derived smooth geometry
Fix some scheme .
The fpqc-site (over ) is the site
The last condition does imply that .
The abbreviation “fpqc” is for fidèlement plat quasi-compacte : faithfully flat and quasi-compact.
Because the collection of fpqc covers of a scheme does not have a small collection of refinements (Stacks project, Tag 0BBK), working with the fpqc topology can be set-theoretically tricky. Indeed, in 1975, Waterhouse gave an example of a functor on schemes that admits no fpqc sheafification. This contradicts many claims in the literature that fpqc sheafification and stackification is functorial (and such claims continue to be made).
Chaper 27.8 in