Galois categories for topological spaces?

Question

Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)?

Recall from SGA 1 that the theory of Galois categories is developed to construct the étale fundamental group of a connected locally noetherian scheme. Given a Galois category $\mathcal{C}$, the main result is that a choice of fiber functor $F$ determines an equivalence between $\mathcal{C}$ and the category of finite $\pi := \operatorname{Aut}(F)$-sets.

Can one prove an analogous result that applies to a path connected+locally path connected topological space? Except in special cases, one can't literally use Galois categories since a topological space can certainly admit connected covers of infinite degree. But maybe we can modify the construct by removing some finiteness conditions?

Ideally, if the answer is "yes", I'd be great to have a written reference developing this idea.

Answer 1

The answer is yes (with mild hypothesis on the space). Moreover the topological situation is simpler, and this was very likely Grothendieck's inspiration.

To see this you need two facts.

First taken from Szamuley's book Galois Groups and Fundamental Groups

Theorem 2.3.4 : Let $X$ be a connected and locally simply connected topological space, and $x \in X$ a base point. The functor $Fib_x$ induces an equivalence of the category of covers of $X$ with the category of left $\pi_1(X, x)$-sets.

Then it is easy to see that you can recover $G$ as the automorphism group of the forgetful functor from $G$-sets to sets, so you are done.

If you can read french you can also have a look at Douady&Douady's nice book Algèbre et théories galoisiennes, or at Bourbaki's recent book Topologie algébrique.

Answer 2

To expand on Niels' answer: given a connected semilocally simply connected topological space $X$ and a point $x\in X$, the fiber functor $F_x:\mathrm{Cov}_X\rightarrow\mathrm{Set}$ from the category of covering spaces of X yields an equivalence of categories $F_x:\mathrm{Cov}_X\xrightarrow{\sim}\pi_1(X,x)\mathrm{Set}$ with permutation representations of the fundamental group $\pi_1(X,x)$ by sending a covering space $\pi:Y\rightarrow X$ to the permutation representation of $\pi_1(X,x)$ on the fiber $\pi^{-1}(x)$ induced by monodromy. However, $\mathrm{Cov}_X$ is very much not a Galois category: according to SGA1 or Stacks, a Galois category is equivalent to the category of finite (continuous) permutation representations of a profinite group, whereas $\mathrm{Cov}_X$ is equivalent to the category of possibly infinite permutation representations of a (discrete) group. I suppose $\mathrm{Cov}_X$ could be called an infinite Galois category, although this conflicts somewhat with the notion of infinite Galois category used to define the pro-étale fundamental group.

On the other hand if you want to work with an honest Galois category: given a connected topological space $X$ and a point $x\in X$, the fiber functor the fiber functor $F_x:\mathrm{FCov}_X\rightarrow\mathrm{Fin}$ from the category of finite covering spaces of X yields an equivalence of categories $F_x:\mathrm{FCov}_X\xrightarrow{\sim}\widehat{\pi}_1(X,x)\mathrm{Fin}$ with finite (continuous) permutation representations of the profinite completion $\widehat{\pi}_1(X,x)$ of the fundamental group. Note that we do not need to assume that $X$ is semilocally simply connected for this to work: in the presence of a universal cover of $X$ the fundamental group is equivalently the automorphism group of the universal cover, but in general the (profinite) fundamental group can be recovered as the automorphism group of the fiber functor.

Why would one want to consider only finite covers of topological spaces? One reason is to formulate comparison theorems: for instance given a scheme $X$ (locally) of finite type over $\mathbb{C}$ and a point $x\in X$ we have an equivalence of Galois categories $\mathrm{FEt}_X\xrightarrow{\sim}\mathrm{FCov}_{X(\mathbb{C})}$, but this is obviously false at the level of infinite covering spaces.