I'm studying a little homotopical algebra in Brown's article [see link below].

You can easily notice that Theorem 3 (page 430) and Proposition 3 (in the following page) imply that one can internalize the notion of "pi_1 acting on the fibers of a covering", idea which dates back, if I'm not wrong, to Quillen's "Homotopical Algebra".

This could be the starting point for some natural (?) questions: the action of pi_1 on the fibers of a covering is worth to be studied because of Galois' theory of coverings (in fact the philosophy is that of Grothendieck's Galois Theory: Galois groups "are" homotopy groups).

Now allow me to state the 64 thousand dollar question:

can we recover Galois' theory of coverings in a suitable model/fibrant category?

I.e., can we classify subgroups(*) of the fundamental group(*) of the base space of a fiber space(*), finding an (anti-)monotone bijection(*) between the lattice of intermediate objects between the base and a suitable "universal"(**) covering?

My two cents: classically, we know very well what to do and how do do it. Here we certainly have enough informations about how to internalize each ingredient:

a. Subgroups of a group object are (iso classes of) group mono to that object;

b. pi_1(B) = Omega(B) = pullback obtained exploiting a path object for B, which is doing externally what pi_1(B) did internally (it is a group which acts on the fibers of a fibration, Omega(Omega(B)) is abelian, ...);

c. A fiber/coverng space is a fibration (here and in (b) one needs a pointed fibrant/model category);

d. Antimonotone bijections are Galois' equivalences: here one looks to the subobjects poset of Omega(B), and to the posetal category C_B, having as objects fibrations with base B (the order is defined by: X < Y iff one fibers over the other - a choice we are rather forced to, just because classically it is so).

The problem is that we lack something forcing C_B to admit a top element.

Another question which is still in the handwaving zone:

In studying classical Galois theory, I found really bothering that the splitting field of a field is only a weak limit (any two splitting fields are isomorphic, but not with a unique iso). All the same, it is really annoying to notice that the universal covering of (even a good) space is a weak limit. What if the localization functor killed this ambiguity "contracting" the groupoid of isomorphisms between different universal coverings, in passing to the homotopy category? Is there a way to write it down without using theology?

Try to meet up this challenge: example 1.1.1.1 in Higher Topos Theory by Lurie suggests (not so coincidentally?) that "being homotopic" in Grp means to be conjugate; now, any two splitting fields are conjugate, am I wrong?

http://www.math.uni-hamburg.de/home/schreiber/Abstract%20homotopy%20theory%20and%20generalized%20sheaf%20cohomology.pdf