In the hope this won't turn into a trivial problem (I couldn't find a similar discussion here), here's my question.
I'm studying a little homotopical algebra in this article by Brown. 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 (at least those marked with "#"):
- Subgroups of a group object are (iso classes of) group mono to that object;
- $\pi_1(B)=\Omega B$ = 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, ...);
- A fiber/coverng space is a fibration (here and in (2) one needs a pointed fibrant/model category);
- 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 seems to be 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 188.8.131.52 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?