@SamNead, I want the equality to be either a homeomorphism or a homotopy equivalence.

Your "wordier" summary is very helpful :) Thank you. I wrote a comment here before thanking you, but I think someone deleted it because I was simply "thanking" and not adding to the discussion, or whatever. Which is annoying, but fine, heh

So perhaps when people say "we want to understand the absolute Gal group", they also mean to understand the action on other arithmetic objects. Which happen to be vector spaces anyway. Thank you!

I edited that back to Z/ell, thank you!

In Proposition 36, page 54 of my recent paper on a singularity detection algorithm, I summarised what I understood from Robert Bryant's answer.

This was very succinct and nice, thank you so much!