I’m going to do a change in plan.
Galois Theory: Let F be a field. In some sense the “universal” Galois group is where is the algebraic closure, since given any algebraic extension we have that . In fact, there is a bijective correspondence between subgroups of the Galois group and algebraic extensions (this is just loosely speaking to show a connection later on, I’m not being careful about finiteness and things). In this case the we have an inverse corrolation. As the fields get bigger, the groups get smaller.
Covering spaces: For suggestive notation, let’s denote to mean Y is a covering of X. Then if X has sufficiently nice conditions, we have that there is a universal cover with covering map . Then we have that where is the group of “deck transformations,” i.e. the automorphisms such that . Now any other cover will “sit below” the universal one, in that the covering will have a factoring . Moreover . Just as in the Galois case, there is a bijective correspondence between conjugacy classes of subgroups of and isomorphism classes of coverings. This time in a sense it is not reversed, though it depends on how you want to look at it.
I found the similarities of these two situations very strange. There must be something deeper. All field extensions are in bijective correspondence to subgroups of the Galois group of the “largest one,” and all (iso classes of) coverings are in bijective correspondence with (up to conjugacy) subgroups of the fundamental group.
It turns out that after some hunting, there is a huge deep field called “the theory of descent” or something similar. It all looks so fascinating, but it is just too far astray from what I’m studying for me to actually learn right now. I thought I could dip a toe in or something and report back my findings, but there doesn’t seem to be any good introductions to the subject or any hope for quickly seeing some of the ideas. So, after a few days of hunting, I’m changing my plans and am going to look for something new to go on about (possibly back to the prime and localization that I built up, then left for dead?).
Actually, if anyone knows of a place to learn some of this stuff, it would be greatly appreciated if you let me know!