TITLE: What is the purpose of the flat/fppf/fpqc topologies?
QUESTION [48 upvotes]: There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully.
As someone who originally started in topology/complex geometry, the étale topology makes some sense to me. It's sort of like the complex topology, in that there are enough "open sets" that things like the inverse function theorem work.
But I don't really understand what these other more "exotic" topologies represent. From the (naive) background of topology, it sounds like we're just "adding more open sets" to our topology (which I know isn't right, since this isn't a topology in the standard sense).
So what do they represent? What do they do for us?

REPLY [21 votes]: The question is what kind of structure you are interested in. For example, the étale topology sees more than you might like as a topologist/complex geometer. If you look at Spec(k) where k is not algebraically closed, the étale topology will see the Galois group. As a topologist/complex geometer you might also like the Nisnevich topology. It has enough covers to make a smooth variety look locally like affine space, but it doesn't see the "arithmetic stuff" that only depends on the ground field and not on the variety itself. But if you are interested in the properties of the ground field, too, you will choose the étale topology.
As for the flat/fppf/fpqc topologies, I think you already got some very convincing answers. I can't add anything to them. Maybe this is rather an overlong comment than an answer.