with a transitive action of $\pi _1(X, x)$, for any base point $x \in X$.

On the other hand, in covering space theory, we have to pick a basepoint of $X$ to talk about the fundamental group, and the correspondence is natural (in the English sense) only after we fix such a basepoint. The analogous situation in Galois theory is that in fact, $G$ is not canonically associated to $k$ either. Instead, we must fix an embedding of $k$ into a separably closed field, which is the same as picking a geometric point of $\operatorname{Spec}k$.

To push the analogy further, there is a sense in which a separable field extension is a “covering space”. By definition, a map $p: Y \to X$ is a covering space iff there is some space $X'$ and a map $f: X' \to X$ such that

$f$ is surjective;

restricted to each component of $X'$, the map $f$ is the inclusion of an open subspace; and

the pullback $Y \times _X X'$ is a disjoint union of copies of $X'$, and the map $f^*p: Y \times _{X} X' \to X'$ is the map sending each copy isomorphically onto $X'$.

This is just the local triviality condition phrased in a fancy way.

The idea is to think of the (surjective!) morphism $\operatorname{Spec}\bar{k} \to \operatorname{Spec}k$ as an inclusion of an “open subset”, and an extension $K/k$ is separable iff the pullback to $\operatorname{Spec}\bar{k}$ is a disjoint union of copies of $\operatorname{Spec}\bar{k}$, i.e. $K \otimes _k \bar{k} = \bar{k} \times \cdots \times \bar{k}$. Of course, we can also replace $\bar{k}$ by some appropriate finite Galois extension if we want to remain in the finite world.

By using an appropriate generalized notion of “open subset”, hence “covering space”, we will generalize the notion of fundamental group to arbitrary schemes, of which Galois theory and covering space theory (of complex algebraic surfaces) are special cases.