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.



Galois categories for topological spaces?
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](http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521888509 "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](http://store.cassini.fr/nouvelle-bibliotheque-mathematique/15-algebre-et-theories-galoisiennes.html "Algèbre et théories galoisiennes"), or at Bourbaki's recent book [Topologie algébrique](https://www.springer.com/fr/book/9783662493601 "Topologie algébrique").


