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.